Date

December 12 (Thu), 10:30--12:00, 2024

Place

Room 006, RIMS

Speaker

Xuanzhong Dai (RIMS)

Title

Chiral differential operators on base affine space

Abstract

　 Let G be an algebraic group and U be its maximal unipotent subgroup. In this talk, we will explore the global sections of chiral differential operators on the base affine space G/U, with a focus on introducing lifting formulas that transform functions on the cotangent bundle into global sections. A notable aspect of this framework is the mysterious Weyl group action, known as the Gelfand-Graev action, on the ring of differential operators on G/U. This action arises through algebra automorphisms rather than the natural geometric action of the Weyl group on the variety G/U. If time permits, we will also examine its extension to the realm of chiral differential operators.

Date

December 5 (Thu), 10:30--12:00, 2024

Place

Room 006, RIMS

Speaker

Simon Wood (Cardiff University / RIMS)

Title

Grothendieck-Verdier duality as a generalisation of rigidity

Abstract

　 Many of the best studied types of tensor categories enjoy a notion of duality called rigidity (which among other nice properties implies that the tensor product is exact, if the underlying category is abelian). However, not all sources of interesting tensor categories obligingly produce rigid ones. In particular, the natural notion of duality for a category of modules over a vertex operator algebra is Grothendieck-Verdier duality, which will be the focus of this talk. I will discuss some recent results on tensor categories with Grothendieck-Verdier duality structures, module categories over these and why these results are encouraging for conformal field theory.

1st talk

Date

November 28 (Thu), 10:30--12:00, 2024

Place

Room 206, RIMS

Speaker

Tasuki Kinjo (RIMS)

Title

Cohomological Hall algebras for 3-Calabi-Yau categories

Abstract

　 Around 1990, Ringel famously introduced an algebraic structure on the space of functions over the moduli space of objects in finitary categories during his study of quantum groups. This algebra is now called the Ringel-Hall algebra.
Recently, an analogous construction of the algebraic structure on the homology of the moduli space of objects in an abelian category has been intensively studied. Such an algebra is called the cohomological Hall algebra (CoHA). For an abelian category with homological dimension less than or equal to two, Kapranov and Vasserot constructed the CoHA using derived algebraic geometry. For an abelian category with homological dimension three and a Calabi-Yau structure, Kontsevich and Soibelman conjectured that one can construct the CoHA using the critical cohomology of the moduli space, and they verified this for the category of representations over the Jacobi algebra associated with a quiver with potential.
In this talk, I will explain a general construction of the cohomological Hall algebra associated with 3-Calabi-Yau categories and how it recovers all known constructions of the CoHA. If time permits, I will explain how this construction can be used to define a bialgebra structure on the homology of the moduli space of objects in 2-Calabi-Yau categories. This talk is based on joint work (https://arxiv.org/abs/2406.12838) with Hyeonjun Park and Pavel Safronov, as well as on another collaboration in progress with Ben Davison.

2nd talk

Date

November 28 (Thu), 14:30--16:00, 2024

Place

Room 206, RIMS

Speaker

Ting Xue (University of Melbourne)

Title

Character sheaves, affine Springer fibres, and d-Harish-Chandra series

Abstract

　 We discuss cuspidal character sheaves in the setting of cyclically graded Lie algebras. Via a nearby cycle construction representations of Hecke algebras of complex reflection groups at roots of unity enter their description. We explain a conjectural level-rank (Koszul) duality arising from connections with Lusztig-Yun’s work, where Fourier transforms of character sheaves are related to representations of trigonometric double affine Hecke algebras. We will also discuss connections with d-Harish-Chandra series introduced by Broué-Malle-Michel and Oblomkov-Yun’s construction of rational Cherednik algebra modules using affine Springer fibres.

This is based on joint work with various co-authors, Grinberg, Liu, Trinh, Tsai, and Vilonen.

Date

November 21 (Thu), 14:30--16:00, 2024

Place

Room 006, RIMS

Speaker

Kari Vilonen (University of Melbourne)

Title

Representations of real groups and Hodge theory

Abstract

　 I will explain how Beilinson-Bernstein localization theory can be extended to mixed Hodge modules. A while back Schmid that the global sections functor takes mixed Hodge modules on a flag manifold to mixed Hodge structures. We cannot completely prove this conjecture but can prove enough of it so that it has implications for representation theory of real groups. For example one gets a Hodge theoretic characterization of unitarity.

1st talk

Date

November 14 (Thu), 10:30--12:00, 2024

Place

Room 006, RIMS

Speaker

Sven Möller (Universität Hamburg)

Title

Equivalence Relations on Vertex Operator Algebras: Genus and Witt Equivalence (joint work with Brandon C. Rayhaun)

Abstract

　 We define and investigate several notions of equivalence of VOAs. In doing so, we see that each notion of equivalence for even lattices (genus, Witt and rational equivalence) has two generalisations to VOAs, one being a more “classical” analogue, and one being a more honest “quantum” analogue.
We study the relations between these various notions. For example, we give a proof that two VOAs in the same hyperbolic genus (as defined by Moriwaki) are in the same bulk genus (as defined by Höhn). We also propose a programme for (partially) classifying c=32, holomorphic VOAs, and we conjecture a Siegel–Weil identity that computes the “average” torus partition function of an ensemble of chiral CFTs defined by any hyperbolic genus, and interpret this formula physically in terms of disorder-averaged holography.
We then study Witt and rational (or orbifold) equivalence. This boils down to the question when two VOAs can be related by topological manipulations (like conformal extensions and subalgebras). We argue that Witt equivalence is necessary for two theories to be related by topological manipulations, and we conjecture that it is also sufficient. We give proofs in various special cases.
We use the notion of Witt equivalence to argue, assuming the conjectural classification of unitary, c=1 RCFTs, that all of the finite global symmetries of the SU(2)_1 Wess-Zumino-Witten model are invertible. Finally, we sketch a “quantum Galois theory” for chiral CFTs, which generalizes prior mathematical literature by incorporating non-invertible symmetries; we illustrate this non-invertible Galois theory in the context of the monster CFT, for which we produce a Fibonacci symmetry.

2nd talk

Date

November 14 (Thu), 14:30--16:00, 2024

Place

Room 006, RIMS

Speaker

Andrea E. V. Ferrari (Deutsches Elektronen-Synchrotron / University of Edinburgh)

Title

Geometric remarks on boundary vertex algebras of A-twisted 3d N=4 abelian gauge theories

Abstract

　 The so-called 4d SCFT/2d VOA correspondence has been the source of a fruitful interaction between physics and representation theory. One cornerstone of this interaction is the conjecture that associated varieties of VOAs arising from 4d SCFTs are isomorphic to Higgs branches, and therefore the VOAs are quasi-lisse. More recently, similar conjectures have been proposed for VOAs supported at the boundary of topologically twisted 3d N=4 theories. In this talk I will discuss these conjectures in the special case of abelian 3d N=4 gauge theories, whose Higgs branches are hypertoric varieties. I will relate some special cases to results known in the mathematical literature, and discuss various implications for free field realisations. If time will permit, I will comment on expected connections to Coulomb branches and symplectic duality.

1st talk

Date

November 7 (Thu), 10:30--12:00, 2024

Place

Room 006, RIMS

Speaker

Simon Lentner (University of Hamburg)

Title

Deforming the Triplet Algebra by SL2-connections (in particular irregular ones)

Abstract

　 Given a vertex operator algebra with a group G of symmetries, there is the notion of g-twisted modules. But moreover for G a Lie group there is (or should be) a notion of twisted modules by a G-connection. For connections with regular singularity, this recovers the g-twisted modules, with g the monodromy, but we can also study irregular singularities. In this talk we discuss some basic questions in this endeavour, and we give interesting results for the triplet vertex algebra, whose category of representations is equivalent to the small quantum group of sl2. In particular, for a regular singularity with nilpotent monodromy g we have previously found twisted modules to be large tilting modules, and for an irregular singularity we find in our upcoming work Whittaker modules.

Both is joint work with B. Feigin.

2nd talk

Date

November 7 (Thu), 14:30--16:00, 2024

Place

Room 006, RIMS

Speaker

Edmund Karasiewicz (National University of Singapore)

Title

The stable wave front set of theta representations

Abstract

　 The Fourier coefficients of theta functions have featured prominently in numerous number theory applications and constructions in the Langlands program. For example, they play an important role in the recent work of Friedberg-Ginzburg generalizing the theta correspondence to higher covering groups. For their construction one wants to know the wave front set of the theta representations, i.e. the largest nilpotent orbit with nonvanishing Fourier coefficient.
To investigate these Fourier coefficients it can be valuable to study the analogous local question. In this talk we consider local theta representations and describe how to compute their stable wave front set. This is joint work with Emile Okada and Runze Wang.

Date

October 24 (Thu), 10:30--12:00, 2024

Place

Room 006, RIMS

Speaker

Jethro van Ekeren (IMPA / RIMS)

Title

Higher chiral homology and rationality of vertex algebras

Abstract

　 In this talk I will describe some recent results on chiral homology of chiral algebras associated with conformal vertex algebras. In particular we establish a criterion for rationality of the vertex algebra in terms of vanishing of degree 1 chiral homology of the

Date

October 3 (Thu), 10:30--12:00, 2024

Place

Room 006, RIMS

Speaker

Vyacheslav Futorny (SUSTech International Center for Mathematics)

Title

Differential operators on algebraic varieties, their invariants and representations

Abstract

　 We will discuss two problems related to algebras of differential operators on algebraic varieties: the birational equivalence problem for the subalgebras of invariant differential operators by a finite group, and constructions of irreducible representations of the Lie algebras of polynomial vector fields on algebraic varieties.

Date

October 3 (Thu), 14:30--16:00, 2024

Place

Room 006, RIMS

Speaker

Iryna Kashuba (SUSTech International Center for Mathematics)

Title

Free Jordan algebras and Tits-Kantor-Koecher categories

Abstract

　 We discuss several equivalent conjectures for the dimensions and character of the homogenous components of the free Jordan algebras. Consequently we consider the category, where the main object of our conjectures, the Tits-Kantor-Koecher construction of the free Jordan algebra, lives, and study free algebras in this category.

Date

September 19, 10:30--12:00, 2024

Place

Room 006, RIMS

Speaker

Shigenori Nakatsuka (RIMS)

Title

Affine W-algebras of classical Lie types

Abstract

　 The webs of W-algebras introduced by Prochazka-Rapcak in physics provide rich perspectives on the ``hidden hierarchy” among the type-A W-superalgebras with hook-type W-superalgebras as building blocks. One such perspective tells that W-superalgebras in type A should be obtained from the affine vertex superalgebras through reduction by stages (partial reductions) associated with hook-type partitions. As Morgan conjectured in the finite setting, there is a very good chance that two affine W-algebras are connected through partial reductions along nilpotent orbit degenerations in general. In this talk, we discuss how to see such a phenomenon through screening operators when nilpotent orbit degenerations are ``basic” in the case of classical Lie types and explain some applications to their representation theory. The talk is based on joint works with Creutzig-Fasquel-Linshaw (arXiv:2403.08121), Fasquel-Fehily-Fursman (arXiv:2408.13785), and Fasquel-Kovalchuk (in preparation).

Date

July 4 (Thu), 10:30--12:00, 2024

Place

Room 006, RIMS
Zoom (Meeting ID: 938 9341 5540, Passcode: 808973)

Speaker

Bernard Leclerc (Université de Caen Normandie)

Title

Shifted quantum affine algebras and cluster algebras

Abstract

　 This is a report on a joint work with Christof Geiss and David Hernandez (arXiv:2401.04616). Shifted quantum affine algebras have been introduced by Finkelberg and Tsymbaliuk in their study of quantized K-theoretic Coulomb branches of certain quiver gauge theories. Their representation theory has been studied by Hernandez, who constructed a category O containing finite-dimensional and infinite-dimensional representations. We introduce a new class of infinite rank cluster algebras associated with A-D-E root systems, and show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the category O of the corresponding shifted quantum affine algebras. In this isomorphism, the cluster variables of the initial seed are mapped to certain Q-variables and the most interesting first step mutations are instances of the QQ-system relations studied recently by Frenkel and Hernandez (arXiv:2312.13256). We conjecture that the images of all cluster monomials are classes of simple objects of O. We prove the conjecture in type A_1. We also show that it holds for the subcategory C of O whose objects are the finite-dimensional representations.

Date

Jun 20 (Thu), 10:30--12:00, 2024

Place

Room 006, RIMS
Zoom (Meeting ID: 872 7089 6474, Passcode: 428037)

Speaker

Hiroyuki Minamoto (Osaka Metropolitan University)

Title

Quiver Heisenberg algebras: a cubic analogue of preprojective algebras

Abstract

　 In this joint work with Martin Herschend, we study a certain class of central extensions of preprojective algebras of quivers under the name quiver Heisenberg algebras (QHA). There are several classes of algebras introduced before by different researchers from different view points, which have the QHA as a special case. While these have mainly been studied in characteristic zero, we also study the case of positive characteristic. Our results show that the QHA is closely related to the representation theory of the corresponding path algebra in a similar way to the preprojective algebra.
Among other things, one of our main results is that the QHA provides an exact sequence of bimodules over the path algebra of a quiver, which can be called the universal Auslander-Reiten sequence. Moreover, we show that the QHA provides minimal left and right approximations with respect to the powers of the radical functor. Consequently, we obtain a description of the QHA as a module over the path algebra, which in the Dynkin case, gives a categorification (as well as a generalization to the positive characteristic case) of the dimension formula by Etingof-Rains.

Date

May 30 (Thu), 15:00--16:30, 2024

Place

Zoom (Meeting ID: 893 4748 0444, Passcode: 949775)

Speaker

Vladyslav Zveryk (Jagiellonian University in Kraków)

Title

Dynkin automorphisms of Gaudin algebras

Abstract

　 The goal of the talk is to present some results on the automorphisms of generalized Gaudin algebras induced from Dynkin automorphisms of simple Lie algebras. I will start by giving an overview of the construction of generalized Gaudin algebras via the Feigin-Frenkel center, as well as a recollection of the basic properties of Dynkin automorphisms of simple Lie groups. Using the geometry of opers whose rings of functions are isomorphic to various generalized Gaudin algebras, we will link generalized Gaudin algebras associated with a simple Lie algebra and its subalgebra fixed by a Dynkin automorphism. One application of this linkage is a new proof of the global version of Jantzen’s twining formula. The talk will be based on arxiv:2311.11872.

Date

May 27 (Mon), 10:30--12:00, 2024

Place

Room 110, RIMS

Speaker

Sebastian Schlegel Mejia (MPIM)

Title

BPS algebras and generalised Kac—Moody algeberas from 2-Calabi-Yau categories

Abstract

　 The talk is a report on joint work with Ben Davison and Lucien Hennecart. I will explain how to associate a generalised Kac—Moody algebra to any sufficiently geometric 2-Calabi—Yau category and relate it to the BPS algebra and thus Cohomological Hall algebra (CoHA) of the category. In particular, we get a structure theorem for the CoHA in terms of intersection cohomology of moduli spaces. I will focus on the important special case of the category of representations of preprojective algebra of a quiver. Time permitting I will explain applications in non-abelian Hodge theory for stacks and enumerative geometry of sheaves on K3 surfaces.

Comment: This is a joint meeting with Algebraic Geometry Seminar.

Date

May 9 (Thu), 10:30--12:00, 2024

Place

Room 006, RIMS
Zoom (Meeting ID: 871 0112 0615, Passcode: 004003)

Speaker

Naoya Hiramae (Kyoto)

Title

Relationship between $p$-hyperfocal subgroups and $\tau$-tilting finiteness of group algebras

Abstract

　 Demonet--Iyama--Jasso introduced a new class of finite dimensional algebras, $\tau$-tilting finite algebras. $\tau$-tilting finiteness of algebras relates to brick finiteness, functorially finiteness of torsion classes, and connectivity of silting complexes. In the context of modular representation theory of finite groups, Eisele--Janssens--Raedschelders showed that group algebras of tame type are $\tau$-tilting finite. Given the classical result that the representation type (representation finite, tame, or wild) of group algebras is determined by their $p$-Sylow subgroups, where $p$ denotes the characteristic of the ground field, it is natural to ask what controls $\tau$-tilting finiteness of group algebras. In this talk, we will see that $\tau$-tilting finiteness of group algebras is determined by so-called $p$-hyperfocal subgroups in some cases. This talk is based on a joint work with Yuta Kozakai.

Date

January 18 (Thu), 10:30--12:00, 2024

Place

Room 006, RIMS
Zoom (Meeting ID: 852 3158 7731, Passcode: 058298)

Speaker

Wenda Fang (RIMS)

Title

Generalized AKS scheme of integrability via vertex algebra

Abstract

　 As in the attached file.

Date

January 12 (Fri), 10:30--12:00, 2024

Place

Room 006, RIMS
Zoom (Meeting ID: 842 2380 6155, Passcode: 725768)

Speaker

Kari Vilonen (Melbourne)

Title

Unitary representations of real groups and localization theory for Hodge modules

Abstract

　 We explain how mixed Hodge modules can be used to understand unitary representations of real groups as was conjectured by the speaker with Wilfried Schmid a while back. Along the way we obtain vanishing results for mixed Hodge modules on flag manifolds and a version of Saito's Kodaira vanishing theorem for twisted mixed Hodge modules. The proofs will depend on deformation and wall crossing arguments as well as the explicit determination of the associated graded of certain mixed Hodge modules. This is joint work with Dougal Davis.

Date

December 14 (Thu), 10:30--12:00, 2023

Place

Room 006, RIMS
Zoom (Meeting ID: 867 1000 5800, Passcode: 539591)

Speaker

Henry Liu (Kavli IPMU)

Title

Vertex algebras from wall-crossing

Abstract

　 In recent work on “universal” wall-crossing formulas, Joyce obtains a geometric construction of a vertex algebra structure on the homology groups of moduli stacks parameterizing objects in certain abelian categories. I will review this construction and explain how it may be extended to some sort of equivariant K-homology groups, producing multiplicative and equivariant versions of vertex algebras. These vertex algebras are compatible with cohomological/K-theoretic Hall algebras, of the same moduli stacks, and I will speculate a bit on the role they may play in controlling enumerative invariants, via wall-crossing or otherwise.

Date

November 30(Thu), 10:30--12:00, 2023

Place

Room 006, RIMS
Zoom (Meeting ID: 836 9024 3172, Passcode: 561711)

Speaker

Shigeo Koshitani (Chiba University)

Title

Representation theory of finite groups for a case of wild representation type

Abstract

　 For a given finite p-group P where p is a prime, one can ask such as; Can we know all finite groups G with a Sylow p-subgroup P? As a kind of generalization of this, in modular representation theory of finite groups there is a famous conjecture, called "Donovan's conjecture" due to Peter Donovan, which says that for a given finite p-group P there should be (up to Morita equivalence) only finitely many block algebras B of finite groups such that P is isomorphic to defect groups of B. We will be discussing mainly one particular interesting case of wild representation type such that for this case even a bit stronger conjecture called "Puig's conjecture" due to Llouis Puig does hold. This is joint work with Caroline Lassueur and Benjamin Sambale.

Date

November 16 (Thu), 10:30--12:00, 2023

Place

Room 006, RIMS
Zoom (Meeting ID: 837 0595 2753, Passcode: 449403)

Speaker

Shun Furihata (RIMS)

Title

On the Beem-Nair Conjecture

Abstract

　 The class $\mathcal{S}$ theory is a class of four-dimensional $\mathcal{N}=2$ super CFT, and a class of the corresponding vertex operator algebras is called the chiral algebras of class $\mathcal{S}$. In connection with this theory, Beem and Nair constructed a symplectic open immersion from the cover of the Kostant–Toda lattice associated to $G$ into the universal centralizer of $G$ for a simple linear algebraic group $G$. They expected that a free field realization of the chiral universal centralizer of $G$ at the critical level will be obtained by the chiralization of this immersion. In this talk, I want to explain the details of this conjecture, and verify that it is true by constructing an embedding from the chiral universal centralizer of $G$ into an appropriate vertex operator algebra at any level.

Date

July 20 (Thu), 10:30--12:00, 2023

Place

Room 006, RIMS
Zoom (Meeting ID : 811 7995 8722, Passcode : 529193)

Speaker

Myungho Kim (Kyung Hee University)

Title

Affinizations, R-matrices and reflection functors

Abstract

　 Kashiwara and Park provided a way to construct functors from the category of finite-dimensional modules over a quiver Hecke algebra to the category of finite-dimensional modules over another quiver Hecke algebra. One of the key notions in the construction is the affinization of a simple module over the (codomain) quiver Hecke algebra. In this talk I will explain the notion of affinizations in the abelian monoidal categories with certain conditions, which provides a way to construct functors from the category of finite-dimensional modules over a quiver Hecke algebra to these categories. One of the main examples of such categories is the localization of a subcategory of the finite-dimensional modules over a quiver Hecke algebra. In particular, we construct the functors $S_i$ between the localizations of the subcategories $C_{s_iw_0}$ and $C_{*,s_i}$ for arbitrary finite type quiver Hecke algebras. They are called the reflection functors since they recover the Saito reflections on crystals. This is a joint work with Masaki Kashiwara, Se-jin Oh and Euiyong Park (arxiv:2304.00238).

Date

June 12 (Mon), 10:30--12:00, 2023

Room

Room 110, RIMS
Zoom (Meeting ID: 823 1020 3999 Passcode: 794766)

Speaker

Yuki Hirano (Tokyo University of Agriculture and Technology)

Title

Mutations of noncommutative crepant resolutions in geometric invariant theory

Abstract

　 For a generic quasi-symmetric representation X of a reductive group G, the G IT quotient stack [X(L)//G] for a generic polarization L is a (stacky) crepa nt resolution of the affine quotient X/G. Halpern-Leistner and Sam proved th at the GIT quotients [X(L)//G] are all derived equivalent, which proved Bond al-Orlov conjecture for [X(L)//G]. One of the key ingredient of Halpern-Leis tner--Sam’s work is a magic window, which is shown to be equivalent to the derived category of the GIT quotient [X(L)//G]. A magic window is also equiv alent to the derived category of a noncommutative crepant resolution (NCCR) of X/G, which is the endomorphism algebra End(M) of a certain module M over X/G. In this talk, we explain that the modules giving NCCR of X/G are relate d by certain operations called exchanges, and in the case when G is a torus, the modules are related by Iyama--Wemyss mutations. If time permits, I will explain that certain autoequivalences of a Calabi-Yau hypersurface correspo nd to the compositions of Iyama--Wemyss mutations via matrix factorizations.

Date

June 1 (Thu), 10:30--12:00, 2023

Room

Room 006, RIMS
Zoom (Meeting ID：834 2356 4125 Passcode: 223798)

Speaker

Masamune Hattori (Nagoya)

Title

Elliptic quantum group and its generalization

Abstract

　 We introduce a family of dynamical Hopf algebroids depending on a complex parameter q, a formal parameter p, a set of structure functions sati sfying the so-called Ding-Iohara condition, and a finite root system. If the set of function is set to be certain theta functions, then our family recov ers the elliptic algebras for untwisted affine Lie algebras introduced by Ji mbo, Konno, Odake and Shiraishi(1999). Also, taking p → 0 for in the case t he root system of type A_l, we recover the Hopf algebras of type A_l introdu ced by Ding and Iohara as a generalization of Drinfeld quantum affine algebr as. Thus, our Hopf algebroid can be regarded as a dynamical analogue of the Ding-Iohara quantum algebras. As a byproduct, we obtain an extension of the Ding-Iohara quantum algebras to those of non-simply-laced type. This talk is based on the joint work with Shintaro Yanagida (arXiv:2210.02777).

Date

May 25 (Thu), 10:30--12:00, 2023

Room

Room 006, RIMS

Speaker

Bohan Li (Yau Mathematical Sciences Center, Tsinghua)

Title

On low rank 4d N=2 SCFTs and VOAs

Abstract

　 Argyres and collaborators have given a nice and explicit classification of 4d $\mathcal{N}=2$ SCFTs at rank one. Most of those theories are engineered by class-$S$ theory with irregular singularities. In this talk, I will give a universal formula for the rank of theory so that a complete search is possible. Various physical quantities of those theories, such as the central charges, flavor symmetry, associated vertex operator algebra and Higgs branch, etc can be computed. One of interesting consequence of our results is the prediction of many new isomorphism of vertex operator algebra. This talk will be based on the joint work with Dan Xie and Wenbin Yan.

Date

May 18 (Thu), 14:30--16:00, 2023

Room

Room 006, RIMS
Zoom (Meeting ID : 832 8055 2567, Passcode : 513212)

Speaker

Ivan Cherednik (University of North Carolina at Chapel Hill, RIMS)

Title

Compactified Jacobians of plane curve singularities

Abstract

　 They are generally affine Springer fibers. However in type A, the compactified Jacobians can be introduced directly in terms of plane curve singularities; they are projective connected varieties, which will be explained and extended to any ranks. Their conjectural relation to the L-functions of such singularities will be the main topic. The corresponding motivic superpolynomials can be used to define certain refined rho-invariants of algebraic knots. There are many interesting questions here. We will discuss what is known now and what can be expected.

Date

May 18 (Thu), 10:30--12:00, 2023

Room

Room 006, RIMS
Zoom (Meeting ID : 832 8055 2567, Passcode : 513212)

Speaker

Lewis Topley (University of Bath)

Title

Dirac reduction and the orbit method

Abstract

　 Namikawa has shown that conic symplectic singularities admit a very nice Poisson deformation theory: there exists a universal Poisson deformation over an affine base such that every filtered Poisson deformation can be obtained via base change. A remarkable theorem of Losev shows that filtered quantizations of these varieties are classified by precisely the same data, and he used this to construct a version of the orbit method. For complex simple algebraic group G, he defines a natural embedding from the set of coadjoint orbits of G to the set of completely prime primitive ideals of U(g). Losev conjectured that when G is classical the image of this map should consist of the ideals obtained from one dimensional representations of finite W-algebras via Skryabin’s equivalence. In this talk I will describe my recent proof of this conjecture. One of the main tools is Dirac reduction which allows us to obtain Yangian-type presentations of the semiclassical finite W-algebra, building on the work of Brundan and Kleshchev.

Date

April 27 (Thu) 14:30--16:00, 2023

Room

RIMS 006

Speaker

Peter Fiebig (University of Erlangen Nuremberg)

Title

Representations and Binomial Coefficients

Abstract

　 The talk is motivated by the „generational phenomena” that occur in the representation theory of algebraic groups in positive characteristics. The representation theory of quantum groups is known to provide a first step approximation to modular representations. Lusztig was the first to suggest that there should be „algebraic structures” that provide further steps towards modular representations beyond quantum groups. None of these structures are known today, even though some candidates have been suggested by several authors. In the talk I want to motivate the generational idea and then introduce a model category that makes the proximity of modular and quantum representations quite transparent. Using this category I want to show that the generational problem seems to be closely connected to finding generalizations of binomial coefficients.

Date

April 27 (Thu) 10:30--12:00, 2023

Room

RIMS 006, RIMS

Speaker

Shunsuke Tsuchioka (Tokyo Institute of Technology)

Title

A vertex operator reformulation of the Kanade-Russell conjecture modulo 9

Abstract

　 We reformulate the Kanade-Russell conjecture modulo 9 via the vertex operators for the level 3 standard modules of type D^{(3)}_4. Along the same line, we arrive at three partition theorems which may be regarded as an A^{(2)}_4 analog of the conjecture. One had been proved by Andrews-van Ekeren-Heluani and we point out that the others are easily proved from their results.

MINI-COURSE

Date

January 23 (Mon), January 25 (Wed), January 27 (Fri)
10:30--12:00, 2023

Room

003, Department of Botany Annexe
---Caution: The seminar room is not in the RIMS building, but in the building in the botanic garden next to RIMS.
If you don’t know the place, join us at the entrance of RIMS at 10:20 on Monday, January 23.

Speaker

Alexei Latyntsev (University of Southern Denmark, Denmark)

Title

Cohomological Hall algebras

Date

December 15 (Thu) 10:30--12:00, 2022

Room

RIMS 110
Zoom (Meeting ID：880 8640 6545 Passcode: 225908)

Speaker

Ryo Ohkawa (RIMS, Osaka Metropolitan University)

Title

Wall-crossing for vortex partition function and handsaw quiver varierty

Abstract

　 A_1型handsaw quiver variety上の積分により定まる分配関数を調べる. この分配関数の関数等式を壁越え現象の解析により証明した. さらに分配関数の明示公式から, 変数変換により多重超幾何級数が得られる. これにより梶原変換公式の有理版をふくむ多重超幾何級数の変換公式に別証明を 与えた. またグラスマン多様体上の積分についての類似の計算や, より一般に枠付き 箙表現について定式化できる等式についても紹介する.

Date

December 8 (Thu) 10:30--12:00, 2022

Room

Zoom (Meeting ID: 839 8417 9220; Passcode: 549080)

Speaker

Jiuzu Hong (University of North Carolina at Chapel Hill)

Title

BD Schubert varieties of parahoric group schemes and global Demazure modules of twisted current algebras

Abstract

　 It is well-known that there is a duality between affine Demazure modules and the spaces of sections of line bundles on Schubert varieties in affine Grassmannians. This should be regarded as a local theory. In this talk, I will explain an algebraic theory of global Demazure modules of twisted current algebras. Moreover, these modules are dual to the spaces of sections of line bundles on Beilinson-Drinfeld Schubert varieties of certain parahoric groups schemes, where the factorizations of global Demazure modules are compatible with the factorizations of line bundles. This generalizes the work of Dumanski-Feigin-Finkelberg in the untwisted setting. In order to establish this duality, following the works of Zhu, we prove the flatness of BD Schubert varieties, and establish factorizable and equivariant structures on the rigidified line bundles over BD Grassmannians of these parahoric group schemes. This talk will be based on the joint with Huanhuan Yu.

MINI-COURSE

Date

November 14 (Mon), November 15 (Tue), November 16 (Wed), November 21 (Mon), November 22 (Tue),
10:30--12:00, 2022
[additional] November 21 (Mon) 14:00--15:30,
November 22 (Tue) 14:00--15:30

Room

RIMS 402 (November 14 (Mon), November 15 (Tue), November 21 (Mon), November 22 (Tue)) and
RIMS 006 (November 16 (Wed))
[additional] RIMS 204 (November 21 (Mon) 14:00--15:30),
RIMS 006 (November 22 (Tue) 14:00--15:30)

Speaker

Gurbir Dhillon (Yale)

Title

Representation theory of affine W-algebras (mini-course)

Abstract

　 We describe some recent and ongoing developments in the study of highest weight representations of affine W-algebras, with an emphasis on new techniques and intuitions coming from the local geometric Langlands program. We will describe what is known in the principal case, including the geometric representation theory (Steinberg--Whittaker sheaves), and some related conjectures of Aganagic--Frenkel--Okounkov,Feigin--Gainutdinov--Semikhatov--Tipunin, Frenkel--Kac--Wakimoto, and Gaitsgory--Lurie. We will also outline some related expectations and conjectures for the non-principal cases.

Date

October 27th (Thu), 10:30--12:00, 2022

Room

Zoom (Meeting ID: 864 4086 0937, Passcode: 152734)

Speaker

Xiao Zheng (The Chinese University of Hong Kong)

Title

Elliptic hypertoric varieties and mirror symmetry

Abstract

　 In this talk, we introduce an elliptic analogue of hypertoric varieties defined using a procedure similar to holomorphic symplectic reduction. We will also introduce a version of Hikita’s isomorphism relating the torus fixed loci in the deformation space of an elliptic hypertoric variety and the equivariant elliptic cohomology of the dual additive hypertoric variety. If time permits, we will also discuss the 2d mirror symmetry between elliptic hypertoric varieties and Dolbeault hypertoric manifolds originally defined by Hausel and Proudfoot. These are based on joint works with Conan Leung, and Ziming Ma.

Date

October 6th (Thu), 16:30--18:00, 2022

Room

Zoom (meeting ID：875 9851 3818 Passcode: 455438)

Speaker

Oscar Kivinen (EPFL)

Title

3d mirror symmetry and affine Springer fibers

Abstract

　 Recent advances in mirror duality for line operators in 3d N=4 SUSY QFTs, especially for pure gauge theories or theories with adjoint matter, suggest a) new analogs of Hilbert schemes of points on the plane “outside type A” b) a co nstruction of (quasi-)coherent sheaves on these varieties starting from affine Springer fibers. We study this construction, a quantization thereof, and pose a number of open problems. Based on joint work with Gorsky and Oblomkov.

Date

August 4th (Thu), 10:30--12:00, 2022

Room

Zoom (meeting ID：896 4691 3847, passcode: 943553)

Speaker

Yuya Mizuno (Osaka Metropolitan University)

Title

Fans and polytopes in tilting theory

Abstract

　 For a finite dimensional algebra $A$, the 2-term silting complexes of $A$ give a simplicial complex $\Delta(A)$, fan $\sigma(A)$ and polytope $P(A)$, which are called the g-simplicial complex, g-fan and g-polytope, respectively. We study several properties of these objects from the viewpoint of the representation theory and combinatorics. In particular, we give a tilting theoretic interpretation of the $h$-vectors and the Dehn-Sommerville equations of $\Delta(A)$. Moreover, we give a characterization of the convexity of $P(A)$ and we explain a representation theoretic interpretation of the dual polytope of $P(A)$ in terms of simple minded collections. This is joint work with Aoki-Higashitani-Iyama-Kase (arXiv:2203.15213).

Date

June 24th (Fri), 14:30--16:00, 2022

Room

Zoom (access information will appear in https://kyoto-u-edu.zoom.us/j/83954592913
meeting ID：839 5459 2913
passcode：529494

Speaker

Yoshiyuki Kimura (Osaka Metropolitan University)

Title

Cluster twist automorphisms and compatible Poisson structures

Abstract

　 We introduce cluster twist automorphisms for (upper) cluster algebras and cluster Poisson algebras (with coefficients). We study their existence and their compatibility with Poisson structures and we also show that the cluster twist automorphisms always permute well-behaved bases for cluster algebras. This is a joint work with Fan Qin and Qiaoling Wei.

Date

June 16 (Thu), 10:30--12:00, 2022

Room

Zoom (access information will appear in https://kyoto-u-edu.zoom.us/j/81651182256
meeting ID：816 5118 2256
passcode：109268

Speaker

Yuchen Fu (RIMS)

Title

Kazhdan-Lusztig Equivalence at the Iwahori Level

Abstract

　 We construct an equivalence between Iwahori-integrable representations of affine Lie algebras and representations of the mixed quantum group, thus confirm a conjecture by Gaitsgory. Our proof utilizes factorization methods: we show that both sides are equivalent to algebraic/topological factorization modules over a certain factorization algebra, which can then be compared via Riemann-Hilbert. On the quantum group side this is achieved via general machinery of homotopical algebra, whereas the affine side requires inputs from the theory of (renormalized) ind-coherent sheaves as well as compatibility with global Langlands over P1. This is joint work with Lin Chen.

Date

February 3 (Thu), 10:00--11:30, 2022

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4811

Speaker

Angela Gibney (University of Pennsylvania)

Title

Vector bundles on the moduli space of curves from vertex operator algebras

Abstract

　 Algebraic structures like vector bundles, their sections, ranks, and characteristic classes, give information about spaces on which they are defined. The stack parametrizing families of stable n-pointed curves of genus g, and  the space that (coarsely) represents it, give insight into curves and their degenerations, are prototypes for moduli of higher dimensional varieties, and are interesting objects of study in their own right. Vertex operator algebras and their representation theory, have had a profound influence on mathematics and mathematical physics, playing a particularly important role in understanding conformal field theories, finite group theory, and invariants in topology.   In this talk I will discuss vector bundles on moduli of curves defined by certain representations of VOAs.

Date

January 27 (Thu), 10:30--12:00, 2022

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4806

Speaker

Zijun Zhou (Kavli IPMU)

Title

Virtual Coulomb branch and quantum K-theory

Abstract

　 In this talk, I will introduce a virtual variant of the quantized Coulomb branch constructed by Braverman-Finkelberg-Nakajima, where the convolution product is modified by a virtual intersection. The resulting virtual Coulomb branch acts on the moduli space of quasimaps into the holomorphic symplectic quotient T^*N///G. When G is abelian, over the torus fixed points, this representation is a Verma module. The vertex function, a K-theoretic enumerative invariant introduced by A. Okounkov, can be expressed as a Whittaker function of the algebra. The construction also provides a description of the quantum q-difference module. As an application, this gives a proof of the invariance of the quantum q-difference module under variation of GIT.

Date

January 13 (Thu), 17:00--18:30, 2022

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4793

Speaker

Vladimir Dotsenko (University of Strasbourg)

Title

A new proof of positivity of DT invariants of symmetric quivers

Abstract

　 I shall talk about a new interpretation of refined Donaldson-Thomas invariants of symmetric quivers, in particular re-proving their positivity (conjectured by Kontsevich and Soibelman, and proved by Efimov). This interpretation has two key ingredients. The first is a certain Lie (super-)algebra, for which we have two interpretations, in the context of Koszul duality theory and in the context of vertex Lie algebras. The second is an action of the Weyl algebra of polynomial differential operators on that Lie algebra, for which the characters of components of the space of generators give precisely the refined DT invariants. This is a joint project with Evgeny Feigin and Markus Reineke, partially relying on my recent work with Sergey Mozgovoy.

Date

December 9 (Thu), 21:00--22:30, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4760

Speaker

Joshua Sussan　(City University of New York)

Title

p-DG structures in higher representation theory

Abstract

　 One of the goals of the categorification program is to construct a homological invariant of 3-manifolds coming from the higher representation theory of quantum groups.  The WRT 3-manifold invariant uses quantum groups at a root of unity. p-DG theory was introduced by Khovanov as a means to categorify objects at prime roots of unity.  We will review this machinery and show how to construct categorifications of certain representations of quantum sl(2) at prime roots of unity.

Date

November 25 (Thu), 16:00--17:30, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4762

Speaker

Kenichi Shimizu (Shibaura Institute of Technology)

Title

Nakayama functors for Frobenius tensor categories

Abstract

　 This talk is based on my joint work with Taiki Shibata. The Nakayama functor is an important notion in the representation theory of finite-dimensional algebras. Fuchs, Schaumann, and Schweigert pointed out that the Nakayama functor has a certain universal property and, by using this property, defined the Nakayama functor for finite abelian categories. As they also pointed out, such an abstract treatment of the Nakayama functor turned out to be very useful for proving general results on finite tensor categories. In this talk, I will explain how and when one can define the Nakayama functor for a locally finite abelian category. Let A be a locally finite abelian category. Technical difficulty is that there is no endofunctor on A satisfying the same universal property as in the finite case. Such an endofunctor on A exists if, for example, A is the category of finite-dimensional comodules over a semiperfect coalgebra. This observation allows us to define the Nakayama functor for Frobenius tensor categories in the sense of Andruskiewitsch, Cuadra and Etingof. Using the Nakayama functor, one can prove some general results on Frobenius tensor categories in the same way as the finite case.

Date

November 11 (Thu), 16:00--17:30, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4757

Speaker

Ryo Sato (RIMS)

Title

Feigin-Semikhatov Duality in W-superalgebras

Abstract

　 W-superalgebras are a large class of vertex superalgebras which generalize affine Lie superalgebras and the Virasoro algebras. Recently, D. Gaiotto and M. Rapčák found dualities between certain hook-type W-superalgebras in relation to certain four-dimensional supersymmetric gauge theories. A large part of their conjecture is proved by T. Creutzig and A. Linshaw, and a specific subclass (Feigin-Semikhatov duality) is done by T. Creutzig, N. Genra, and S. Nakatsuka in a different way. In this talk I will talk about a monoidal correspondence of representations induced by the latter duality and relative semi-infinite cohomology. This talk is based on a joint work with T. Creutzig, N. Genra, and S. Nakatsuka.

Date

July 29 (Thu), 15:00--16:30, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4695

Speaker

Shinji Koshida (Chuo University)

Title

The quantum group dual of the first-row modules for the generic Virasoro VOA

Abstract

　 In several examples it has been observed that a module category of a vertex operator algebra (VOA) is equivalent to a category of representations of some quantum group. In this talk, we discuss such a duality in the case of the Virasoro VOA at generic central charge. We do not address the category of all modules of the generic Virasoro VOA, but we consider the infinitely many modules from the first row of the Kac table. Building on an explicit quantum group method of Coulomb gas integrals, we give a new proof of the fusion rules and the analyticity of compositions of intertwining operators. Crucially, we prove the associativity of the intertwining operators among the first-row modules, and find that the associativity is governed by the 6j-symbols of the quantum group Uq(sl2). This talk is based on a joint work with Kalle Kytölä.

Date

July 15 (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4669

Speaker

Gaëtan Borot　(Humboldt-Universität zu Berlin)

Title

Whittaker vectors for W-algebras from topological recursion

Abstract

　 Inspired by Alday-Gaiotto-Tachikawa conjecture in physics, Schiffman-Vasserot and Braverman-Finkelberg-Nakajima showed that, if G is a simple simply-laced Lie group, the partition function of pure N = 2 supersymmetric gauge theories with gauge group G can be reconstructed as the norm of certain Whittaker vectors of principal W(g)-algebras. After reviewing the context, I will explain how such Whittaker vectors (and in principle many more "Whittaker-like" vectors) can be computed by a topological recursion a la Eynard-Orantin, and potential consequences. This is based on joint works with Bouchard, Chidambaram, Creutzig and Noshchenko.

Date

July 8 (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4683

Speaker

Pablo Boixeda Alvarez (Institute for Advanced Study)

Title

The small quantum group and certain affine Springer fibers

Abstract

　 In this talk I will discuss several connections between the small quantum group and a certain affine Springer fiber. In particular I will mainly discuss some relation of the center of the small quantum group and the cohomology of the affine Springer fiber, part of ongoing joint work with R. Bezrukavnikov, P. Shan and E. Vasserot. I will also mention some description of the category in terms of microlocal sheaves as part of ongoing work with R.Bezrukavnikov, M. McBreen and Z. Yun and certain connections of cohomology of the affine Springer fiber and the Hilbert scheme of points of C^2 as part of joint work with O. Kivinen and I. Losev.

Date

July 1 (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4680

Speaker

Syu Kato (Kyoto University)

Title

Categorification of DAHA and Macdonald polynomials

Abstract

　 We exhibit a categorification of the double affine Hecke algebra (DAHA) associated with an untwisted affine root system (except for type G) and its polynomial representation by using the (derived) module category of some Lie superalgebras associated to the root system. This particularly yields a categorification of symmetric Macdonald polynomials. This is a joint work with Anton Khoroshkin and Ievgen Makedonskyi.

Date

June 24 (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4663

Speaker

Andrey Smirnov (University of North Carolina at Chapel Hill)

Title

Quantum difference equations and elliptic stable envelopes

Abstract

　 A remarkable class of differential and q-difference equations emerges naturally in the study of enumerative geometry of quiver varieties. This class includes Knizhnik-Zamolodchikov equations, quantum dynamical equations and other important objects in representation theory. In my talk I overview a geometric approach to these equations based on the theory of elliptic stable envelopes and three-dimensional mirror symmetry. In this approach we use geometric methods to constrain the monodromy of the associated q-difference equations. Then, the equations can be reconstructed from the monodromy via a simple limiting procedure. The three-dimensional mirror symmetry of the elliptic stable envelopes relates the equations associated to a quiver variety with those of symplectic dual variety.

Date

June 17 (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4662

Speaker

Yuto Moriwaki (RIMS)

Title

Code conformal field theory and framed algebra

Abstract

　 It is known that there are 48 Virasoro algebras acting on the "monster conformal field theory". We call conformal field theories with such a property, which are not necessarily chiral, code conformal field theories. Recently, we introduce a notion of a framed algebra, which is a finite-dimensional non-associative algebra, and showed that the category of framed algebras and the category of code conformal field theories are equivalent. We have also constructed a new family of conformal field theories using this equivalence. These conformal field theories are expected to be useful for the study of moduli spaces of conformal field theories.

Date

June 10 (Thu), 16:00--17:30, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4649

Speaker

Alexander Zuevsky (Czech Academy of Sciences)

Title

Reduction cohomology on Riemann surfaces

Abstract

　 We define and compute a cohomology of the space of Jacobi forms based on precise analogues of Zhu reduction formulas derived by Bringmann-Krauel-Tuite. It is shown that the reduction cohomology for Jacobi forms is given by the cohomology of $n$-point connections over a deformed vertex algebra bundle defined on the torus. The reduction cohomology for Jacobi forms for a vertex algebra is determined in terms of the space of analytical continuations of solutions to Knizhnik-Zamolodchikov equations.
A counterpart of the Bott-Segal theorem for the reduction cohomology of Jacobi forms on the torus is proven.
Algebraic, geometrical, and cohomological meanings of reduction formulas is clarified.

Date

May 27 (Thu), 15:00--16:30, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4639

Speaker

Daniel Bruegmann (Max Planck Institute for Mathematics)

Title

Vertex Algebras and Factorization Algebras

Abstract

　 Vertex algebras and factorization algebras are two approaches to chiral conformal field theory. Costello and Gwilliam describe how every holomorphic factorization algebra on the plane of complex numbers satisfying certain assumptions gives rise to a Z-graded vertex algebra. They construct some models of chiral conformal theory as factorization algebras. We attach a factorization algebra to every Z-graded vertex algebra.

Date

May 20 (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4633)

Speaker

Jethro van Ekeren (Universidade Federal Fluminense)

Title

Chiral homology of elliptic curves

Abstract

　 In this talk I will discuss results of an ongoing project (joint with Reimundo Heluani) on the chiral homology of elliptic curves with coefficients in a conformal vertex algebra. Since the work of Y. Zhu it is clear that this homology has important applications to the representation theory of vertex algebras. We construct a flat connection on the first chiral homology over the moduli space, and relate the nodal curve limit with the Hochschild homology of the Zhu algebra. We construct flat sections from self-extensions of modules. Along the way we find interesting links between these structures, associated varieties of vertex algebras, and classical identities of Rogers-Ramanujan type (this last part joint work with George Andrews).

Date

May 13 (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4624)

Speaker

Michael McBreen (The Chinese University of Hong Kong)

Title

Hypertoric Hitchin systems and Kirchoff polynomials

Abstract

　 I will present joint work with Michael Groechenig, which associates a degenerating family of abelian varieties to a graph. On the one hand, it is an algebraisation of the `Dolbeault' hypertoric spaces originally defined by H ausel and Proudfoot. On the other, it is an approximation to the relative compactified Jacobian of a family of curves degenerating to a nodal curve with specified dual graph. We then consider our construction over the p-adic numbers, and compute the p-adic volumes of the fibers. We find they are given by the Kirchoff polynomial of the graph.

Date

May 6 (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4616)

Speaker

Yasuaki Hikida (YITP, Kyoto University)

Title

Generalized Fateev-Zamolodchikov-Zamolodchikov dualities and Gaiotto- Rapcak's VOAs

Abstract

　 About two decades ago, Fateev, Zamolodchikov and Zamolodchikov (FZ Z) conjectured a strong/weak duality between two dimensional conformal field theories, SL(2)/U(1) coset model and sine-Liouville theory. Recently, we ha ve succeeded to generalize the FZZ-duality by extending its original derivat ion done with Schomerus. The generalized FZZ-dualities can be regarded as a conformal field theoretic realization of dualities among VOAs conjectured by Gaiotto and Rapcak via brane junctions in string theory. In this talk, I wi ll explain how to derive the generalized FZZ-dualities and relation to the G aiotto-Rapcak's dualities.

Date

April 22 (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4595)

Speaker

Tatsuki Kuwagaki (Osaka University)

Title

Sheaf quantization: example and construction

Abstract

　 Constructible sheaves have played an important role in the development of representation theory. The topic of this talk is sheaf quantization, which is a geometric refinement of the notion of constructible sheaf (“constructible sheaf (or local system) of 21st century”). I will give an introduction to sheaf quantization and how it is difficult (at present) to construct it in general; I’d like to explain how the ideas from exact WKB analysis, resurgent analysis, and Fukaya category come into the story.

Date

April 15 (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4590)

Speaker

Thomas Creutzig (University of Alberta)

Title

From W-algebras to W-superalgebras

Abstract

　 A W-superalgebra is a vertex superalgebra associated to a Lie superalgebra, g, an invariant bilinear form on g and an even nilpotent element in g. If g is a Lie algebra and f is principal nilpotent then one obtains the principal W-algebra of g. Feigin-Frenkel duality are isomorphisms between principal W-algebras. These isomorphisms somehow generalize to non principal nilpotent elements, however the isomorphism is only between coset subalgebras of W-algebras and W-superalgebras. In my talk I will first introduce the isomorphisms that generalize Feigin-Frenkel duality. I then want to outline a program on how to use the dualities to get correspondences between tensor categories of W-algebra modules and dual W-superalgebra modules.

Date

February 18 (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4575)

Speaker

Hironori Oya (Shibaura Institute of Technology)

Title

Systematic construction of isomorphisms among quantum Grothendieck rings

Abstract

　 A quantum Grothendieck ring of the monoidal category of finite-dimensional modules over a quantum loop algebra Uq(Lg) is a one parameter deformation of the usual Grothendieck ring. It is introduced by Nakajima and Varagnolo-Vasserot in the case when Uq(Lg) is of simply-laced type through a geometric method, and subsequently by Hernandez when Uq(Lg) is of arbitrary untwisted affine type through an algebraic method. In the simply-laced case, quantum Grothendieck rings are known to give an algorithm for calculating q-characters of simple modules, which is an analogue of Kazhdan-Lusztig algorithm.
In this talk, we present a collection of algebra isomorphisms among quantum Grothendieck rings, which respect the (q,t)-characters of simple modules. As a corollary, we obtain new positivity results for the simple (q,t)-characters of non-simple-laced types. Moreover, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur-Weyl dualities, we show that an analogue of Kazhdan-Lusztig algorithm for computing simple q-characters is available when g is of type B.
This result is a vast generalization of our previous work [Hernandez-O, Adv. Math. 347 (2019), 192--272]. Hence, besides the summary of main results, I will explain some details of the proof, focusing on the tools and results obtained after [HO].
This talk is based on a joint work (arxiv:2101.07489) with Ryo Fujita, David Hernandez, and Se-jin Oh.

Date

February 4 (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4565)

Speaker

Xuanzhong Dai (Fudan University)

Title

Chiral de Rham complex on the upper half plane and modular forms

Abstract

　 Chiral de Rham complex constructed by Malikov, Schechtman and Vaintrob in 1998, is a sheaf of vertex algebras on a complex manifold. For any congruence subgroup $\Gamma$, we consider the $\Gamma$-invariant global sections of the chiral de Rham complex on the upper half plane, which are holomorphic at all the cusps. We show that it contains an $N = 2$ superconformal structure and we give an explicit lifting formula from modular forms to it. As an application, the vertex algebra structure modifies the Rankin-Cohen bracket, and the modified bracket with the Eisenstein series involved becomes nontrivial between constant modular forms.

Date

January 28th (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4535)

Speaker

Xuhua He (The Chinese University of Hong Kong)

Title

Flag varieties over semifields

Abstract

　 In 1994, Lusztig developed the theory of total positivity for arbitrary split real reductive groups and their flag manifolds. Later the theory has found important applications in different areas: cluster algebras, higher Teichmuller theory, the theory of amplituhedron in physics, etc. Recently, Lusztig initiated the study of Kac-Moody monoids over arbitrary semifield and their flag manifolds. In the case where the Kac-Moody datum comes from a real reductive group and the semifield is $R_{>0}$, the Kac-Moody monoid over $R_{>0}$ is exactly the totally nonnegative part of the real reductive group. In this talk, I will discuss my joint work with Huanchen Bao on the flag manifolds B(K) over arbitrary semifield K and associated to any Kac-Moody? datum G. We show that B(K) admits a natural action of the Kac-Moody monoid G(K) and admits a decomposition into cells.

Date

January 21 (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4564)

Speaker

Kota Murakami (Kyoto University)

Title

PBW parametrizations and generalized preprojective algebras

Abstract

　 Kashiwara-Saito realized crystal bases of quantum enveloping algebras on irreducible components of varieties of nilpotent modules over preprojective algebras for simply-laced types. Recently, Geiss-Leclerc-Schroer generalized these realizations to non-simply laced types by developing representation theory of a class of 1-Iwanaga-Gorenstein algebras and their preprojective algebras associated with symmetrizable GCMs (=generalized Cartan matrices) and their symmetrizers. In this talk, we relate representation theory of the generalized preprojective algebras with numerical data about the dual canonical bases, so called Lusztig data, for symmetrizable GCMs of finite types. In particular, we realize Mirkovic-Vilonen polytopes from some generic modules over generalized preprojective algebras as a generalization of the work of Baumann-Kamnitzer-Tingley.

Date

January 14 (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4552)

Speaker

Haruhisa Enomoto (Nagoya University)

Title

ICE-closed subcategories of module categories

Abstract

　 In the representation theory of algebras, the study of subcategories of module categories has been one of the main topics, and is related to many areas. Among them, torsion classes and wide subcategories are important and have been studied by many people. In this talk, I will introduce the notion of ICE-closed subcategories of module categories, which are closed under taking Images, Cokernels and Extensions. This class contains both torsion classes and wide subcategories. In the representation category of a Dynkin quiver, they bijectively correspond to rigid representations. For a general finite-dimensional algebra, I will explain how to classify ICE-closed subcategories using the poset structure of torsion classes, or using $\tau$-tilting theory. This talk is based on my joint work with Arashi Sakai (Nagoya).

Date

January 7th (Thu), 10:30--12:00, 2021

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4544)

Speaker

Toshiya Yurikusa (Tohoku University)

Title

Denseness of g-vector fans for tame algebras

Abstract

　 The g-vector fan of a finite dimensional algebra is a simplicial polyhedral fan whose rays are the g-vectors of the indecomposable 2-term presilting complexes. We consider the property that the g-vector fan is dense. We prove that gentle algebras satisfy it by using their surface model (based on a joint work with Toshitaka Aoki). The main ingredients of our proof are the g-vectors of the laminations and their asymptotic behavior under Dehn twists. More generally, using the generic decompositions and twist functors instead of them, we can prove it for tame algebras (based on a joint work with Pierre-Guy Plamondon).

Date

December 24th (Thu), 10:30--12:00, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4536)

Speaker

Sven Moeller (RIMS)

Title

Schellekens' VOAs, Generalised Deep Holes and the Very Strange　Formula

Abstract

　 In this talk I will summarise recent results regarding the classification of strongly rational, holomorphic VOAs (or CFTs) of central charge 24 (based on joint works with Jethro van Ekeren, Gerald Höhn, Ching Hung Lam, Nils Scheithauer and Hiroki Shimakura). First, we show that there is an abstract bijection (without classifying either side) between these VOAs and the generalised deep holes of the Leech lattice VOA. The proof uses a dimension formula obtained by pairing the VOA character with a vector-valued Eisenstein series and an averaged version of Kac's Lie theoretic very strange formula. This is a quantum analogue of the result by Conway, Parker and Sloane (and Borcherds) that the deep holes of the Leech lattice are in bijection with the Niemeier lattices. Then, we explain how this can be used to classify the (exactly 70) strongly rational, holomorphic VOAs of central charge 24 with non-zero weight-one space. (The case of zero weight-one space, which includes the Moonshine module, is more difficult and still open.)

Date

December 3rd (Thu), 10:30--12:00, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4508)

Speaker

Takahiro Nishinaka (Ritsumeikan University)

Title

The Chiral Algebra of Genus-Two Class S Theory

Abstract

　 Vertex operator algebras (VOAs) of class S are those closely related to four-dimensional N=2 superconformal field theories in physics, and give an interesting functor from the category of 2-bordisms to a category of VOAs. In particular, gluing two bordisms corresponds to a certain BRST reduction of VOAs. VOAs in this category are generally associated with a semi-simple Lie algebra and a 2-manifold. In this talk, I will discuss the one associated with sl(2) and genus-two manifold without boundary, arguing that its automorphism group contains an SU(2) sub-group which is unexpected even from physics.

Date

November 26th (Thu), 10:30--12:00, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4504)

Speaker

Dinakar Muthiah (Kavli IPMU)

Title

Equations for affine Grassmannians and their Schubert varieties.

Abstract

　 I will discuss work on a conjectural moduli description of Schubert varieties in the affine Grassmannian and proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on equations defining type A affine Grassmannians. As an application of our ideas, we prove a conjecture of Pappas and Rapoport about nilpotent orbit closures. This involves work with Joel Kamnitzer, Alex Weekes, and Oded Yacobi.

Date

November 12th (Thu), 10:30--12:00, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4474)

Speaker

Yuto Moriwaki (Kavli IPMU)

Title

Two dimensional conformal field theory, current-current deformation and Mass formula

Abstract

　 This talk deals with a deformation of a two dimensional conformal field theory. We introduce a notion of a full vertex algebra, which is a mathematical formulation of a compact two dimensional conformal field theory on $R^2$. We also give examples of full vertex algebras and discuss the relation between vertex algebras and full vertex algebras. Then, we construct a deformation of a full vertex algebra, which serves as a current-current deformation of the conformal field theory in physics. As an application, we consider the deformation of a tensor product of a vertex algebra and some full vertex algebra. Such deformation may produce new vertex algebras. We give a formula which counts a weighted sum of the number of vertex algebras appearing in the deformation.

Date

November 5th (Thu), 10:30--12:00, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4476)

Speaker

並河 良典氏 (RIMS)

Title

ベキ零軌道の普遍被覆と双有理幾何

Abstract

　 複素半単純リー環のベキ零軌道の閉包を正規化した代数多様体は,シンプレクティッ ク特異点になり, そのシンプレクティック特異点解消や {\bf Q}-分解端末化はよく 研究されている. この講演では, ベキ零軌道の普遍被覆に付随したシンプレクティッ ク特異点を考え, 同様の問題を考える. 古典単純リー環の場合には, {\bf Q}-分解端 末化を具体的に構成するアルゴリズムを与えることができる. また異なる{\bf Q}-分 解端末化が何個存在するかの, 明示公式を与えることもできる.

Date

August 6th (Thu), 10:30--12:00, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4426)

Speaker

Ryo Kanda (Osaka City University)

Title

Feigin-Odesskii's elliptic algebras

Abstract

　 This talk is based on joint work with Alex Chirvasitu and S. Paul Smith. Feigin and Odesskii introduced a family of noncommutative graded algebras, which are parametrized by an elliptic curve and some other data, and claimed a number of remarkable results in their series of papers. The family contains all higher dimensional Sklyanin algebras, which have been widely studied and recognized as important examples of Artin-Schelter regular algebras. In this talk, I will explain some properties of Feigin-Odesskii's algebras, including the nature of their point schemes and algebraic properties obtained by using the quantum Yang-Baxter equation.

Date

July 30th (Thu), 10:30--12:00, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4418)

Speaker

Liron Speyer (OIST)

Title

Semisimple Specht modules indexed by bihooks

Abstract

　 I will first give a brief survey of some previous results with Sutton, in which we found a large family of decomposable Specht modules for the Hecke algebra of type B indexed by `bihooks'. We conjectured that outside of some degenerate cases, our family gave all decomposable Specht modules indexed by bihooks. There, our methods largely relied on some hands-on computation with Specht modules, working in the framework of cyclotomic KLR algebras.
I will then move on to discussing a new project with Muth and Sutton, in which we have studied the structure of these Specht modules. By transporting the problem to one for Schur algebras via a Morita equivalence of Kleshchev and Muth, we are able to show that in most characteristics, these Specht modules are in fact semisimple, and give all composition factors (including their grading shifts). In some other small characteristics, we can explicitly determine the structure, including some in which the modules are `almost semisimple'. I will present this story, with some running examples that will help the audience keep track of what's going on.

Date

July 16th (Thu), 10:30--12:00, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4412)

Speaker

Koichi Harada (Department of Physics, The University of Tokyo)

Title

Quantum deformation of Feigin-Semikhatov's W-algebras from quantum toroidal gl(1)

Abstract

　 Quantum toroidal algebras contain many q-W algebras as truncations, but most of them have not been studied in detail. The typical examples among them are the subregular W-algebras of type A. The screening charges were proposed by Feigin and Semikhatov and their q-deformation was also found in the study of quantum toroidal gl(1). Further, the recent work by Gaiotto and Rapcak provides a clue to obtain many q-W algebras by gluing quantum toroidal gl(1)s. In this talk, I will discuss quantum deformation of Feigin-Semikhatov's W-algebras by using quantum toroidal gl(1) and Gaiotto-Rapcak's framework.

Date

July 9th (Thu), 10:30--12:00, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4394)

Speaker

Wataru Yuasa (RIMS, Kyoto University)

Title

The tail of the one-row colored sl(3) Jones polynomial and the Andrews-Gordon type identity

Abstract

　 I will review my works on the one-row colored sl(3) tail of knots and links. The tail is a q-series obtained as a limit of the colored Jones polynomial.
The first topic is the existence of tails of the one-row colored sl(3) colored Jones polynomials for oriented "adequate" links. In the case of sl(2), it showed by Armond and Garoufalidis-Le independently. The second topic is the Andrews-Gordon type identities for (false) theta series obtained from the tail of (2,m)-torus knots and links. It is known that our formula of one-row colored sl(3) tail coincides with the diagonal part of the sl(3) false theta function obtained by Bringmann-Kaszian-Milas.
In this talk, I will also give a quick review on quantum invariants of knots and links.

Date

July 2nd (Thu), 14:00--15:30, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4398)

Speaker

Kanam Park (Kobe University)

Title

A certain generalization of $q$-hypergeometric functions and its related monodromy preserving deformation

Abstract

　 We define a series $\mathcal{F}_{M,N}$ as a certain generalization of $q$-hypergeometric functions. We also study the system of $q$-difference nonlinear equations which admits particular solutions in terms of $\mathcal{F}_{N,M}$. The function $\mathcal{F}_{N,M}$ is a common generalization of $q$-Appell-Lauricella function $\varphi_D$ and the generalized $q$-hypergeometric function ${}_{N+1}\varphi_N$. We construct a Pfaffian system which the function $\mathcal{F}_{N,M}$ satisfies. We derive from the Pfaffian system a monodromy preserving deformation which admits particular solutions in terms of $\mathcal{F}_{N,M}$. In this talk, we will introduce the function $\mathcal{F}_{N,M}$ and its fundamental properties and the system derived from a Pfaffian system which $\mathcal{F}_{M,N}$ satisfies.

Date

June 25th (Thu), 10:30--12:00, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4381 )

Speaker

Ryosuke Kodera (Kobe University)

Title

Affine Yangians and rectangular W-algebras

Abstract

　 W-algebra is a class of vertex algebras attached to a complex reductive Lie algebra, a nilpotent element in the Lie algebra, and a complex number. We consider the case of the general linear Lie algebra $\mathfrak{gl}_N$ with $N=l \times n$ and a nilpotent element whose Jordan form corresponds to the partition $(l^n)$. We call it rectangular W-algebra. Its current algebra (or enveloping algebra) is defined as the associative algebra generated by the Fourier modes of generating fields. The goal of this talk is to construct an algebra homomorphism from the affine Yangian of type A to the current algebra of the rectangular W-algebra. We use the coproduct and the evaluation map of the affine Yangian to construct it. We hope that the homomorphism will be applied to the study of the AGT correspondence for parabolic sheaves and of integrable systems associated with the W-algebra. The talk is based on a joint work with Mamoru Ueda.

Date

June 18th (Thu), 10:30--12:00, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4365 )

Speaker

Yuma Mizuno (Tokyo Institute of Technology)

Title

Difference equations arising from cluster algebras

Abstract

　 The theory of cluster algebras gives powerful tools for systematic studies of discrete dynamical systems. Given a sequence of quiver mutations that preserves the quiver, we obtain a finite set of algebraic relations, yielding a discrete dynamical system. Such a set of algebraic relations is called a T-system. In this talk, I will explain that T-systems are characterized by pairs of matrices that have a certain symplectic property. This generalize a characterization of period 1 quivers, which was given by Fordy and Marsh, to arbitrary mutation sequences. I will also explain the relation between T-systems and Nahm's problem about modular functions, which is one of the main motivations of our study.

Date

June 11th (Thu), 10:30--12:00, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4359 )

Speaker

Shintarou Yanagida (Nagoya)

Title

Derived gluing construction of chiral algebras

Abstract

　 We discuss the gluing construction of class S chiral algebras in derived setting. The gluing construction in non-derived setting was introduced by Arakawa to construct a family of vertex algebras of which the associated varieties give genus 0 Moore-Tachikawa symplectic varieties. Motivated by the higher genus case, we introduce a dg vertex algebra version of the category of Moore-Tachikawa symplectic varieties, where taking associated schemes gives a functor to the category of derived Moore-Tachikawa varieties.
This talk is based on the preprint arXiv:2004.10055.

Date

June 4th (Thu), 10:30--12:00, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4344 )

Speaker

Tsukasa Ishibashi (RIMS)

Title

Algebraic entropy of sign-stable mutation loops

Abstract

　 In the theory of cluster algebra, a mutation loop is a certain equivalence class of a sequence of seed mutations and permutations of indices. They form a group called the cluster modular group, which can be regarded as a combinatorial generalization of the mapping class groups of marked surfaces.
　We introduce a new property of mutation loops which we call the “sign stability”, as a generalization of the pseudo-Anosov property of a mapping class. A sign-stable mutation loop has a numerical invariant which we call the “cluster stretch factor”, in analogy with the stretch factor of a pA mapping class. We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop, and conclude that these two are estimated by the logarithm of the cluster stretch factor. This talk is based on a joint work with Shunsuke Kano.

Date

May 28th (Thu), 10:30--12:00, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4330 )

Speaker

Shigenori Nakatsuka (Kavli IPMU)

Title

On Gaiotto-Rapcak's dualities in W-superalgebras and their affine cosets

Abstract

　 The principal affine W-algebras enjoy the Feigin-Frenkel duality a nd the Goddard-Kent-Olive construction if associated with simply-laced Lie a lgebras. Recently, Gaiotto and Rapcak proposed a generalization of this tria lity (or dualities) among W-superalgebras（and their affine cosets) by using the 4-dimensional N=4 super Yang-Mills theories. In this talk, we prove the Feigin-Frenkel type duality between the Heisenberg cosets of the subregular W-algebras and the principal W-superalgebras for A and B-types. This is a j oint work with T. Creutzig and N. Genra.

Date

May 14th (Thu) 10:30--12:00, 2020

Room

Zoom (access information will appear in https://www.math.kyoto-u.ac.jp/ja/event/seminar/4327 )

Speaker

Hideya Watanabe (RIMS)

Title

Classical weight modules over i-quantum groups

Abstract

　 i-quantum groups are certain coideal subalgebras of quantum groups appearing in the theory of quantum symmetric pairs. Many results concerning quantum groups have been generalized to i-quantum groups. However, representation theory of i-quantum groups is much more difficult than that of quantum groups due to the lack of Chevalley-like generators. In this talk, I introduce the notion of classical weight modules over i-quantum groups, and explain that they can be thought of as counterparts of the weight modules over quantum groups.

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Date

March 9th (Mon) 10:30--12:00, 2020

Room

RIMS 006

Speaker

Hideya Watanabe (RIMS)

Title

Classical weight modules over i-quantum groups

Abstract

　 i-quantum groups are certain coideal subalgebras of quantum groups appearing in the theory of quantum symmetric pairs. Many results concerning quantum groups have been generalized to i-quantum groups. However, representation theory of i-quantum groups is much more difficult than that of quantum groups due to the lack of Chevalley-like generators. In this talk, I introduce the notion of classical weight modules over i-quantum groups, and explain that they can be thought of as counterparts of the weight modules over quantum groups.

Date

January 23th (Thu) 14:00–15:30, 2020

Room

RIMS 006

Speaker

Yota Shamoto (IPMU)

Title

Irregular vertex algebras

Abstract

　 Mainly motivated by Alday-Gaiotto-Tachikawa correspondence and their applications, irregular singularities in conformal field theory are investigated in mathematical physics. We shall initiate an attempt to give a mathematical language of such theory by introducing the notions of coherent state modules and irregular vertex algebras. In this talk, we shall explain the definitions, examples and some fundamental properties. This talk is based on a joint work with Akishi Ikeda at Osaka university.

Date

January 16th (Thu) 10:30–12:00, 2020

Room

RIMS 006

Speaker

Husileng Xiao (Harbin Engineering University)

Title

On representation of the finite W-superalgebras

Abstract

　 In the last decade, Losev found a Poisson geometric realization of the finite W-algebras . This provides him a powerful tool to study the representations of the finite W-algebras. In this talk, I will first introduce my joint work with Bin Shu, which generalize the above realization to the finite W-superalgebra case. Then I will discuss its application to the representations of the finite W-superalgebras.

Date

December 12th (Thu) 10:30–12:00, 2019

Room

RIMS 006

Speaker

Andrew Linshaw (Denver)

Title

Dualities of W-algebras and the W_{\infty}-algebra

Abstract

　 We classify one-parameter vertex algebras that arise as extensions of affine gl_m tensored with a vertex algebra of type W(2,3,\dots), where the extension is generated by 2m fields in a fixed conformal weight which transform as the standard representation of gl_m and its dual. As a consequence, we obtain some new dualities between families of W-algebras and W-superalgebras. We also give a new proof of the coset realization of principal W-algebras of type A that was obtained in my recent work with Arakawa and Creutzig. This is a joint work with T. Creutzig.

Date

November 14th (Thu) 10:30–12:00, 2019

Room

RIMS 006

Speaker

Masahiro Chihara (Kyoto U)

Title

Demazure slices of type $A_{2l}^{(2)}$

Abstract

　 Demazure slices are associated graded pieces of infinite-dimensional version of Demazure modules for affine Lie algebras. In this talk, we review (1) Demazure slices for other types and (2) usual finite dimensional Demazure modules of type $A_{2l}^{(2)}$. Then we explain a relation between graded characters of Demazure slices for $A_{2l}^{(2)}$ and specialized nonsymmetric Macdonald-Koornwinder polynomials.

Date

November 7th (Thu) 10:30–12:00, 2019

Room

RIMS 006

Speaker

Kota Murakami (Kyoto U)

Title

On module categories of preprojective algebras with symmetrizers

Abstract

　 Geiss-Leclerc-Schröer has introduced preprojective algebras for symmetrizable GCMs and their symmetrizers. They are expected to generalize some Lie theoretical aspects of Gelfand-Ponomarev's preprojective algebras. In this talk, we will discuss module categories and some combinatorical invariants of these algebras from a viewpoint of tilting theory.

Date

Oct. 17th (Thu) 10:30–12:00, 2019

Room

RIMS 006

Speaker

Shoma Sugimoto (RIMS Kyoto U)

Title

On the Feigin-Tipunin VOA

Abstract

　 The triplet VOA ($A_1$ type Feigin-Tipunin VOA) is one of the most famous examples of $C_2$-cofinite and irrational VOA. However, there are not much known about the Feigin-Tipunin VOA $W(p)_Q$, the $ADE$ type generalization of the triplet VOA. In this talk, we will give the geometric construction and character formulas of $W(p)_Q$ that conjectured in the paper of Feigin-Tipunin. Moreover, we will give a $W$-algebraic conditioning of $C_2$-cofiniteness of $W(p)_Q$ under the expectable assumptions of simpleness. In the case of $A_2$ type with a fixed $p$, this conditioning enables us to prove the $C_2$-cofiniteness of $W(p)_Q$ much easier than direct calculation.

Date

Oct. 10 (Thu) 10:30–12:00, 2019

Room

RIMS 006

Speaker

Matt Szczesny (Boston University)

Title

Hall algebras of toric varieties over F_1

Abstract

　 Hall algebras of categories of quiver representations and coherent sheaves on smooth projective curves over F_q recover interesting representation-theoretic objects such as quantum groups and their generalizations. I will define and describe the structure of the Hall algebra of coherent sheaves on a projective variety over F_1, with P^2 as the main example. Examples suggest that it should be viewed as a degenerate q->1 limit of its counterpart over F_q.

Date

Oct. 3rd (Thu) 13:30--14:30, 2019

Room

RIMS 006

Speaker

Mamoru Ueda (RIMS Kyoto)

Title

Affine Super Yangian

Date

September 5th (Thu) 10:30--12:00, 2019

Room

RIMS 006

Speaker

Ievgen Makedonskyi (Kyoto)

Title

Peter-Weyl, Howe and Schur-Weyl theorems for current groups

Abstract

　 The classical Peter-Weyl theorem describes the structure of the space of functions on a semi-simple algebraic group. On the level of characters (in type A) this boils down to the Cauchy identity for the products of Schur polynomials. We formulate and prove the analogue of the Peter-Weyl theorem for the current groups. In particular, in type A the corresponding characters identity is governed by the Cauchy identity for the products of q-Whittaker functions. We also formulate and prove a version of the Schur-Weyl theorem for current groups. The link between the Peter-Weyl and Schur-Weyl theorems is provided by the (current version of) Howe duality.

Date

May 23 (Thu) 10:30--12:00, 2019

Room

RIMS 006

Speaker

Kanehisa Takasaki

Title

Hurwitz numbers of Riemann sphere and integrable hierarchies.

Abstract

　 The Hurwitz numbers count the topologically non-equivalent types of finite ramified coverings of a given Riemann surface. When the base Riemann surface is the Riemann sphere, these numbers are known to be related to intersection numbers of the Hodge classes and the psi classes on the moduli space of stable complex curves. On the other hand, the same numbers can be expressed in a genuinely combinatorial form in terms of symmetric groups. The latter expression reveals that the generating functions of a particular class of Hurwitz numbers of the Riemann sphere become tau functions of the KP hierarchy and its relatives. I will review these facts for non-experts of integrable systems.

Date

May 10 (Fri) 16:30--18:00, 2019

Room

RIMS 402

Speaker

Saiei-Jaeyeong Matsubara-Heo (Kobe)

Title

Integral representations of GKZ hypergeometric functions: Gauss-Manin connection, intersection theory, and quadratic relations

Abstract

　 GKZ(Gelfand, Kapranov, Zelevinsky) system is a holonomic system which describes classical hypergeometric systems in a unified manner. In this talk, we realize GKZ system as a Gauss-Manin connection where we treat Euler integral and Laplace integral at the same time. Focusing on regular holonomic case, we give a method of reinterpreting the combinatorics of regular triangulations to the construction of the basis of twisted cycles at "toric infinity". This naturally gives rise to an orthogonal decomposition of the twisted homology group with respect to the intersection pairing. As an application, we give a general quadratic relation of GKZ hypergeometric functions associated to a unimodular triangulation. We also discuss an algorithm of computing cohomology intersection numbers based on a joint work with Nobuki Takayama. The techniques above produce several new quadratic relations of hypergeometric functions of several variables.

Date

April 25 (Thurs) 10:30--12:00, 2019

Room

RIMS 006

Speaker

Anatol Kirillov

Title

Deformed Hecke algebras and quantum cohomology of graph varieties

Abstract

　 For any finite graph I define certain algebras, including deformed Hecke type ones. All these algebras contain commutative subalgebras generated by either additive or multiplicative, or elliptic Dunkl elements. I'm planing to explain why these commutative subalgebras can be identified with classical and (small) quantum cohomology (and that of K-theory) of certain graph varieties, including partial flag and Hessenberg varieties.

Date

Apr 19 (Fri), 14:30--16:00, 2019

Room

RIMS 402

Speaker

Kazuya Kawasetsu (RIMS)

Title

Relaxed highest-weight modules over affine VOAs

Abstract

　 The (chiral) symmetry algebras of 2d conformal field theory are described as vertex operator algebras (VOAs). Among them, C_2-cofinite ones with semisimple module categories correspond to rational conformal field theory, whose characters span a SL_2(\Z)-invariant vector space and whose fusion rules in module categories satisfy the celebrated Verlinde formula.
The affine VOAs are those construted by (and almost the same as) affine Kac-Moody algebras. Among them, non-integrable affine VOAs are important examples of non-C_2-cofinite VOAs. The Verlinde formula for non-integrable (admissible) affine VOAs proposed by T. Creutzig and D. Ridout involves in relaxed highest-weight modules over affine Kac-Moody Lie algebras, which are modules parabolically induced from weight modules over the associated finite-dimensional simple Lie algebras.
In this talk, we briefly review the Creutzig-Ridout theory and discuss recent progress on character formulas and classification of relaxed highest-weight modules. This talk is based on joint works with David Ridout.

Date

Apr 18 (Thu), 10:30--12:00, 2019

Room

RIMS 006

Speaker

Sota Asai (RIMS)

Title

Semibricks in $\tau$-tilting theory

Abstract

　 This talk is based on my paper "Semibricks" in IMRN. In representation theory of a finite-dimensional algebra $A$ over a field $K$, bricks and semibricks are fundamental and useful notions. Here, a brick means an $A$-module whose endomorphism ring is a division $K$-algebra, and a semibrick means a set of bricks which are pairwise Hom-orthogonal, so (semi)bricks are a generalization of (semi)simple modules. I study semibricks from the point of view of $\tau$-tilting theory. I proved that there is a one-to-one correspondence between the support $\tau$-tilting modules and the semibricks satisfying a certain condition called left finitess. Also, I introduced brick labels for the exchange quiver of the support $\tau$-tilting modules by using this bijection. I would like to explain these results and the new perspective of $\tau$-tilting theory given by them.

Date

Apr 11 (Thu), 14:30--16:00, 16:30--18:00, 2019

Room

RIMS 006

Speaker

14:30-16:00 Kazuya Kawasetsu (RIMS)

Title

Relaxed highest-weight modules over affine VOAs

Abstract

　 The (chiral) symmetry algebras of 2d conformal field theory are described as vertex operator algebras (VOAs). Among them, C_2-cofinite ones with semisimple module categories correspond to rational conformal field theory, whose characters span a SL_2(\Z)-invariant vector space and whose fusion rules in module categories satisfy the celebrated Verlinde formula.
The affine VOAs are those construted by (and almost the same as) affine Kac-Moody algebras. Among them, non-integrable affine VOAs are important examples of non-C_2-cofinite VOAs. The Verlinde formula for non-integrable (admissible) affine VOAs proposed by T. Creutzig and D. Ridout involves in relaxed highest-weight modules over affine Kac-Moody Lie algebras, which are modules parabolically induced from weight modules over the associated finite-dimensional simple Lie algebras.
In this talk, we briefly review the Creutzig-Ridout theory and discuss recent progress on character formulas and classification of relaxed highest-weight modules. This talk is based on joint works with David Ridout.

Speaker

16:30-18:00 佐竹郁夫 (香川大学)

Title

楕円ワイル群の不変式論へのアプローチ

Abstract

　 楕円ワイル群の不変式環の生成元として、
(1) アフィンリー環の指標、
(2) (一般化された）ヤコビ形式、
(3) フロベニウス構造に付随する平坦不変式、
がある。 (1) を用いて (2) を構成することができ、 (2) を用いて、フロベニウス構造を記述することで (3) を構成でき、 従って (1) と (3) の関係を記述できる (arXiv:1708.03875)。 その後、楕円ルート系のコクセター変換を用いた (3) の特徴づけを見出した。 この特徴づけの立場から (1) のアフィンリー環の指標、ワイル分母が どのような性質、特徴づけをもつのかを述べる。

Date

Apr 4 (Thu), 14:00--18:00, 2019

Room

RIMS 006

Speaker

14:00-15:00 Kari Vilonen

Title

Character sheaves for graded Lie algebras, geometric theory

Abstract

　 　 In recent joint work with Grinberg and Xue we studied a nearby cycle construction which forms a basis for Springer theory in the symmetric space setting. In consequent work with Xue we applied this theory to classify character sheaves in the symmetric space setting. In this talk we extend the geometric theory to the more general case of graded Lie algebras.

Speaker

15:30-16:30 Ting Xue

Title

Graded Lie algebras, character sheaves, and DAHA representations

Abstract

　 In their recent work, Lusztig and Yun construct representations of certain graded double affine Hecke algebras (DAHA) using geometry of graded Lie algebras. In joint work (in progress) with Vilonen we study the geometry of graded Lie algebras from another point of view. More precisely, we classify character sheaves in the setting of graded Lie algberas, where representations of Hecke algebras associated with complex reflection groups enter the story. We will explain some conjectures arising from the connection between the above two works, which relate finite dimensional irreducible representations of graded DAHA to irreducible representations of Hecke algebras. If time permits, we will also explain a Schur-Weyl duality conjecture arising from the geometric construction of rational Cherednik algebra modules of Oblomkov and Yun using affine Springer fibres.

Speaker

17:00-18:00 Bernard Leclerc

Title

A twofold generalization of Gabriel's theorem

Abstract

　 (joint work with C. Geiss and J. Schröer)
By a celebrated theorem of Gabriel, isomorphism classes of indecomposable representations of an A-D-E quiver are in one-to-one correspondence with positive roots of the corresponding root system.
Let C be a Cartan matrix (of type A, B, C, ..., F4, G2), and let D be a symmetrizer for C (i.e. a diagonal matrix with positive integer entries such that DC is symmetric). Fix an orientation of the Dynkin diagram corresponding to C, and an arbitrary field F. To this datum, in joint work with Geiss and Schröer we have introduced an F-algebra H and studied its representation theory. When C is of A-D-E type and the symmetrizer is equal to k times the identity matrix, H is isomorphic to the path algebra over the truncated polynomial ring F[t]/(t^k) of the quiver corresponding to C and the fixed orientation. I will present a twofold generalization of Gabriel's theorem in this situation. Namely there are two bijections:
(1) between isoclasses of indecomposable rigid locally free modules in rep(H) and positive roots of C;
(2) between isoclasses of bricks in rep(H) and positive roots of the tranpose of C.
I will also sketch how this generalizes to symmetrizable generalized Cartan matrices C.

Date

[changed] Jan 31 (Thu), 15:30--17:00, 2019

Room

RIMS 006

Speaker

Jethro van Ekeren (Universidade Federal Fluminense - Brazil)

Title

Minimal Models, Arc spaces, and Rogers-Ramanujan Identities

Abstract

　 (Joint work with Reimundo Heluani.) With each vertex algebra one may canonically associate two affine schemes: its 'singular support' and its 'associated scheme'. Let us call a vertex algebra 'classically free' if its singular support coincides with the arc space of its associated scheme. Motivated by questions from the theory of chiral algebras and the geometric Langlands program, we study classical freeness of Virasoro minimal models and simple affine vertex algebras. We show that the minimal models of type (2, 2k+1), the so called 'boundary minimal models', are classically free and all others are not. The coordinate rings of the two schemes in question are naturally graded and the isomorphism yields an equality of graded dimensions which recovers the celebrated Rogers-Ramanujan identity.

Date

Jan 31 (Thurs), 10:30--12:00, 2019

Room

RIMS 006

Speaker

Euiyong Park (University of Seoul)

Title

Localization for quiver Hecke algebras

Abstract

　 In this talk, I explain my recent work with Masaki Kashiwara, Myungho Kim and Se-jin Oh on a generalization of the localization procedure for monoidal categories developed in [S.-J. Kang, M. Kashiwara and M. Kim, Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras, Invent. Math. 211 (2018), no. 2, 591-685]. Let $R$ be a quiver Hecke algebra of arbitrary symmetrizable type and $R$-gmod the category of finite-dimensional graded $R$-modules. For an element w of the Weyl group, $C_w$ is the subcategory of $R$-gmod which categorifies the quantum unipotent coordinate algebra $A_q(n(w))$. We introduce the notions of braiders and a real commuting family of braiders, and produce a localization procedure which is applicable more general cases. We then construct the localization $\tilde{C}_w$ of $C_w$ by adding the inverses of simple modules which correspond to the frozen variables in the quantum cluster algebra $A_q(n(w))$. The localization $\tilde{C}_w$ is left rigid and it is conjectured that $\tilde{C}_w$ is rigid.

Date

Jan 25 (Fri), 16:30--18:00, 2019

Room

RIMS 402

Speaker

Anatol Kirillov

Title

Rigged Configurations and parabolic Kostka polynomials

Abstract

　 I'm planning to explain several applications of a Fermionic Formula for parabolic Kostka polynomials discovered by the author of this talk in 1982/83, to give proofs of a number of conjectures (and their generalization) in Combinatorics and Representation Theory (for type A ) concerning q-Kostka polynomials, such as the Gupta--Brylinski , Generalized Saturation, Combinatorial Unimodality, Polynomiality Conjectures, and new interpretation of Littlewood--Richardson and Kronecker numbers.

CANCELLATION

Date

Jan 24 (Thu), 10:30--12:00, 2019

Room

RIMS 006

Speaker

佐竹郁夫 (香川大学)

Title

楕円ワイル群の不変式論へのアプローチ

Abstract

　 楕円ワイル群の不変式環の生成元として、
(1) アフィンリー環の指標、
(2) (一般化された）ヤコビ形式、
(3) フロベニウス構造に付随する平坦不変式、
がある。 (1) を用いて (2) を構成することができ、 (2) を用いて、フロベニウス構造を記述することで (3) を構成でき、 従って (1) と (3) の関係を記述できる (arXiv:1708.03875)。 その後、楕円ルート系のコクセター変換を用いた (3) の特徴づけを見出した。 この特徴づけの立場から (1) のアフィンリー環の指標、ワイル分母が どのような性質、特徴づけをもつのかを述べる。

Date

Jan 24 (Thu), 14:00--15:30, 2019

Room

RIMS 006

Speaker

Se-jin Oh (Ewha Woman's University)

Title

Two families if commutation classes and categories attached to them

Abstract

　 In this talk, I will introduce two families of commutation classes of the same simply-laced finite type and their combinatorial features. The one is called "adapted classes" and the another is called "twisted adapted classes". Surprisingly, they encode information of categries of finite dimensional modules over Langlads dual quantum affine algebras in an interesting way. Using the generalized Schur-Weyl daulity constructed by Kang-Kashiwara-Kim, we can construct simplicity-preserving correspondences between "heart" subcategories of finite dimensional modules over Langlads dual quantum affine algebras in various ways. The correspondece between Langlads dual quantum affine algebras was obervated by Frenkel-Hernandez and is not well-understood yet. This is joint work with Kashiwara, Kim, Suh and Scrimshaw.

Date

Dec 21 (Fri), 17:00--18:30, 2018

Room

RIMS 402

Speaker

Hironori Oya (Shibaura Institute of Technology)

Title

Quantum Grothendieck ring isomorphisms for quantum affine algebras of type A and B

Abstract

　 In this talk, I present a ring isomorphism between ``$t$- deformed'' Grothendieck rings (=quantum Grothendieck rings) of finite- dimensional module categories of quantum affine algebras of type $\ mathrm{A}_{2n-1}^{(1)}$ and $\mathrm{B}_n^{(1)}$. This isomorphism implies several new positivity properties of $(q, t)$-characters of simple modules of type $\mathrm{B}_n^{(1)}$. Moreover, it specializes at $t = 1$ to the isomorphism between usual Grothendieck rings obtained by Kashiwara, Kim and Oh via generalized quantum affine Schur-Weyl dualities. This coincidence gives an affirmative answer to Hernandez's conjecture (2002) for type $\mathrm{B}_n^{(1)}$ : the $(q, t)$- characters of simple modules specialize to their actual $q$-characters. Hence, in this case, the multiplicities of simple modules in standard modules are given by the evaluation of certain analogues of Kazhdan- Lusztig polynomials whose coefficients are positive. If time permits, we discuss a refinement of description of our isomorphism.
This talk is based on a joint work with David Hernandez.

Date

Nov 29 (Thu), 16:30--18:00, 2018

Room

Room 110 Department of Mathematics, Kyoto Universtity

Speaker

Hitoshi Konno (Tokyo University of Marine Science and Technology)

Title

Elliptic Quantum Group and Elliptic Stable Envelopes

Abstract

　 楕円量子群U_{q,p}(g)の表現を用いて, Okounkovらが提唱する同変楕円コホモロジー 上のellipt ic stable envelopesを導出する方法を, g=￥hat{sl}_Nの場合(対応するquiver vari etyは一般 旗多様体の余接束)について紹介する. この応用として, 楕円q-KZ方程式の楕円超幾 何積分解の 導出や, Okounkovのvertex functionとの関係, ￥hat{sl}_N型楕円面模型のnested B ethe ansat z との関係, Gelfand-Tsetlin基底の明示式の導出, 同変楕円コホモロジーの固定点 類上に構成 される楕円量子群の有限次元表現について議論する.

References; H.Konno, ``Elliptic Weight Functions and Elliptic q-KZ Equation'', J.Int.Systems 2 (2017), H.Konno, ``Elliptic Stable Envelopes and Finite-dimensional Representations of Elliptic Quantum Group'', J.Int.Systems 3 (2018).

Date

Nov 16 (Fri), 16:30--18:00, 2018

Room

RIMS 402

Speaker

柴田大樹 (岡山理科大）

Title

Quasireductive supergroups and their representations

Abstract

　 スーパー代数群は代数群のスーパー類似物として定義される．これは単なる一般化ではなく，代数群の（モジュラー）表現論への応用があるなど興味深い対象である．これまでスーパー代数群の研究は具体的な対象を扱ったものばかりであったが，V. Serganova (2011) はスーパー・シュヴァレー群や queer スーパー群 Q(n) を含む準簡約スーパー群 (quasireductive supergroups) という概念を導入し体系的な研究を行った．特に標数ゼロの代数閉体上定義された準簡約スーパー群 G について，そのスーパー・リー代数 Lie(G) の有限次元既約表現を構成している． 本講演ではホップ代数的アプローチによるスーパー代数群の構造論および表現論の研究手法を紹介する．表現論に関して，体上定義された準簡約スーパー群 G の有理表現圏と，あるスーパー余可換ホップ代数 hy(G) の``可積分''表現圏とが圏同値であることを説明し，実際に G の既約表現を構成する．基礎体が標数ゼロの代数閉体であれば，これは Serganova の構成した既約表現と一致している．また G が良いスーパー放物部分群を持つときに，Kempf コホモロジー消滅定理のスーパー類似などが成り立つことも述べたい．

Room

Kyoto University North Campus
Maskawa Building for Education and Research Room 507
(North direction from the RIMS main building)

Date

Nov 8 (Thu), 13:00--14:15, 2018

Speaker

Myungho Kim (Kyung Hee University)

Title

Laurent phenomenon and simple modules over symmetric quiver Hecke algebras

Abstract

　 The unipotent quantum coordinate ring $A_q(n(w))$ is isomorphic to the Grothendieck ring of a monoidal category $C_w$ consisting of some finite dimensional graded modules over a quiver Hecke algebra. Moreover $A_q(n(w))$ has a quantum cluster algebra structure, and it is shown that the cluster monomials are classes of real simple modules in $C_w$. In this talk, I will present some interesting consequences of this "monoidal categorification" of $A_q(n(w))$ with the Laurent phenomenon of the cluster algebras. This is a joint work with Masaki Kashiwara.

Date

Nov 8 (Thu), 14:30--16:15, 2018

Speaker

Shunsuke Tsuchioka (Tokyo Institute of Technology)

Title

A local characterization of $B_2$ regular crystals

Abstract

　 Stembridge characterizes regular crystals associated with a simply-laced GCM in terms of local graph-theoretic quantities. We give a similar axiomatization for $B_2$ regular crystals and thus for regular crystals of finite GCM except $G_2$ and affine GCM except $A^{(1)}_1, G^{(1)}_2, A^{(2)}_2, D^{(3)}_4$. As we will explain in detail the previous studies, finding a set of local axioms that characterizes $B_2$ regular crystals has been an open problem.

Date

Nov 8 (Thu), 16:30--18:00, 2018

Speaker

Toshiyuki Tanisaki (Osaka City University)

Title

ある誘導表現の計算

Abstract

　 $G$を複素数体上の単純代数群， $T$, $B$を$G$の極大トーラス，およびBorel部分群とする． また，$G$, $B$, $T$の座標環を$A(G)$, $A(B)$, $A(T)$で表す． $A(B)$を随伴作用に関して$B$加群と思い， $G$への誘導表現$Ind(A(B))$を考える． このとき，幾何的議論により $Ind(A(B))=A(G)\otimes_{A(T)^W}A(T)$, $R^{>0}Ind(A(B))=0$ がわかる． 講演者は，1のベキ根での量子群においても 同様の事が成り立つことを予想した． 講演では，$A$型の場合のこの予想の証明を述べる予定である． なお，この予想が正しければ， 1のベキ根での量子旗多様体上のD加群の理論がほぼ完成し， ベキ根での量子群の表現に関するLusztigの予想が 正しいことがわかる．

Date

Oct 26 (Fri), 16:30--18:00, 2018

Room

RIMS 402

Speaker

Dinakar Muthiah (IPMU)

Title

Toward double affine flag varieties and Grassmannians

Abstract

　 Recently there has been a growing interest in double affine Grassmannians, especially because of their relationship with Coulomb branches of quiver gauge theories. However, not much has been said about double affine flag varieties. I will discuss some results toward understanding double affine flag varieties and Grassmannians (and their Schubert subvarieties) from the point of view of $p$-adic Kac-Moody groups. I will discuss Hecke algebras, Bruhat order, and Kazhdan-Lusztig polynomials in this setting. This includes work joint with Daniel Orr and joint with Manish Patnaik.

Date

Oct 19 (Fri), 16:30--18:00, 2018

Room

RIMS 402

Speaker

Kazuya Kawasetsu (Melbourne)

Title

Modular linear differential equations of fourth order and minimal W-algebras

Abstract

　 Modular linear differential equations are differential equations invariant under modular transformations. They play important roles in the study of 2D conformal field theory, vertex operator algebras and modular forms. For example, characters of lisse (C_2-cofinite) vertex algebras and more generally, those of quasi-lisse vertex algebras, satisfy modular linear differential equations. Moreover, they have also used in the attempt to classify lisse vertex algebras from their characters. In this talk, we study a certain family of modular linear differential equations of fourth order and discuss which vertex operator algebras satisfy the differential equations.

Date

Oct 15 (Mon), 16:30--18:00, 2018

Room

京都大学北部総合教育研究棟 507号室

Speaker

池田 曉志氏 (IPMU)

Title

2重次数付きGinzburg Calabi-Yau dg代数とルート系のq-変形, そしてその導来圏上のq-安定性条件の空間について

Abstract

　 この講演では, まずはquiverに対するGinzburg Calabi-Yau dg代数 の構成を 2重次数付きで行うことで, その代数上の2重次数付きdg加群の導来圏が, quiverに付 随した ルート系のq-変形の圏化を与えることについて説明する. 特に, Seidel-Thomasの球 面捻り 関手がHecke環を通してこのルート系のq-変形に作用していることについて見る. ま た, 特に quiverがアファインADE型の時には, この構成がクライン特異点のC^*-同変層の導来 圏を記述 していることについて説明し, 次数付きpreprojective代数との関係についても述べ る. 次に これらのdg代数の導来圏の上の自然なBridgeland安定性条件だと思われるq-安定性条 件を導入し, 特にquiverがADE型の場合には, q-安定性条件の中心電荷がCherednikのKZ接続(Hecke 環 がモノドロミーとして現れる)の水平切断を与えるという予想(A型の時は定理)につい て述べる.

Date

Oct 5 (Fri), 16:30--18:00, 2018

Room

RIMS 402

Speaker

Yuri Billig (Carleton)

Title

Representations of Lie algebras of vector fields on affine algebraic varieties.

Abstract

　 Very little has been known about representation theory of Lie algebras of polynomial vector fields on affine algebraic varieties beyond the cases of affine space and a torus. We study a category of representations of the Lie algebras of vector fields on a smooth algebraic variety X that admit a compatible action of the algebra of polynomial functions on X. We construct simple modules in this category and state a conjecture on the general structure of such modules. This is a joint work with Slava Futorny and Jonathan Nilsson.

Date

Oct 3 (Wed), 10:30--12:00, 2018

Room

Room 108 3rd building, Department of Mathematics

Speaker

Vasily Kryov

Title

Drinfeld-Gaitsgory interpolation Grassmannian and geometric Satake equivalence

Abstract

　 This talk is a review of the paper arxiv.org/abs/1805.07721. Let G be a reductive complex algebraic group. Recall that a geometric Satake isomorphism is an equivalence between the category of G(O)-equivariant perverse sheaves on the affine Grassmannian for G and the category of finite dimensional repre- sentations of the Langlands dual group Gˇ. They are equivalent as Tannakian categories, the fiber functor sends a perverse sheaf to its global cohomology. It follows from the above that for any perverse sheaf P there exists an action of the Lie algebra of Gˇ on the global cohomology of P. We will explain how to construct this action explicitly. To do so, we will describe a geometric construction of the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra U(nˇ). Using this construction, we will provide the desired action. It will be obtained via a cospecialization morphism for a certain one-parametric deformation of the affine Grassmannian of G.
If time permits, we will discuss some possible generalizations of our constr uction of the action, in particular, we will discuss the relation of the def ormation mentioned above with the Drinfeld-Gaitsgory deformation considered in their paper on Braden’s theorem.

Date

Aug 17 (Fri), 16:30--18:00, 2018

Room

RIMS 402

Speaker

Oleksandr Tsymbaliuk (Yale)

Title

Coulomb branches, shifted quantum algebras and modified q-Toda systems

Abstract

　 In the recent series of papers by Braverman-Finkelberg-Nakajima a mathematical construction of the Coulomb branches of 3d N=4 quiver gauge theories was proposed (the latter are supposed to be symplectic dual to the corresponding well-understood Higgs branches). They can be also realized as slices in the affine Grassmannian and therefore admit a multiplication.
In the current talk, we shall discuss the quantizations of these Coulomb branches and their K-theoretic analogues, and the (conjectural) down-to-earth realization of these quantizations via shifted Yangians and shifted quantum affine algebras. Those admit a coproduct quantizing the aforementioned multiplication of slices. In type A, they also act on equivariant cohomology/K-theory of parabolic Laumon spaces.
As another interesting application, the shifted quantum affine algebras in the simplest case of sl(2) give rise to a new family of 3^{n-2} q-Toda systems of sl(n), generalizing the well-known one due to Etingof and Sevostyanov. If time permits, we shall explain how to obtain 3^{rk(g)-1} modified q-Toda systems for any simple Lie algebra g.
This talk is based on the joint works with M. Finkelberg and R. Gonin.

Date

July 16 (Mon), 10:30--12:00, 2018

Room

RIMS 402

Speaker

Arturo Pianzola

Title

A prescient SGA3: Applications of reductive group schemes to infinite dimensional Lie theory

Abstract

　 Many interesting infinite dimensional Lie (super conformal) algebras can be thought as being "finite dimensional" when viewed, not as algebras over the given base field, but rather as algebras over their centroids (usually a Laurent polynomial ring). From this point of view, the algebras in question look like "twisted forms" of simpler objects which with one is familiar. The quintessential example of this type of behaviour is given by the affine Kac-Moody Lie algebras. Once the twisted form point of view is embraced,the theory of torsors and reductive group schemes developed by Demazure and Grothendieck [SGA3] arises naturally. The talk will explain these concepts and connections.

Date

June 18 (Mon), 16:00--18:00, 2018

Room

Room 108, Department of Math, Kyoto Univ.

Speaker

Leonid Rybnikov (Higher School of Economics / Tokyo)

Title

Gaudin model and crystals (joint work with Iva Halacheva, Joel Kamnitzer and Alex Weekes)

Abstract

　 Drinfeld-Kohno theorem relates the monodromy of KZ equation to the braid group action on a tensor product of $U_q(\mathfrak{g})$-modules by R-matrices. The KZ equation depends on the parameter $\kappa$ such that $q=\exp(\frac{\pi i}{\kappa})$. We describe the limit Drinfeld-Kohno correspondence when $\kappa\to 0$ along the imaginary line. On the KZ side this limit is the Gaudin integrable magnet chain while on the quantum group side the limit is a $\mathfrak{g}$-crystal. Namely, we construct a bijection between the set of solutions of the algebraic Bethe ansatz for the Gaudin model and the corresponding $\mathfrak{g}$-crystal, which preserves the natural cactus group action on these sets. This can be regarded as the $\kappa\to 0$ limit of the Drinfeld-Kohno theorem. If time allows I will also dicuss some conjectural generalizations of this result relating it to works of Losev and Bonnafe on cacti and Kazhdan-Lusztig cells.

Date

June 8 (Fri), 16:30--18:00, 2018

Room

Room 402, RIMS

Speaker

Ivan Cherednik (Chapel Hill / Kyoto)

Title

Jacobian factors in any ranks and DAHA superpolynomials

Abstract

　 The theory of the moduli spaces of torsion free sheaves in any ranks over singular curves is quite a challenge, including nodal curves and rk=2 (Gieseker, Bertram, others). Its local counterpart is the theory of affine Springer fibers for non-reduced (germs of) singular curves, which is unsettled too.
For type A and in the nil-elliptic case, these fibers can be identified with the Jacobian factors, which are simple to define projective(!) varieties, though this approach was not extended to higher ranks as well. For plane curve singularities (spectral curves are of this kind in type A), there is a strong support: the corresponding geometric superpolynomials are expected to coincide with the DAHA superpolynomials colored by columns, and through them to be connected with any other theories of superpolynomials, including the original (uncolored) ones due to Khovanov-Rozansky.
I will define in this talk Jacobian factors in any ranks and state their connection with the DAHA superpolynomials. This is joint with Ian Philipp. The connection conjecture was checked in many cases and a general proof seems doable (at least in the motivic setting, to be explained).

Date

6月1日 (Fri), 13:00--14:30, 2018

Room

Room 402, RIMS

Speaker

Yusuke Ohkubo氏 (Tokyo)

Title

DIM代数の特異ベクトルとAGT対応から現れる一般化Macdonald多項式

Abstract

　 W-infinity代数のq-変形としてDing-Iohara-Miki代数と呼ばれるHopf代数がある。こ の代数の自 由場表現から得られるある代数がAGT対応のq-変形版で本質的な役割を果たす。本公 演ではこの 代数の表現論とAGT対応、またその際に用いられる一般化されたMacdonald多項式（AF LT基底、固 定点基底のq-変形版）の性質について説明する。特に、その代数の特異ベクトルが特 殊なヤング 図の組を持った一般化Macdonald多項式と一致することについて述べる。この一致は 従来から知 られている通常のMacdonald多項式と変形W代数の特異ベクトルとの一致のある種の一 般化とみな すことができる。

Date

6月1日 (Fri), 15:00--16:00, 2018

Room

Room 402, RIMS

Speaker

Tianshu Liu氏 (Melbourne)

Title

Affine osp(1|2) and its coset construction

Abstract

　 Conformal field theory is an essential tool of modern mathematical physics with applications to string theory and to the critical behaviour of statistical lattice models. The symmetries of a conformal field theory include all angle-preserving transformations. In two dimensions, these transformations generate the Virasoro algebra, a powerful symmetry that allows one to calculate observable quantities analytically. The construction of one family of conformal field theories from the affine Kac-Moody algebra sl(2) were proposed by Kent in 1986 as a means of generalising the coset construction to non-unitary Virasoro minimal models, these are known as the Wess-Zumino-Witten models at admissible levels. This talk aims to illustrate, with the example of the affine Kac-Moody superalgebra osp(1|2) at admissible levels, how the representation theory of a vertex operator superalgebra can be studied through a coset construction. The method allows us to determine key aspects of the theory, including its module characters, modular transformations and fusion rules.

Date

6月1日 (Fri), 16:30--18:00, 2018

Room

Room 402, RIMS

Speaker

David Ridout氏 (Melbourne)

Title

Relaxed modules over affine vertex operator algebras

Abstract

　 Some of the most important non-rational VOAs are the admissible level affine ones. Their simple modules in category O have been classified, but this omits many physically necessary simple modules. Motivated by modularity (and physical) consistency, we report on work towards a classification of simple modules in category R, the category of "relaxed" highest-weight modules.
Joint work with Kazuya Kawasetsu.

Date

5月18日 (Fri), 16:30--18:00, 2018

Room

Room 402, RIMS

Speaker

Ivan Ip (Kyoto)

Title

Positive Peter-Weyl Theorem

Abstract

　 I will explain the Peter-Weyl Theorem for split real quantum groups of type An, generalizing the previous result in the case of Uq(sl(2,R)). I will talk about the necessary ingredients needed to state and proof the theorem, including the GNS representation of C*-algebra, Heisenberg double construction, and cluster realization of positive representations. This is a joint work with Gus Schrader and Alexander Shapiro.

Date

5月17日 (Thu), 10:30--12:00, 2018

Room

Room 006, RIMS

Speaker

Thomas Creutzig氏 (Alberta/RIMS)

Title

S-duality in the example of the large N=4 superconformal algebra

Abstract

　 There is a very rich interplay between certain supersymmetric four-dimensional gauge theories, the quantum geometric Langlands program and vertex algebras. The key ingredient for all three are master chiral algebras that serve as functors between representation categories. I will present a few theorems concerning a family of vertex superalgebras called the large N=4 superconformal algebras at central charge -6 and explain how they confirm some conjectures originating from physics and geometry.

Date

4月19日 (Thu), 10:30--12:00, 2018

Room

Room 006, RIMS

Speaker

Ievgen Makedonskyi氏 (Kyoto U)

Title

Vertex algebras and coordinate rings of semi-infinite flags

Abstract

　 The direct sum of irreducible level one integrable representations of affne Kac-Moody Lie algebra of (affne) type ADE carries a structure of P/Q-graded vertex operator algebra. There exists a filtration on these modules due to Kato and Loktev such that the corresponding graded vector space is a direct sum of global Weyl modules. The associated graded space with respect to the dual filtration is isomorphic to the homogenous coordinate ring of semi-infinite flag variety. We describe the ring structure in terms of vertex operators and endow the homogenous coordinate ring with a structure of P/Q-graded vertex operator algebra. We use the vertex algebra approach to derive semiinfinite Pluecker-type relations in the homogeneous coordinate ring.

Date

4月6日 (Fri), 16:30--18:00, 2018

Room

Room 402, RIMS

Speaker

Kari Vilonen氏 (Melbourne)

Title

Koszul duality for real groups

Abstract

　 I will begin by explaining the Langlands/Vogan duality for real groups. After that I will discuss its categorical version. This is joint work with Roman Bezrukavnikov.

Date

3月9日 (Fri), 16:30--18:00, 2018

Room

Room 402, RIMS

Speaker

Shigenori Nakatsuka氏 (U. Tokyo)

Title

Geometric construction of integrable Hamiltonian hierarchies associated with the classical affine W-algebras.

Abstract

　 The Drinfeld-Sokolov hierarchies are integrable Hamiltonian hierarchies associated with the principal classical affine W-algebras. Feigin-Frenkel realized these hierarchies geometically using a geometric interpretation of the free field realization of principal classsical affine W-algebras. In this talk, we consider a generalization of this result to some classical affine W-algebras which are not principal. We also obtain a characterization of the corresponding Hamiltonians as the set of conservative quantities of some differential equations which are the affine Toda equations in the principal cases.

Date

2月23日 (Fri), 16:30--18:00, 2018

Room

Room 402, RIMS

Speaker

Takahiro Nagaoka氏 (Kyoto U)

Title

The universal Poisson deformation space of hypertoric varieties.

Abstract

　 Hypertoric variety $Y(A, \alpha)$ is a (holomorphic) symplectic variety, which is defined as Hamiltonian reduction of complex vector space by torus action. By definition, there exists projective morphism $\pi:Y(A, \alpha) \to Y(A, 0)$, and for generic $\alpha$, this gives a symplectic resolution of affine hypertoric variety $Y(A, 0)$. In general, for conical symplectic variety and it's symplectic resolution, Namikawa showed the existence of universal Poisson deformation space of them. We construct universal Poisson deformation space of hypertoric varieties $Y(A, \alpha)$, $Y(A, 0)$. We will explain this construction and concrete description of Namikawa-Weyl group action in this case. If time permits, We will also talk about some classification results of affine hypertoric variety. This talk is based on my master thesis.

Date

2月16日 (Fri), 14:00--15:30, 2018

Room

Room 402, RIMS

Speaker

Ian Le氏 (Perimeter Institute)

Title

An introduction to higher Teichmuller theory

Abstract

　 Let S be a topological surface. Teichmuller space parameterizes the different ways of giving S the structure of a Riemann surface. Uniformization tells us that any Riemann surface can realized as a quotient of the upper-half-plane by a subgroup of PSL(2,R). Thus Teichmuller space to be viewed as a space of representations of the fundamental group of S into PSL(2,R). We will explain how cluster algebras and the theory of total positivity give an approach to Teichmuller theory which recovers classical ideas (like hyperbolic geometry, measured laminations, and quadratic differentials) while also permitting a generalization to PSL(n,R).

Date

2月16日 (Fri), 15:45--16:45, 2018

Room

Room 402, RIMS

Speaker

Gus Schrader氏 (Columbia University)

Title

Cluster algebra realization of quantum groups and their positive representations

Abstract

　 I will speak about recent joint work with Alexander Shapiro in which we develop a cluster realization of the quantum group U_q(sl(n)) using quantized moduli spaces of framed local systems on marked surfaces. I will also discuss the notion of a positive representation of a quantum cluster algebra, and explain how the positive representations of U_q(sl(n)) introduced by Frenkel and Ip can be studied in the cluster algebra framework.

Date

2月16日 (Fri), 17:00--18:00, 2018

Room

Room 402, RIMS

Speaker

Alexander Shapiro氏 (UC Berkeley)

Title

Positive representations of quantum groups & modular functor

Abstract

　 Igor Frenkel and Ivan Ip conjectured that positive representations of quantum groups are closed under tensor products. This conjecture happens to be closely related to the so-called modular functor conjecture by Fock and Goncharov. I will speak about joint works with Gus Schrader (some of which are works in progress) where we prove the above conjectures.

Date

2月2日 (Fri), 16:30--18:00, 2018

Room

Room 402, RIMS

Speaker

Myungho Kim氏 (Kyung Hee University)

Title

Monoidal categorification of cluster algebras

Abstract

　 In this talk, I will explain our work on the monoidal categorification of the quantum coordinate ring $A_q(n(w))$ of the unipotent subgroup associated with a symmetric Kac-Moody algebra $g$ and an element $w$ of the Weyl group. This is a joint work with Seok-Jin Kang, Masaki Kashiwara, and Se-jin Oh.
The notion of monodical categorification of cluster algebras was introduced by Hernandez and Leclerc: an abelian monodical category $C$ is called a monodical categorification of a cluster algebra $A$ if the Grothendieck ring of $C$ is isomorphic to $A$ and the cluster monomials of $A$ belong to the classes of real simple objects of $C$.
The existence of a monodical categorification of a cluster algebra $A$ implies several nice properites of $A$ in a natural way, for example, the positivity of the coefficients of the expansion of cluster monomials with respect to an arbitrary cluster.
Our main result is that a subcategory $C_w$ of category of finite-dimensional graded modules over the symmetric quiver Hecke algebra is a monodical categorification of the (quantum) cluster algebra $A_q(n(w))$. Combining the results of Khovanov-Lauda, Rouquier and Varagnolo-Vasserot, we conclude that the cluster monomials of $A_q(n(w))$ belongs to the upper global basis (dual canonical basis). It answers the conjecture by Kimura and Geiss-Leclerc-Schröer, which can be also regarded as a sharpened version of a question asked by Fomin-Zelevinsky.

Date

12月22日 (Fri), 10:30--12:00, 2017

Room

Room 402, RIMS

Speaker

疋田辰之氏 (RIMS)

Title

Periodic modules for hypertoric varieties

Abstract

　 Lusztig defined certain representations of affine Hecke alg ebras called periodic modules using some periodic hyperplane arrangements an d constructed canonical bases for them. Also he gave a geometric interpretat ion using equivariant K-theory of Slodowy varieties. I will explain an analo gue of such combinatorics for hypertoric varieties and give some geometric a nd representation theoretic applications.

Date

12月22日 (Fri), 13:00--14:30, 2017

Room

Room 402, RIMS

Speaker

桑原敏郎氏 (筑波大)

Title

Mackey’s formula for cyclotomic Hecke algebras and rational Cher ednik algebras of type G(r,1,n)

Abstract

　 The restriction/induction functors play an important role f or the representation theory of cyclotomic Hecke algebras and rational Cherednik algebras of type G(r,1,n). In this talk, we discuss an analog of Mackey’s formula for two parabolic su balgebras of the cyclotomic Hecke algebras and the rational Cherednik algebras.

Date

12月22日 (Fri), 14:45--16:15, 2017

Room

Room 402, RIMS

Speaker

Konstanze Rietsch氏 (King’s College London)

Title

Mirror symmetry for some homogeneous spaces

Abstract

　 I will give an overview of results on mirror symmetry for G /P, including Grassmannians and Dubrovin/Givental style mirror symmetry in t he presence of a torus action.

Date

12月22日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

Shahn Majid氏 (Queen Mary University of London)

Title

Double-Bosonization and dual bases of quantum groups ℂq[SL( 2)] and ℂq[SL(3)]

Abstract

　 The talk is based on my recent work with Ryan Aziz. We find a dual version of a previous double-bosonisation theorem whereby each finit e-dimensional braided-Hopf algebra B in the category of comodules of a coqua sitriangular Hopf algebra A has an associated coquasitriangular Hopf algebra coDA(B). As an application we find new generators for ℂq[SL(2)] reduc ed at q a primitive odd root of unity with the remarkable property that thei r monomials are essentially a dual basis to the standard PBW basis of the re duced Drinfeld-Jimbo quantum enveloping algebra uq(𝔰𝔩(2)). O ur methods apply in principle for general ℂq[G] as we demonstrate for ℂq[SL(3)] at certain odd roots of unity.

Date

12月1日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

Andrei Okounkov氏 (Columbia University/Kyoto University)

Title

Quasimaps counts and Bethe eigenfunctions

Abstract

　 I will explain several aspects of what is done in a paper with the same title, joint with Mina Aganagic.

Date

11月17日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

柳田　伸太郎氏 (名古屋大学)

Title

Elliptic Hall algebra over $\mathbb{F}_1$

Abstract

　 This talk is motivated by the recent work of Morton and Samuelson which states that the Turaev skein algebra for torus is isomorphic to a specialization of the elliptic Hall algebra. In this talk we introduce the category $B_q$ which is an $\mathbb{F}_1$-analogue of the category of coherent sheaves over an elliptic curve. Although our category is not an abelian category, even nor an additive category, it is an example of so-called belian and quasi-exact category in the sense of Deitmar. Then we can consider the Hall algebra $U_q$ associated to $B_{q}$ using Szczesny's construction of Hall algebra for monoid representations. The main statement is that $U_{q}$ is isomorphic to the Turaev skein algebra of torus. Thus our construction gives the `B-side counter-part' of the torus skein algebra directly, not replying on an `bi-hand' specialization process on Hall algebra.　

Date

10月27日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

Kari Vilonen氏 (Melbourne)

Title

Springer theory for symmetric spaces

Abstract

　 A year ago I spoke on Springer theory for symmetric spaces in the special case of SL(n,R). In this talk I will discuss the general case. We start with a detailed discussion of a nearby cycle construction which plays a crucial role in the theory. This part is joint work with Grinberg and Xue. After that I will explain how using the geometric input one obtains the Springer theory in general. This part is joint work with Xue.

Date

10月20日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

渡邉 英也氏 (東工大)

Title

Representation theory of quantum symmetric pairs and Kazhdan-Lusztig bases

Abstract

　 In 2013, Huanchen Bao and Weiqiang Wang discovered the Schur -Weyl-type duality between some quantum symmetric pair coideal subalgebras $U^{\jmath}$ and the Hecke algebra $H$ (with unequal parameter) of type B. Namely, they equipped the $d$-th tensor power of the vector representation $V$ of $U_q(\mathfrak{sl}_n)$ with a $(U^{\jmath},H)$-bimodule structure which satisfies the double centralizer property. In this talk, we investigate the bimodule structure of $V^{\otimes d}$ and see that its $\jmath$-canonical basis (introduced by Bao and Wang) coincides with its (parabolic) Kazhdan-Lusztig basis. Time permitting, we will see how this result relates to the Lusztig's periodic $W$-graphs. This talk is partially based on a joint work with Bao and Wang.

Date

10月6日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

藤田 遼氏 (京都大学)

Title

Affine highest weight categories for quantum loop algebras of Dynk in types

Abstract

　 For a Dynkin quiver $Q$ (i.e. Dynkin graph of a simple Lie a lgebra $\mathfrak{g}$ of type ADE with an orientation), Hernandez-Leclerc defined a monoidal subcategory $\mathcal{C}_{Q}$ of the category of finite-dimensional modules over the quantum loop algebra associated with $\mathfrak{g}$. They proved that its Grothendiek ring is isomorphic to the coordinate algebra of the maximal unipotent subgroup associated with $\mathfrak{g}$ and that the classes of simple modules correspond to the dual canonical basis elements. In this talk, we see that a "central completion" of the category $\mathcal{C}_{Q}$ has a structure of affine highest weight category. We rely on Nakajima's geometric method using the equivariant K-theory of graded quiver varieties. As an application, we conclude that Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor gives a monoidal equivalence between the Hernandez-Leclerc category $\mathcal{C}_{Q]$ and the category of finite-dimensional modules over the quiver Hecke (KLR) algebra associated with $Q$, assuming the simpleness of poles of normalized R-matrices for type E.

Date

8月4日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

Prof. Evgeny Mukhin (IUPUI)

Title

The Universal Differential Operator

Abstract

　 In this expository talk I will review various facts around Gaudin model associated to gl(n). I will focus on the universal differential operator which plays the central role in many important constructions and which was neglected for many years.

Date

7月28日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

大矢浩徳氏 (東京大学・数理)

Title

The Chamber Ansatz formulae for quantum unipotent cells

Abstract

　 Berenstein, Fomin and Zelevinsky introduced biregular automorphisms, called twist automorphisms, on unipotent cells in their study of total positivity criteria. These automorphisms are essentially used for describing the inverses of specific embeddings of tori into unipotent cells. The resulting descriptions are called the Chamber Ansatz.
In this talk, we consider a quantum analogue of their setting. First, we construct the twist automorphisms on arbitrary quantum unipotent cells and study their compatibility with the dual canonical bases. Next, we provide quantum analogues of the Chamber Ansatz formulae. We also discuss the relation between our results and the quantum cluster algebra structures on quantum unipotent cells introduced by Geiss-Leclerc-Schröer and Goodearl-Yakimov.
A part of this talk is based on joint work with Yoshiyuki Kimura.

Date

7月21日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

元良直輝氏 (RIMS)

Title

Coproducts for W-algebras in A-type

Abstract

　 The (affine) W-algebras are vertex algebras defined by generalized Drinfeld-Sokolov reductions associated with Lie algebras and nilpotent orbits. Using Zhu's functor, they can be associated with finite W-algebras, in particular, with trancations of shifted Yangians in A-type due to Brundan-Kleshchev. We will introduce the "coproduct" structure on W-algebras in A-type, which is analogue of finite case. In our construction, we use the Wakimoto representations.

Date

6月16日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

時本一樹氏 (RIMS)

Title

Affinoids in the Lubin-Tate perfectoid space and special cases of the local Langlands correspondence

Abstract

　 Let F be a non-archimedean local field. The non-abelian Lubin-Tate theory asserts that the local Langlands correspondence for GL_n(F) and the local Jacquet-Langlands correspondence are realized in the cohomology of the Lubin-Tate tower. Motivated by this theory, Boyarchenko-Weinstein and Imai-Tsushima constructed affinoid subspaces of the Lubin-Tate perfectoid space (a certain limit space of the tower) and proved that the cohomology of the reduction of each affinoid realizes the two correspondences for certain representations.
In this talk, I will discuss a similar result for some other representations.

Date

6月9日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

Sergey Loktev氏 (Higher School of Economics)

Title

Weyl modules: one variable vs. multivariable

Abstract

　 Weyl modules over current algebras are defined as universal highest weight modules. They are well-defined for current algebras in any number of variables.
We discuss which properties of one-variable Weyl modules can be generalized for the multivariable case.

Date

5月26日 (Fri), 15:30--unfixed, 2017

Room

Room 402, RIMS

Speaker

Ivan Cherednik氏 (Chapell Hill/RIMS)

Title

DAHA approach to plane curve singularities

Abstract

　 I will present a recent conjecture that connects the geometry of compactified Jacobians of unibranch plane singularities with the DAHA-superpolynomials of algebraic knots. This is directly related to p-adic orbital integrals (in Fundamental Lemma) and theory of affine Springer fibers (the anisotropic case, type A). This is based on certain partition of the flagged Jacobian factors (new objects, to be defined from scratch); there are connections with the Kazhdan-Lusztig dimension formulas (their 1988 paper), and the works by Bezrukavnikov, Lusztig-Smelt and Piontkowski.
The DAHA-superpolynomials are expected to coincide with the stable Khovanov-Rozansky polynomials of algebraic knots. They depend on the paarameters a,q,t; for instance, a=-1, q=t result in Alexander polynomials, which can be directly expressed via the corresponding singularities (without any Jacobian factors). When a=0, q=1, t=1/p the DAHA-superpolynomials conjecturally coincide with the p-adic orbital integrals. Our conjecture readily implies that the orbital integrals in type A depend only on the topological (not just analytic!) type of singularity.
This is joint with Ivan Danilenko and Ian Philipp. I may skip some details concerning the DAHA construction (the first hour), but the geometric superpolynomials will be defined in full and from scratch. This is an entirely local theory; the definitions are not too involved.

Date

5月19日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

Liron Speyer氏 (大阪大学・情報)

Title

Specht modules for the KLR algebras of type C

Abstract

　 The KLR algebras were introduced almost a decade ago to cate gorify the negative half of a quantum group. In type A, Brundan and Kleshchev showed that cyclot omic quotients of KLR algebras are isomorphic to cyclotomic Hecke algebras, which has spurred on the development of their graded representation theory, in particular with a theory of Specht modules. We will report on recent joint work with Susumu Ariki and E uiyong Park, in which we have defined a family of Specht modules for the KLR algebras in type C. We will outline some of their basic properties and explain why they are int eresting objects to study. We will finally discuss how we used these Specht modules to classify which cyclotomic quotients of the KLR algebras of type C are semisimple.

Date

5月12日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

平野 雄貴氏 (京都大学)

Title

Faithful actions from hyperplane arrangements

Abstract

　 Donovan--Wemyss associates to a flopping contraction $f:X\to Y$ of 3-folds a group action from the fundamental group of the complement of a complex hyperplane arrangement on the derived category of coherent sheaves on $X$. We show that this action is faithful when $f$ is a crepant resolution by studying tilting modules over noncommutative crepant resolutions of $Y$. This is a joint work with M.Wemyss.

Date

4月21日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

白石勇貴氏 (京都大学)

Title

拡大カスピダルワイル群の不変式論と軌道空間上のフロベニウス構造

Abstract

　 本研究は高橋篤史先生との共同研究です。 拡大カスピダルワイル群と呼ばれる、星状コクセター ディンキン図形に付随したワイル群をアフィン化し、 更に1次元分とある方向への作用で拡大した群を 考えます。
ルーイエンガの不変式論により、この群による 軌道空間は複素多様体の構造を持ち、とある予想的 条件（性質Pと呼ぶ）の下で、ドゥブロヴィンの フロベニウス構造が入る事を紹介します。
また時間が有れば、このフロベニウス構造と、 ある種の特異点や正則関数に対する齋藤の原始形式の 理論から構成されるフロベニウス構造との同型について 説明します。

Date

4月14日 (Fri), 16:20--17:50, 2017

Room

Room 402, RIMS

Speaker

Yosuke Morita氏 (Kyoto U)

Title

Homogeneous spaces that do not model any compact manifold

Abstract

　 A manifold is said to be locally modelled on a homogeneous space G/H if it is obtained by patching open sets of G/H by left translations of elements of G. A typical example is a Clifford-Klein form, namely, a quotient of G/H by a discrete subgroup of G acting properly and freely on G/H. Since T. Kobayashi's work in the late 1980s, a number of obstructions to the existence of compact manifolds locally modelled on G/H (or compact Clifford-Klein forms of G/H) has been found. I will explain an obstruction arising from the comparison of relative Lie algebra cohomology and de Rham cohomology.

Date

4月7日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

加藤周氏 (京大・理)

Title

非対称Macdonald多項式の特殊化の間の双対性について

Abstract

　 $ADE$型のルート系に対しては対応するアフィン・リー環の レベル$1$の(適切な)Demazure加群の指標と非対称Macdonald多項式の$t = 0$ への特殊化が一致することが知られている(Sanderson-Ion)。
この講演では$ADE$型のルート系の非対称Macdonald多項式の$t = \infty$への 特殊化も類似の解釈を持つことを説明し、$t = 0$への特殊化との関係を議論 する。講演内容はEvgeny Feigin, Ievgen Makedonskyi氏との共同研究arXiv:1703.04 108の第５節とAppendix (及びよく知られた結果)に基づく。

Date

1月20日 January 20(Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

Ivan Ip氏 (京都大学)

Title

Cluster realization of Uq(g) and factorization of universal R matrix

Abstract

　 For each simple Lie algebra g, I will talk about a new　 presentation of an embedding of Uq(g) into certain quantum torus algebra, described by a quiver diagram, using the previous construction of positive representations of split real quantum groups. We will discuss its relation to cluster structure of G-local system described recently by Le, and a factorization of the universal R matrix which corresponds to a sequence of quiver mutations giving the half-Dehn twist of the triangulation of a twice- punctured disk with two marked points. This generalizes the well-known result of Faddeev for type A1 and the recent work of Schrader-Shapiro for type An.

Date

1月13日 January 13(Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

八尋耕平氏 (東大数理)

Title

D-modules on partial flag varieties and intertwining functors

Abstract

　 Beilinson and Bernstein provided a relationship between the category of D-modules on the full flag variety and a category of representations of semisimple Lie algebras. They introduced intertwining functors for D-modules on the full flag variety and gave a proof of Casselman submodule theorem using them. In this talk, we discuss the case of partial flag varieties. We show that in some cases intertwining functors are equivalences of derived categories. We also discuss the behavior of global sections under the intertwining operators.

Date

1月6日 January 6(Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

鈴木咲衣氏 (RIMS)

Title

The universal quantum invariant and colored ideal triangulation

Abstract

　 The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal R-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal R-matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal R-matrix. On the other hand, R. Kashaev showed that the Heisenberg double of a finite dimensional Hopf algebra has the canonical element (the S-tensor) satisfying the pentagon relation. In this talk we reconstruct the universal quantum invariant using the Heisenberg double, and extend it to an invariant for colored singular triangulations of topological spaces, especially for colored ideal triangulations of tangle complements. In this construction, a copy of the S-tensor is attached to each tetrahedron, and the invariance under the colored Pachner (2; 3) moves is shown by the pentagon relation of the S-tensor.

Date

12月9日 December 9(Fri), 16:30--18:00, 2016

Room

Room 402, RIMS

Speaker

佐藤 敬志氏 (京大理)

Title

Hessenberg varieties and hyperplane arrangements

Abstract

　 A Hessenberg variety is a subvariety of a flag variety determined by a "good" subset of the positive root system. By the way, a subset of the positive root system gives a hyperplane arrangement in the Lie algebra of a maximal torus. Similarly to a flag variety, the chambers of this arrangement denote a cell decomposition of the (regular nilpotent) Hessenberg variety. By this relation between a Hessenberg variety and a hyperplane arrangement, we describe the cohomology ring of the (regular nilpotent) Hessenberg variety in terms of the subarrangement and show that its Poincaré polynomial has two expressions like the Borel's work on flag varieties. This is a joint work with Takuro Abe, Tatsuya Horiguchi, Mikiya Masuda, and Satoshi Murai.

Date

11月25日 November 25(Fri), 16:30--18:00, 2016

Room

Room 402, RIMS

Speaker

大島 芳樹氏 (IPMU)

Title

Determinant formula for parabolic Verma modules of Lie superalgebras

Abstract

　 We give a determinant formula for parabolic Verma modules of contragredient finite-dimensional Lie superalgebras assuming that the Levi component is contained in the even part. Our formula generalizes previous results of Jantzen for parabolic Verma modules of (non-super) Lie algebras, and of Kac concerning (non-parabolic) Verma modules for Lie superalgebras. This is a joint work with Masahito Yamazaki.

Date

11月18日 November 18(Fri), 16:30--18:00, 2016

Room

Room 402, RIMS

Speaker

Evgeny Feigin 氏(Higher School of Economics)

Title

Generalized Weyl modules and nonsymmetric Macdonald polynomials

Abstract

　 We define a family of modules over the Iwahori subalgebra of an affine Kac-Moody Lie algebra, generalizing classical Weyl modules. The modules in the family are labeled by integral weights of the underlying finite-dimensional algebra. We describe the representation theoretical and combinatorial properties of the generalized Weyl modules. In particular, we show that they serve as categorification of various specializations of the nonsymmetric Macdonald polynomials and of the Orr-Shimozono combinatorial formula.

Date

11月11日 November 11(Fri), 16:30--18:00, 2016

Room

Room 402, RIMS

Speaker

Benoit Collins氏（京大・理)

Title

Positivity for the dual of the Temperley-Lieb basis

Abstract

　 A problem raised by Vaughan Jones is to consider the basis dual to the canonical basis of the Temperley-Lieb algebra for non-degenerate loop values, and investigate the coefficients of this basis element in the original basis. For example, the dual of the identity element is a multiple of the Jones Wenzl projection, and computing it is an important problem for which some formulas have been given recently (e.g. by Morrisson). The goal of this talk is to describe a new combinatorial formula for all of these coefficients. As a byproduct, we solve one question of Jones and prove that all these coefficients are never zero for real parameters \ge 2, and we compute their sign. Our strategy relies on identifying these coefficients with the Weingarten function of the free orthogonal quantum group, and on developing quantum integration techniques. I will spend some time on recalling definitions and properties of some objects that are less well-known, such as Weingarten functions and free orthogonal quantum groups. This talk is based on joint work with Mike Brannan, arXiv:1608.03885.

Attention

いつもと曜日・お部屋が異なります
Date and room are different from the usual

Date

10月27日 October 27(Thu), 16:30--18:00, 2016

Room

Dept. of Math. 305

Speaker

Thorge Jensen氏(MPI/RIMS)

Title

The p-canonical basis of Hecke algebras

Abstract

　 Motivated by open problems in modular representation theory, we describe a positive characteristic analogue of the Kazhdan-Lusztig basis of the Hecke algebra of a crystallographic Coxeter system and investigate some of its properties. After giving several examples, we will mention recent results about p-cells.

Date

10月21日 October 21(Fri), 16:30--18:00, 2016

Room

RIMS 402

Speaker

Leonardo Patimo氏（MPI）

Title

The Hard Lefschetz Theorem in Positive Characteristic for the Flag Varieties

Abstract

　 Hodge theoretic properties of the Flag Varieties (in characteristic 0) are a fundamental ingredient in the proof of the Kazhdan-Lusztig conjectures. Investigating Hodge theoretic properties in positive characteristic could lead to a better understanding of Lusztig's conjecture on algebraic groups in positve characteristic. As a first step in this direction, in this talk we prove, for any flag variety, that the Hard Lefschetz Theorem holds in characteristic p if p is larger than the number of positive roots.

Date

10月7日 October 7(Fri), 16:30--18:00, 2016

Room

RIMS 402

Speaker

藤田遼氏（京大・理)

Title

Tilting modules of affine quasi-hereditary algebras

Abstract

　 We discuss tilting modules of affine quasi-hereditary algebras. We present an existence theorem of indecomposable tilting modules when the algebra has a large center and use it to deduce a criterion for an exact functor between two affine highest weight categories to give an equivalence. As an application, we prove that the Arakawa-Suzuki functor gives a fully faithful embedding of a block of the deformed BGG category of glm into the module category of a suitable completion of degenerate affine Hecke algebra of $\mathop{GL}_n$.

Date

9月30日 September 30(Fri), 15:00--18:00, 2016

Room

RIMS 402

Speaker

杉山 聡 氏（東大数理）

Title

Fukaya category in the Koszul duality theory

Abstract

　 We compute all the Ext groups, composition of them, and their higher structure of the simple modules of a path algebra with relations over a tree type quiver $A$ by using the Fukaya categories of some exact Riemann surfaces. This is nothing but a computation of an $A_\infty$- Koszul dual $A^!$ of $A$. In this talk, (i) we review the theory of Fukaya categories, (which appeared in symplectic geometry, mainly in the context of Homological Mirror Symmetry), then (ii) we study the "abstract" method of computing Koszul dual via Fukaya categories, finally (iii) we see the three examples of the computation (which are enough convincing so that they let us imagine the full proof).

Date

9月23日 September 23(Fri), 16:30--18:00, 2016

Room

RIMS 402

Speaker

小寺 諒介氏 (京大・理)

Abstract

　 We prove that the quantized Coulomb branches associated with framed quiver gauge theory of Jordan type are isomorphic to spherical Cherednik algebras.

Attention

いつもと曜日・お部屋が異なります
Date and room are different from the usual

Date

8月8日 August 8(Mon), 16:30--18:00, 2016

Room

Room 307,Research Building No.4, Kyoto University
（京都大学総合研究4号館307号室）

Speaker

Yung-Ning Peng氏（National Central University）

Title

PARABOLIC PRESENTATION OF THE SUPER YANGIAN $Y_{M|N}$ AND ITS APPLICATION

Abstract

　 The super Yangian Y_{M|N} associated to the general linear Lie superalgebra gl_{M|N}, defined by Nazarov, is a super analogue of the classical Yangian algebra Y_{N} associated to gl_{N}. It can be described in two different manners: the RTT presentation and the Drinfeld's presentation. In this talk, we introduce a series of presentations of Y_{M|N}, depending on a composition \mu of M +N and a 0^{M}1^{N} -sequence s. Our presentation covers both the RTT presentation and Drinfeld's presentation as special examples by taking \mu = (M + N) or \mu = (1^{M+N}), and a lot of new presentations that never appeared before. Moreover, we will discuss about some application of our result.

Date

7月15日 July 15(Fri), 16:30--18:00, 2016

Room

RIMS Room 402

Speaker

Hans Franzen氏（Bonn）

Title

Classical and orientifold Donaldson-Thomas invariants as Chow groups

Abstract

　 We show that the primitive part of Kontsevich-Soibelman's Cohomological Hall algebra of a quiver can be identified with Chow groups of moduli spaces of stable quiver representations. This shows that the Donaldson-Thomas invariants agree with the dimensions of these Chow groups. A similar method also applies for Young's orientifold DT invariants which are an analog of classical DT invariants for orthogonal/symplectic groups. We show that these invariants can be identified with Chow groups of moduli spaces of $\sigma$-stable self-dual representations.

Date

7月8日 July 8(Fri), 16:30--18:00, 2016

Room

RIMS Room 402

Speaker

林拓磨氏（東大数理）

Title

A description of principal series representations of SU(1,1) over ${\mathbb{Z}}\left[1/2\right]$-algebras

Abstract

　 Principal series representations are Hilbert representations of real reductive Lie groups obtained by parabolic inductions. Their associated (${\mathfrak{g}},K$)-modules are known to be obtained from the corresponding parabolic induction of ($\mathfrak{g},K$)-modules. Hence they satisfy a universal property. If we start with the Lie group SU(1,1) the associated (${\mathfrak{g}},K$)-modules to the principal series representations have an explicit description. In particular, they are defined over commutative rings. In this talk, I will prove that these ($ {\mathfrak{g}},K$)-modules over ${\mathbb{Z}}\left[1/2\right]$-algebras enjoy a similar universal property as well.

Date

6月17日 June 17(Fri), 16:30--18:00, 2016

Room

RIMS Room 402

Speaker

田中雄一郎氏（東大数理）

Title

Visible actions of compact Lie groups on complex spherical varieties

Abstract

　 With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the theory of visible actions on complex manifolds.

In this talk we consider visible actions of a compact real form U of a connected complex reductive algebraic group G on spherical varieties. Here a connected complex G-variety X is said to be spherical if a Borel subgroup of G has an open orbit on X. The sphericity implies the multiplicity-freeness property of the space of polynomials on X. We firstly give a proof of the visibility for affine homogeneous spherical varieties, and then show the visibility for general spherical varieties by using the method of induction of visible actions. A prototype of the method of induction was introduced by Kobayashi (2005) for the case of complex spherical nilpotent orbits of type A, and recently extended by A. Sasaki (2016) to the case of arbitrary type. Our proof is highly motivated by those earlier results.

Date

6月3日 June 3(Fri), 16:30--18:00, 2016

Room

RIMS Room 402

Speaker

渡部正樹氏（東大数理）

Title

Kraskiewicz-Pragacz modules and positivity properties of Schubert polynomials

Abstract

　 Kraskiewicz-Pragacz modules are certain family of modules over the upper triangular Lie algebra whose characters are Schubert polynomials. Due to this property, some problems on Schubert-positivities of polynomials are closely related with the class of modules having filtrations with successive quotients being KP modules. In this talk I will explain my result which give a characterization of such modules, in terms of certain Ext groups, using the methods of highest weight categories. As applications of such a characterization we obtain a representation-theoretic proof (other than the classical geometric proof) for the positivity of the products of Schubert polynomials, as well as a new result generalizing the positivity of plethysms of Schur functions to Schubert polynomials.

Date

5月20日 May 20(Fri), 16:30--18:00, 2016

Room

RIMS Room 402

Speaker

渡邉英也氏（東工大）

Title

Combinatorial formulas expressing periodic R-polynomials and periodic Kazhdan-Lusztig polynomials

Abstract

　 Periodic Kazhdan-Lusztig polynomials naturally appear in the representation theory of affine Hecke algebras and affine quantum groups. They are computed from periodic R-polynomials. In this talk, we will give a combinatorial formula expressing periodic R-polynomials by using the "doubled" Bruhat graph. Then, a combinatorial formula for periodic KL-polynomials can be constructed from this formula. Time permitting, we will briefly explain how periodic Kazhdan-Lusztig polynomials appear in the representation theory of affine quantum groups.

Date

5月13日 May 13(Fri), 16:30--18:00, 2016

Room

RIMS Room 402

Speaker

Bea Schumann氏 (Cologne and Tokyo)

Title

Homological description of crystal structures on quiver varieties

Abstract

　 We explain a crystal isomorphism in finite types between the explicit crystal structure on Lusztig's parametrisation of the canonical basis obtained by Reineke in terms of representations of quivers and the geometric construction of crystal bases obtained by Kashiwara and Saito in terms of quiver varieties. Using the interplay between the representation theory of the Dynkin quiver and the representation theory of the preprojective algebra, we thereby compute the actions of the Kashiwara operators on the irreducible components of the quiver varieties.

Date

5月6日 May 6(Fri), 16:30--18:00, 2016

Room

RIMS Room 402

Speaker

Hiraku Nakajima 氏（RIMS）

Title

Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type A

Abstract

　 Cherkis bow varieties are found in the ADHM type description of instantons on the Taub-NUT space. They were originally given in terms of Nahm's equations, but I will give their quiver description, which are useful for analysis of their properties. As an application, I will explain that they are Coulomb branches of quiver gauge theories of affine type A. This is a joint work with Yuuya Takayama.

Date

4月22日 April 22(Fri), 16:30--18:00, 2016

Room

RIMS Room 402

Speaker

跡部発 氏（京都大学）

Title

Local theta correspondence of tempered representations and Langlands parameters

Abstract

　 局所テータ対応について、2つの問題がある。
一つは与えられた既約表現に対して、そのテータリフトはいつ nonzero になるか という問題。もう一つは、テータリフトが nonzero の時、その唯一の既約商は どのような表現かという問題。本講演では、p 進体上の unitary dual pair に おいて、与えられた既約表現が緩増加である時に、これら2つの問題に Langlands 対応の言葉で答えを与える。
なお、本研究は Wee Teck Gan 氏との共同研究である。

Date

4月15日 April 15(Fri), 16:30--18:00, 2016

Room

RIMS Room 402

Speaker

Naoki Genra氏 (RIMS)

Title

Screening operators for W-algebras

Abstract

　 We show that the (affine) W-algebras for generic levels are constructed as intersections of kernels of screening operators.
As applications, we prove two conjectures.
First, Fateev-Lukyanov's WBn-algebras are isomorphic to the W-algebras for osp(1,2n) and its regular nilpotent element.
Second, Feigin-Semikhatov's W^(2)_{n}-algebras are isomorphic to the W-algebras for sln and its subregular nilpotent element.

Date

4月7日 April 7(Thu), 16:30--18:00, 2016

Room

RIMS Room 109

Speaker

Shintaro Yanagida氏 (Nagoya)

Title

K-theoretic AGT relation

Abstract

　 We sutdy a geometric action of the deformed Virasoro algebra on the torus equivariant K groups of instantont moduli spaces on the complex plane.
The main ingredient is the analysis of K-theoretic stable envelopes. As a consequence, we can prove some parts of the conjectures on K-theoretic AGT correspondence proposed in the previous collaboration with Awata, B. Feigin, Hoshino, Kanai and Shiraishi.

Date

1月22日 January 22(Fri), 16:30--18:00, 2016

Room

RIMS Room 402

Speaker

土岡俊介氏（東大数理）

Title

On a general Schur's partition identity

Abstract

　 We will talk on a generalization of Schur's partition identity which is a kind of Rogers-Ramanujan type identity. Our identity comes from K\"{u}lshammer-Olsson-Robinson theory of generalized blocks and the Fock space representations of quantum affine algebras due to Kashiwara-Miwa-Petersen-Yung. This is a joint work with Masaki Watanabe (University of Tokyo).

Attention

いつもと曜日・お時間・お部屋が異なります
Date and room are different from the usual

Date

12月21日 December 21(Mon), 15:00--16:30, 2015

Room

RIMS Room 110

Speaker

Constantin Teleman 氏 (UC Berkeley)

Title

Gauge Theory in 2 and 3 dimensions and categorical representations.

Abstract

　 I will introduce the notion of a categorified (topological) representation of a compact Lie group G, which is the mathematical counter-part to a topological boundary condition for (pure) 3-dimensional gauge theory. The main examples come from the Gromov-Witten theories of compact symplectic manifolds with Hamiltonian group action. The character theory of these representations is captured, in the spirit of quantum mechanics, by the holomorphic symplectic geometry of a certain manifold, now recognised as the `Coulomb branch’ of the pure 3D gauge theory. Twisted versions of Gromov-Witten theory relate to gauge theory with `matter’. The theory gives a clean account of some aspects of the gauged (non-linear!) Sigma-model and the appearance of the Toda integrable system.

Attention

いつもと曜日・お部屋が異なります
Date and room are different from the usual

Date

12月3日 December 3(Thu), 16:30--18:00, 2015

Room

RIMS Room 206 [changed!]

Speaker

松本 拓也 氏 (名古屋大学)

Title

中心拡大されたリースーパー代数sl(2|2)に付随する 量子アファイン代数について (Quantum affine algebra associated with the centrally extended Lie superalgebra sl(2|2))

Abstract

　 リースーパー代数sl(2|2)は全ての超リー代数の中で唯一2次元の 普遍中心拡大を持つ特殊なものであるが， 超弦理論におけるゲージ/重力対応や1次元ハバード模型における 対称性として現れ，様々な物理的文脈で重要な役割を果たしている． よって，その代数構造を詳しく調べることは重要であると思われる． 今回は，その中心拡大されたsl(2|2)代数に付随する量子群と 無限次元への拡張である量子アファイン代数を紹介したい． また，これまで知られていたヤンギアン代数への退化も議論したい． 本講演はN. Beisert氏(ETH)とW. Galleas氏(DESY)との共同研究 http://arxiv.org/abs/1102.5700 に基づきます．

Date

11月13日 November 13(Fri), 15:30--16:30, 16:45 - 17:45, 2015

Room

RIMS Room 402

Speaker

入谷 寛 氏 (京都大学)

Title

A Fock sheaf for Givental quantization

Abstract

　 Genus-zero Gromov-Witten (GW) theory defines a generalized variation of Hodge structure (sometimes called semi-infinite Hodge structure). For a given generalized variation of Hodge structure, we define a sheaf of Fock spaces on the base of the Hodge variation. The Fock sheaf is locally modeled on the quantization formalism of Givental, and higher-genus GW potentials can be regarded as a section of the Fock sheaf. This formalism gives a framework to discuss the modularity and the crepant transformation conjecture in higher-genus GW theory; for example, we observe that the total descendant GW potentials of compact toric orbifolds X are "modular" with respect to a certain subgroup of the group of autoequivalences of D(X). This is based on joint work with Tom Coates.

Date

11月6日 November 6(Fri), 16:30--18:00, 2015

Room

RIMS Room 402

Speaker

佐藤僚氏 (東大)

Title

Kazama-Suzuki coset construction and logarithmic extensions of weight modules

Abstract

　 Kazama-Suzukiコセット構成とは，アフィンLie代数のsmoothな加群と荷電フェル ミオンFock加群をテンソルした空間に互いに可換な$\mathcal{N}=2$超共形代数 とHeisenberg Lie代数の作用を構成する手法である．この手法によっ て，$A_{1}^{(1)}$型アフィンLie代数の既約ユニタリ（=可積分）最高ウェイト 表現から$\mathcal{N}=2$超共形代数の全ての既約ユニタリ最高ウェイト加群が 得られることはよく知られている．本講演では，非ユニタリな場合にもこの構成 が適切な加群圏の間にアーベル圏としての圏同値を与えることを解説する．特 に，Virasoro代数の$L_{0}$作用素が非対角に作用する（対数的）な加群につい て取り扱う．

Date

10月2日 October 2(Fri), 16:30--18:00, 2015

Room

RIMS Room 402

Speaker

渡邉英也(Hideya Watanabe)氏 (東工大)

Date

9月25日 September 25(Fri), 16:30--18:00, 2015

Room

RIMS Room 402

Speaker

Simon Wood氏 (Australian National University)

Date

8月7日 August 7(Fri), 16:30--18:00, 2015

Room

RIMS Room 402

Speaker

Oren Ben-Bassat 氏 (Oxford)

Date

7月10日 July 10(Fri), 16:30--18:00, 2015

Room

RIMS Room 402

Speaker

桑原敏郎 Toshiro Kuwabara 氏 (High School of Economics, Russia)

Date

6月12日 June 12(Fri), 16:30--18:00, 2015

Room

RIMS Room 402

Speaker

池田 曉志 氏 (東大カブリIPMU)

Attention

いつもとお時間・場所が違います
Date and Place are different from the usual

Date

6月4日 June 4(Thu) 13:30--15:00, 2015

Room

RIMS Room 006(basement)

Speaker

Ivan Ip氏 (Kyoto University SGU)

Date

5月29日 May 29(Fri), 16:30--18:00, 2015

Room

Room RIMS Room 402

Speaker

Bin Shu氏 (East China Normal University)

Title

Finite W-superalgebras and existence of Kac-Weisfeiler modules for basic Lie superalgebras in positive chatacteristic

Abstract

　 In this talk, we will introduce finite W-superalgebras for basic Lie superalgebras associated with even nilpotent elements. We will then present the PBW theorem and other properties fo them. We will finally discuss the existence of the so-called Kac-Weisfeiler modules for basic Lie superalgebras in positive characteristic. This is a joint work with Yang Zeng.

Date

5月22日 May 22(Fri), 16:30--18:00, 2015

Room

Room RIMS Room 402

Speaker

Anton Evseev氏 (University of Birmingham)

Title

RoCK blocks, wreath products and KLR algebras

Abstract

　 The so-called RoCK (or Rouquier) blocks play an important role in representation theory of symmetric groups over a finite field of characteristic $p$, as well as of Hecke algebras at roots of unity. Turner has conjectured that a certain idempotent truncation of a RoCK block is Morita equivalent to the principal block $B_0$ of the wreath product $S_p\wr S_d$ of symmetric groups, where $d$ is the "weight" of the block. The talk will outline a proof of this conjecture, which generalizes a result of Chuang-Kessar proved for $d < p$. The proof uses an isomorphism between a Hecke algebra at a root of unity and a cyclotomic Khovanov-Lauda-Rouquier algebra, the resulting grading on the Hecke algebra and the ideas behind a construction of R-matrices for modules over KLR algebras due to Kang-Kashiwara-Kim.

Attention

いつもと時間・場所が違います

Date

5月14日 May 14(Thu), 13:30--15:00, 2015

Room

Room RIMS Room 006

Speaker

Kentaro Wada氏 (Sinsyu)

Title

New realization of cyclotomic q-Schur algebras

Abstract

　 G(r,1,n)型の複素鏡映群に付随した cyclotomic q-Schur 代数は Ariki-Koike 代数の quasi-hereditary cover の1つである。r=1の場合，cyclotomic q-Schur 代数は古典的な q-Schur 代数そのものであり, q-Schur 代数は一般線形リー代 数に付随する量子群の商代数であることが知られている。量子群の Hopf代数と しての構造と普遍R-行列によって，(nを全て動かした)q-Schur 代数の加群圏上 にはモノイダル構造が定まる。r>1の場合には，そのような構造は知られていな いが，Rouquier-Shan- Varagnolo-Vasserot によって得られているアファイン一 般線形リー代数のアファイン放物型圏O との関係等によって，cyclotomic q-Schur 代数の加群圏上にもモノイダル構造が定まることが期待される。この講 演ではその可能性の1つについてお話ししたい。 まず，r個に分けられた一般線形リー代数のカルタンデータに付随してリー代 数 g を導入する。r=1 の場合は一般線形リー代数に付随したカレントリー代数 そのものであり，r>1 の場合，リー代数 g はカレントリー代数の filtered deformation になっている。すると q=1 の場合の cyclotomic q-Schur 代数 は，このリー代数 g の普遍包絡代数の商代数となる。次に，リー代数 g の普遍 包絡代数の q-類似として代数 U_q を導入し，cyclotomic q-Schur 代数を U_q の商代数として実現する。その後，（まだ分かっていないことが多いが) リー代 数 g やその q-類似 U_q の表現論について可能な限り説明したいと思います。

Date

5月8日 May 8(Fri), 16:30--18:00, 2015

Room

Room RIMS Room 402

Speaker

Yoshihiro Takeyama氏 (Tsukuba University)

Title

A deformation of affine Hecke algebra and integrable stochastic particle system

Abstract

　 We introduce a deformation of the affine Hecke algebra of type $GL$ with four parameters. Making use of its representation on the space of polynomials, we can construct a discrete analogue of integration operators satisfying the braid relations. It determines a difference operator which can be regarded as a discretization of the Hamiltonian of the one-dimensional delta Bose gas. By specializing the parameters of the discrete Hamiltonian, we get the transition rate matrix of an integrable stochastic particle system called (a continuous time limit of ) the $q-$Hahn system.

Attention

今回は連続講演でいつもと時間・場所が違います

Date

4月16日 Apr 16(Thu) 10:30--12:00, 13:30--15:00, 2015

Room

RIMS Room 006(地階)

Date

4月17日 Apr 17(Fri) 15:00-18:00

Room

RIMS Room 402

Speaker

脇本実氏 (九州大学)

Date

2月6日 Feb 6(Fri) 16:30--18:00, 2015

Room

RIMS Room 402

Speaker

Hironori Oya氏 (University of Tokyo)

Date

1月30日 Jan 30(Fri) 16:30--18:00, 2015

Room

RIMS Room 402

Speaker

Oleksandr Tsymbaliuk氏 (Simons Center and RIMS)

Attention

Date and Place are different from the usual.

Date

Room

Room 110, Building No.3

Speaker

Anthony Henderson氏 (University of Sydney)

Date

1月9日 Jan 9(Fri) 16:30--18:00, 2015

Room

RIMS 402

Speaker

石井基裕氏 (東北大・情報)

Attention

Time and Place are different from the usual.

Date

11月27日 Nov 27(Thu) 16:30--18:00, 2014

Room

Dept. Math. 3rd build. 109

Speaker

中島啓氏 (京大・数理研)

Date

11月21日 Nov 21(Fri) 17:00--18:30, 2014

Room

Room 402, RIMS

Speaker

廣惠一稀氏(Kazuki Hiroe) (城西大学)

Attention

Time is different from the usual.

Date

10月24日 Oct 24(Fri) 16:00--18:00, 2014

Room

Room 402, RIMS

Speaker

Nikolai Vavilov 氏 (State University of Saint-Petersburg)

Attention

Time and Place are different from the usual.

Date

7月17日 July 17(Thu) 16:00--18:00, 2014

Room

Room 108, Building No.3, Kyoto University

Speaker

Andrei Okounkov氏 (Columbia University)

Attention

There will be two informal talks on the elliptic quantum groups by the same speaker at (1) July 11(Fri) 14:30 - 15:30 and (2) July 12(Sat) 10:00 - 13:00.

Date

7月11日 July 11(Fri) 16:30--18:00, 2014

Room

Rims 402, Kyoto University

Speaker

Hitoshi Konno 氏 (Tokyo University of Marine Science and Technology)

Attention

!Caution There are two talks and the first talk starts at 13:00!

Date

6月20日(金) June 20(Fri) 13:00--14:30, 2014

Room

Room 402 of RIMS, Kyoto University

Speaker

Jethro Van Ekeren氏 (Technische Universität Darmstadt)

Title

Superconformal Blocks

Abstract

　 Let V be a chiral algebra (associated to a vertex algebra) over a family X of complex curves. An important collection of objects associated to V are the spaces of conformal blocks. Roughly speaking these are spaces of sections of V over fibres, whose dependence on the moduli yields a bundle with flat connection over the family. Understanding the structure of conformal blocks in particular cases leads to interesting theorems. Examples include nonabelian theta functions, and Zhu's theorem on modular invariance of vertex algebra characters.

In this talk I will describe joint work with R. Heluani in which we construct superconformal blocks associated to N=2 SUSY vertex algebras living on super-analogues of elliptic curves. The family of supercurves is described as a quotient by the classical Jacobi group, and equivariance of normalised superconformal blocks under this group establishes their transformation under this group as Jacobi forms.

Date

14:45 - 16:15, 2014

Speaker

Alexander P. Veselov氏 (Loughborough, UK and Tokyo, Japan)

Title

Gaudin subalgebras and stable rational curves

Abstract

　 Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno-Drinfeld Lie algebra t_n, associated to A-type hyperplane arrangement. It turns out that Gaudin subalgebras form a smooth algebraic variety isomorphic to the Deligne-Mumford moduli space \bar M_{0,n+1} of stable genus zero curves with n+1 marked points. A real version of this result allows to describe the moduli space of separation coordinates on the unit sphere in terms of geometry of Stasheff polytope. The talk is based on joint works with L. Aguirre and G. Felder and with K. Schoebel.

Date

6月13日 June 13(Fri) 17:00--18:00, 2014

Room

Rims 402, Kyoto University

Speaker

Hiraku Abe氏 (Osaka City University)

Title

Springer多様体のトーラス同変コホモロジー環
(Torus equivariant cohomology ring of Springer varieties)

Abstract

　 谷崎俊之氏によるA型Springer多様体のコホモロジー環の表示は旗多様体のコホモロ ジー環のBorel表示を自然に一般化するものであった.本講演ではこの表示のトーラス 同変版を解説する．すなわち, A型Springer多様体がもつ自然なトーラス作用に関し てその同変コホモロジー環の表示を与える．この際，Springer多様体の同変コホモロ ジーに対称群の表現を構成することが鍵となる．同変コホモロジー理論でよく用いら れる局所化と呼ばれる手法を用いてこの表現を構成する．本研究は大阪市立大学の堀 口達也氏との共同研究である．

Date

6月6日 June 6(Fri) 14:45--16:15 & 16:30--18:00, 2014

Room

Rims 402, Kyoto University

Speaker

Tomoki Nakanishi氏 (Nagoya University)

Title

Cluster algebras, dilogarithm, and Y-systems

Abstract

　 Cluster algebras were introduced by Fomin and Zelevinsky around 2000 as an underlying combinatorial structure in Lie theory. They also (often quite unexpectedly) appear in several branches of mathematics besides representation theory, e.g., hyperbolic geometry and Teichm\"uller theory, Poisson geometry, discrete dynamical systems, exact WKB analysis, etc. In this talk I review the application of cluster algebras to the dilogarithm and Y-systems, based on joint works over the recent years with R. Inoue, O. Iyama, R. Kashaev, B. Keller, A. Kuniba, R. Tateo, J. Suzuki, and S. Stella.

The talk consists of two parts. In the first part, after reviewing some basic properties of cluster algebras, I present the dilogarithm identity associated with any period of seeds in a cluster algebra. In the second part, I explain that this identity is related to the longstanding conjectures on the periodicities of Y-systems and the associated dilogarithm identities in conformal field theory, which arose through the thermodynamic Bethe ansatz approach in 90's. Then, I show how efficiently cluster algebra theory proves these conjectures.

Date

5月23日 May 23(Fri) 16:30--18:00, 2014

Room

RIMS 402, Kyoto University

Speaker

Daisuke Sagaki氏 (Tsukuba University)

Title

Demazure subcrystals of crystal bases of level-zero extremal weight modules over quantum affine algebras

Abstract

　 We give a characterization of the crystal bases of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra. This characterization is given in terms of the initial directions of semi-infinite Lakshmibai-Seshadri paths (SiLS paths), and is established under a suitably normalized isomorphism between the crystal basis of the level-zero extremal weight module and the crystal of SiLS paths. This talk is based on a joint work with Satoshi Naito (arXiv:1404.2436).

Date

5月16日 May 16(Fri) 16:30--18:00, 2014

Room

Rims 402, Kyoto University

Speaker

瀧雅人氏 (理研)

Title

Seiberg Duality, 5d SCFTs and Nekrasov Partition Functions

Abstract

　 We propose an equality between five-dimensional (5d) Nekrasov partition functions that are associated with local del Pezzo surfaces, namely the generating functions of the refined Gopakumar-Vafa invariants. It is known that M-theory compactified on a local del Pezzo Calabi-Yau 3-fold leads to a 5d superconformal field theory (SCFT), and their BPS partition functions are these Nekrasov partition functions. One can expect that the "Picard-Lefschetz transformation" of the 3-folds implies the duality between the compactified M-theories and the resulting 5d SCFTs.
This stringy argument yields conjectural relations between the corresponding Nekrasov partition functions.

Attention

!開始時間がいつもとは異なりますので、ご注意ください!

Date

5月9日 May 9(Fri) 14:30--16:00, 2014

Room

Rims 402, Kyoto University

Speaker

Chul-Hee Lee氏 (SNU)

Title

Kirillov-Reshetikhin modules and the WZW fusion ring

Abstract

　 The Kirillov-Reshetikhin modules form a special class of finite dimensional representations of quantum groups. Their characters are known to satisfy some functional relations called T-systems and Q-systems. In an attempt to calculate the central charges of certain conformal field theories using the dilogarithm function based on the Thermodynamic Bethe Ansatz method, some conjectures about solutions of level restricted version of Q-systems have been proposed. In this talk, I will explain how the WZW fusion ring can be used to answer them and discuss their status and related problems.

Attention

!今回は2コマ講演でいつもより開始時間が早くなっていますのでご注意ください!

Date

4月25日(金) April 25(Fri) 14:45--16:15, 2014

Room

Room 402 of RIMS, Kyoto University

Speaker

Andrei Negut氏(RIMS, Columbia University)

Title

Quantum toroidal $gl_n$ and its shuffle presentation

Abstract

　 We will discuss the Feigin-Odesskii shuffle algebra presentation of the quantum toroidal gl_n algebra. This will allow us to identify many copies of quantum affine gl_n sitting inside the quantum toroidal, each corresponding to a choice of rational slope.
In particular, the universal R-matrix of quantum toroidal gl_n decomposes as a product of universal R-matrices for quantum affine gl_n, in a way reminiscent of the Khoroshkin-Tolstoy factorization for affine types. The role of positive roots in the direction of the affinization is played by these rational slopes.

Date

4月25日(金) April 25(Fri) 16:30--, 2014

Room

Room 402 of RIMS, Kyoto University

Speaker

Ivan Cherednik氏(RIMS, UNC at Chapel Hill)

Title

Generalized Rogers-Ramanujan identities and Nil-DAHA

Abstract

　 The core application of Nil-DAHA so far is the construction of the global Q-Whittaker functions and Dunkl operators in the Q-Toda theory and its nonsymmetric variant, including a surprising application to the PBW-filtration (counting the minimal number of creation opertors). As Boris Feigin and the speaker demonstrated, this new theory is closely related to coset algebras and can be used to define Rogers-Ramanujan sums of modular type associated with any root systems. The sums we obtain quantize the constant Y-systems (of type $RxA_n$ for any reduced root systems R). This involves dilogarithms, the so-called Nahm Conjecture and a lot of interesting RT, arithmetic and physics, though the talk will be mainly focused on the main construction (practically from scratch).

Attention

!通常セミナーは金曜日ですが、この回は変則的に木曜日となります!

Date

4月17日 April 17(Thu) 16:30--18:00, 2014

Room

Rims 402, Kyoto University

Speaker

名古屋創氏 (立教大)

Title

On the tau function of the sixth Painleve equation from Virasoro conformal field theory

Abstract

　 An explicit asymptotic expansion of the tau function of the sixth Painleve equation was discovered by Gamayun, Iorgov and Lisovyy [arxiv:1207.0787]. I will explain that their series expansion of the tau function can be derived from Virasoro conformal field theory. I note that the same approach was done by Iorgov, Lisovyy and Teschner [arxiv:1401.6104]. I will begin by reviewing known results of the tau function, fundamentals of Virasoro conformal field theory, the connection problem of conformal blocks, and then I will explain how to obtain the fundamental solution to the linear problem of PVI and the tau function from Virasoro conformal field theory. My talk is based on a joint work with Hiroe, Jimbo and Sakai.

Date

4月11日 April 11(Fri) 16:30--18:00, 2014

Room

Rims 402, Kyoto University

Speaker

Kari Vilonen氏 (Northwestern University)

Title

Langlands duality for real groups

Abstract

　 For real reductive groups the Langlands duality, as refined by Vogan, acquires a symmetry and both sides of the duality can be viewed as representations of reductive groups. Lifting this duality to the level of categories is a conjecture of Soergel. I will discuss this conjecture and its proof in the case when on one side of the duality the group is quasi-split. This is joint work with Roman Bezrukavnikov.

Date

4月4日 April 4(Fri) 16:30--18:00, 2014

Room

Rims 402, Kyoto University

Speaker

Seok-Jin Kang氏 (Seoul National University)

Title

Cyclotomic categorification theorem and 2-representation theory

Abstract

　 The khovanov-Lauda-Rouquier algebras and their cyclotomic quotients provide categorification of the negativehalf of quantum groups and their integrable highest weight modules. We will discuss the motivation and basic ideas of these categorification theorems and possible future developments.

Most materials are based on the joint work with Masaki Kashiwara.

Date

2月14日 February 14(Fri) 16:30--18:00, 2014

Room

Rims 402, Kyoto University

Speaker

Myngho Kim氏 (KIAS)

Title

Symmetric quiver Hecke algebras and R-matrices for quantum affine algebras

Abstract

　 In this talk, I will introduce a family of functors between the category of finite-dimensional graded $R(n)-$modules and the category of finite-dimensional $U_q'(g)-$modules. Here, $R(n)$ is a symmetric quiver Hecke algebra and $U_q'(g)$ is a quantum affine algebra. We call these functors the ``quantum affine Schur-Weyl duality functors''. As an example, I will explain how one can lift the ring homomorphism discovered by Hernandez-Leclerc to a categorical level. This is a joint work with Seok-Jin Kang and Masaki Kashiwara.

Date

2月7日 February 7(Fri) 16:30--18:00, 2014

Room

Rims 402, Kyoto University

Speaker

池田岳氏 (岡山理科大)

Title

Pfaffian sum formula for the symplectic Grassmannians

Abstract

　 The classical Giambelli formula for the Grassmannian expresses a Schubert class as the determinant of a matrix whose entries are Chern classes of the universal quotient bundle. We seek for a Giambelli-type formula for the isotropic Grassmannians of a symplectic vector space. For the Lagrangian case, i.e. the case when the maximal dimensional isotropic subspaces are considered, P.Pragacz proved a formula which expresses a Schubert class as a single Pfaffian, which is nothing but Schur's Q-function. The torus equivariant analogue of Pragacz's formula has been proved by M. Kazarian, and myself in different context. The non-maximal and non-equivariant cases were studied by A.Buch, A.Kresch, and H.Tamvakis. They proved a Giambelli-type formula for any Schubert class written in terms of Young's raising operators.

We study the non-maximal and equivariant cases. Our formula expresses any torus equivariant Schubert class as a *sum* of the Pfaffians whose entries are equivariantly modified Chern classes of the quotient bundle. As a corollary, we obtain a proof of E. Wilson's conjectural formula. This is joint work with Tomoo Matsumura.

Date

11月1日 November 1(Fri) 16:30--18:00, 2013

Room

Rims 402, Kyoto University

Speaker

廣惠一希氏 (城西大学)

Title

Riemann球面上の線型微分方程式と箙の表現

Abstract

　 Crawley-BoeveyはFuchs型微分方程式と星型箙の表現が対応 することを用いて、加法的Deligne-Simpson問題を解決した。 これを拡張して不確定特異点を持つ方程式と箙の表現との対応が Boalchによって特別な場合に与えられ、さらに山川大亮氏（東工大） と講演者との共同研究で一般化された。
これらを元にして、本講演では加法的Delinge-Simpson問題を 不確定特異点をもつ微分方程式に対して良い条件下で定式化し、 微分方程式と箙の表現との対応のこの問題への応用についてお話ししたい。

Date

10月11日 October 11(Fri) 16:30--18:00, 2013

Room

Rims 402, Kyoto University

Speaker

山川大亮氏 (東京工業大学)

Title

有理型接続のモジュライ空間と箙多様体

Abstract

　 本講演では，Boalchによってアナウンスされ， 廣惠一希氏（城西大学）との共同研究によって証明が得られた， ある種の（射影直線上定義された）有理型接続のモジュライ空間が 特別な場合に箙多様体と複素シンプレクティック多様体として 同型になるという結果を紹介する． これは対数型接続に関するCrawley-Boeveyの結果を拡張するものであり， 現れる箙はより複雑なものになる． また時間が許せば，関連する話題として モノドロミー保存変形のWeyl群対称性についても触れる．

Date

10月4日 October 4(Fri) 16:30--18:00, 2013

Room

Rims 402, Kyoto University

Speaker

Anne Moreau氏 (Universit\'{e} de Poitiers)

Title

The arc space of spherical varieties and motivic integration.

Abstract

　 In this talk, we will be interested in the motivic integral over the arc space of a complex Q-Gorensein spherical G-variety X where G is a reductive connected group. We gave a formula for the stringy E-function of X in term of its colored fan, which generalizes that of Batyrev for the toric case. As an application, we obtain a smoothness criterion for locally factorial horospherical varieties and we conjecture that this criterion still holds for any locally factorial spherical variety. All this is based on joint works with Victor Batyrev.

Date

7月26日 July 26(Fri) 15:30--17:00, 2013

Room

Rims 402, Kyoto University

Speaker

Toshiro Kuwabara氏 (Higher School of Economics)

Title

BRST cohomologies for rational Cherednik algebras

Abstract

　 Quantization of Kleinian singularities can be realized as two different quantum Hamiltonian reductions. They are known as rational Cherednik algebras (symplectic reflection algebras) and finite W-algebras. Losev showed that these two quantizations are isomorphic by using realization of these algebras in terms of deformation-quantization. One can define a cohomology theory associated with Hamiltonian reduction, which is known as BRST cohomologies. In this talk, we see that higher BRST cohomologies corresponding to the rational Cherednik algebras do not vanish, while ones corresponding to the finite W-algebras vanish. Moreover, we see that the higher BRST cohomologies can be determined explicitly. To determine the higher cohomologies, we use the realization as deformation-quantization algebras and affinity properties of these sheaves of deformation-quantization algebras.

Date

7月19日 July 19(Fri) 16:30--18:00, 2013

Room

Rims 402, Kyoto University

Speaker

奥村将成氏(東大数理）

Title

頂点代数と同変リー亜代数コホモロジー

Abstract

　 Lian-Linshaw は，Malikov-Schechtman-Vaintrob が導入したカイラルドラーム 複体を可微分多様体の場合に詳しく取り扱った．その後 Lian-Linshaw-Song は， カイラルドラーム複体のある部分複体を用いて，リー群の作用を持つ可微分多様 体の同変コホモロジーの頂点代数類似物を構成した．本講演では，彼らが用いた 複体を，リー亜代数を用いて一般化し，同変リー亜代数コホモロジーの頂点代数 類似物を構成する．同変リー亜代数コホモロジーは，リー群の作用を持つ可微分 多様体の同変コホモロジーだけでなく，同変ポアソンコホモロジーも含む概念で あり，その頂点代数類似物も得られている．また，ある特別な複体を導入し，そ の性質を調べる．そこで得られた性質を用いて，変形リー亜代数と呼ばれる，リ ー環の多様体上の無限小作用を反映して得られるリー亜代数に対し計算を行う

Date

6月21日 June 21(Fri) 14:45--16:15, 2013
[attention]お時間が変更になりました

Room

Rims 402, Kyoto University

Speaker

井上玲氏(千葉大)

Title

クラスター代数と結び目の複素体積

Abstract

　 クラスター代数を用いて結び目の複素体積を定式化 する方法を紹介する。 特に、クラスター代数の特徴的な操作であるmutationを使ってR 作用素を構成し、 結び目補空間の理想四面体分割を調べる。 本講演は樋上和弘氏(九州大学)との共同研究に基づく。

Date

5月31日 May 31(Fri) 16:30--18:00, 2013

Room

Rims 402, Kyoto University

Speaker

森真樹氏(東大・数理)

Title

非整数次数のHecke代数の表現圏とセルラー構造

Abstract

　 Hecke代数のスーパー化であるHecke-Cliffordスーパー代数のモジュラー既約表 現は、 BrundanとKleshchev及び土岡によりLie代数の圏化を用いて分類された。 一方講演者は一般化されたセルラー代数の構造を用いて、より具体的かつ初等的 に これらの既約表現を構成することに成功した。この証明の中で、 「非整数次数のHecke代数の表現圏」が有効に使われたのでそれを紹介したい。 これはDeligneが構成した、自然数とは限らないtに対する「t次対称群の表現圏」 の自然な拡張である。

Date

5月22日(Wed)(May 22) 14:45--16:15, 2013
[attention]いつもとお時間が違います

Room

Rims 006, Kyoto University
[attention]いつもとお部屋が違います

Speaker

中筋麻貴氏(上智大)

Title

Hecke algebraとIwahori fixed vector

Abstract

　 $p$-進群の不分岐主系列表現のintertwining 作用素の明示公式を得るために， Hecke algebraを用いる手法がRogawski(1985)によって報告されている． Rogawskiの目的はHecke algebraの既約表現の分類であったが，これは Casselman基底にIntertwining作用素を作用させたIwahori fixed vectorをもつ$p$-進群の既約表現の分類と同値である． 本講演では，Rogawskiのアイデアから得られるいくつかのIwahori部分群の特性関数に関する結果と， これらを応用することによって得られるIwahori fixed vectorの基底の明示公式について話す．

Date

4月26日(Fri)(April 26) 16:30--18:00, 2013

Room

Rims 402, Kyoto University

Speaker

岡田　崇 氏 (小山高専)

Title

Quintic periods and stability conditions via homological mirror symmetry

Abstract

　 For the Fermat quintic Calabi-Yau threefold and the theory of stability conditions [Bri07], there have been two natural aims. One is that we should define central charges of stability conditions by quintic periods involving Gamma functions [CdGP] without losing quantum corrections. The other is that for well-motivated stability conditions on a derived Fukaya-type category, stable objects should be Lagrangians. For the Fermat quintic Calabi-Yau threefold, we discuss these aims with the simplest homological mirror symmetry in [Oka09,FutUed], taking advantages of derived categories of representations of tensor products of quivers.

Date

4月19日(Fri)(April 19) 16:30--18:00, 2013

Room

Rims 402, Kyoto University

Speaker

尾角正人氏 (大阪市立大学)

Title

PBW bases of the nilpotent subalgebra of U_q(g) and quantized algebra of functions

Abstract

　 For a finite-dimensional simple Lie algebra g, let U^+_q(g) be the positive part of the quantized universal enveloping algebra, and A_q(g) be the quantized algebra of functions. We show that the transition matrix of the PBW bases of U^+_q(g) coincides with the intertwiner between the irreducible A_q(g)-modules labeled by two different reduced expressions of the longest element of the Weyl group of g. This generalizes the earlier result by Sergeev on A_2 related to the tetrahedron equation and endows a new representation theoretical interpretation with the recent solution to the 3D reflection equation for C_2. Our proof is based on a realization of U^+_q(g) in a quotient ring of A_q(g).

This is a joint work with Atsuo Kuniba and Yasuhiko Yamada.

Date

2月22日(Fri)(February 22) 16:30--18:00, 2013

Room

Rims 402, Kyoto University

Speaker

Weiqiang Wang氏 (Virginia)

Title

The structures of the centers of Hecke algebras

Abstract

　 We will establish a precise connection between the centers of Hecke algebras associated to the symmetric groups and the ring of symmetric functions, quantizing the classical Frobenius characteristic map. This leads to an answer to a question of Lascoux on identification of several remarkable bases of the centers with bases of symmetric functions. In addition, we will describe a remarkable filtered algebra structure on such a center, which in its classical limit is intimately related to the cohomology ring of Hilbert scheme of points on the affine plane. This is based on joint work with Jinkui Wan (Beijing) and Andrew Francis (Sydney).

Date

2月8日(Fri)(February 15) 16:30--18:00, 2013

Room

Rims 402, Kyoto University

Speaker

Alexander Premet氏 (Manchester)

Title

Derived subalgebras of centralizers and completely prime primitive ideals.

Abstract

　 Let g be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic 0. In my talk I am going to explain how to classify the primitive ideals I of U(g) whose associated variety occurs with multiplicity 1 in the associated cycle AC(I). The classification is based on the detailed study of the abelian quotients g_e/[g_e, g_e] where g_e is the centraliser of a nilpotent element e in g.

Date

2月1日(Fri)(February 1) 16:30--18:00, 2013

Room

Rims 402, Kyoto University

Speaker

柏原正樹氏 (京大・数理研)

Date

1月18日(Fri)(January 18) 16:30--18:00, 2013

Room

Rims 402, Kyoto University

Speaker

Simon Goodwin氏 (Birmingham)

Title

Representation theory of finite W-algebras

Abstract

　 There has been a great deal of recent research interest in finite W-algebras motivated by important connection with primitive ideals of universal enveloping algebras and applications in mathematical physics.
There have been significant breakthroughs in the rerpesentation theory of finite W-algebras due to the research of a variety of mathematicians. In this talk, we will give an overview of the representation theory of finite W-algebras focussing on W-algebras associated to classical Lie algebras (joint with J. Brown) and W-algebras associated to general linear Lie superalgebras (joint with J. Brown and J. Brundan).

Date

12月7日(Fri)(December 7) 16:30--18:00, 2012

Room

Rims 402, Kyoto University

Speaker

名古屋創氏 (神戸大)

Title

Quantum Painleve systems from hypergeometric integrals of Euler type

Abstract

　 Euler 型積分表示から, Lie 環 sl_n に付随する超幾何積分の系列を定め, この系列の満たす Schroedinger 方程式として, 量子パンルヴェ系 （モノドロミー保存変形の量子化）が得られるという予想について話す.
この話は, 共形場理論の Knizhnik-Zamolodchikov 方程式が超幾何積分の 系列を解として持つが、逆に超幾何積分の系列から KZ 方程式が復元する という話(Schechtman-Varchenko, Looijenga)の類似である.
KZ 方程式と Euler 型積分表示から得られる量子パンルヴェ系との 関係は, Lie 環が sl_2 のときには分かっている(N).
講演では, 例として Gauss の超幾何から量子 PVI を導出する方法, 量子 PVI と KZ 方程式や BPZ 方程式との関係について話した後, 一般の場合の予想と具体例について話す.

Attention

今回11/30は2コマ講演でいつもより開始時間が早くなっていますのでご注意ください

Date

11月30日(Fri) (November 30) 14:45--16:15, 2012

Room

Room 402 of RIMS, Kyoto University

Speaker

Satoshi Naito (Tokyo Institute of Technology)

Title

Quantum Lakshmibai-Seshadri paths and Ram-Yip's combinatorial formula for Macdonald polynomials

Abstract

　 First, I will explain Ram-Yip's combinatorial formula for Macdonald polynomials, which is described in terms of the so-called alcove walks. Then, I will explain what happens in this formula when we specialize the parameter "t" to $0$. Finally, I will mention the relation between the specialized Macdonald polynomials above and the graded characters of tensor products of level-zero fundamental representations, which can be described in terms of quantum Lakshmibai-Seshadri paths.

Date

11月30日(Fri) (November 30) 16:30--18:00, 2012

Room

Room 402 of RIMS, Kyoto University

Speaker

Christian Kassel (Strasbourg)

Title

Drinfeld twists and finite groups

Abstract

　 Drinfeld twists were introduced by Drinfeld in his work on quasi-Hopf algebras. In joint work with Pierre Guillot (published in IRMN 10 (2010), 1894-1939), after observing that the invariant Drinfeld twists on a Hopf algebra form a group, we determine this group when the Hopf algebra is the algebra of a finite group. The proofs use quantum group techniques and Tannakian theory.

Date

11月16日(Fri)(November 16) 16:30--18:00, 2012

Room

Rims 402, Kyoto University

Speaker

直井克之氏(東大・Kavli IPMU)

Title

BC型量子ループ代数のminimal affinizationについて

Abstract

　 同じ最高ウェイトを持つ量子ループ代数$U_q(L\mathfrak{g})$の有限次元加群の 中で、（適当な半順序に関して）極小なものをminimal affinizationと呼ぶ。
量子ループ代数の有限次元加群について、その有限型部分代 数$U_q(\mathfrak{g})$加群構造を調べることは、古典極限を調べることに帰着 される。
本講演ではBC型のminimal affinizationについて、その古典極限を少し変形して 得られる次数付き極限を用いることで、 指標公式などが得られることについて述べたいと思う。

Date

6月29日(Fri)(June 29) 16:30--18:00, 2012

Room

Rims 402, Kyoto University

Speaker

廣惠一希氏(京大・数研)

Title

行列型１階線形常微分方程式と箙の表現

Abstract

　 W. Crawley-Boeveyは確定特異点型１階行列線形常微分方程式と箙の表現との対 応を発見し、 方程式を既約に実現しうる留数行列の共約類を決定した（加法的Deligne- Simpson問題）。
さらにこの対応は不確定特異点を1点のみ許した方程式に P. Boalchによって拡張されている（正確には高々極の位数３の不分岐不確定特 異点）。
本講演では極の位数が一般の不分岐不確定特異点を任意個許した微分方程式を考え、 これと箙の表現との対応を与える。 またこの応用として、方程式の既約性と箙のルート系との関係を述べたい。

Date

6月22日(Fri)(June 22) 16:30--18:00, 2012

Room

Rims 402, Kyoto University

Speaker

水川裕司氏(防衛大)

Title

環積が作用する確率モデルと多変数 Krawtchouk 多項式

Abstract

　 多変数 Krawtchouk 多項式は R. C. Griffiths によって1971年に定義された離 散直交多項式である． 表現論的な枠組では，この直交多項式は複素鏡映群のなすゲルファントペアの帯 球関数として 得られることが知られている．また，最近特殊関数論サイドからの研究とし て，Grunbaum と Rahman によって直交性を与える必要十分条件が考えられたり，Ilievにより差分方程式 へのLie環論からのアプローチなどが行われている．
本公演では，この多項式によって記述されるいくつかの確率モデルの表現論的な 解釈を試みる． 具体的には Rahamn と Hoare によって考案された Poker dice ゲームや，古典 的な Ehrenfest の拡散モデルを取り上げたい．

Date

6月8日(Fri)(June 8) 16:30--18:00, 2012

Room

Rims 402, Kyoto University

Speaker

柳田伸太郎氏(RIMS)

Title

On Hall algebra of complexes

Abstract

　 The topic of my talk is the Hall algebra of 2-periodic complexes, which is recently introduced by T. Bridgeland. I will discuss its properties and relation to auto-equivalences of derived category. I shall also mention the connection of this theory and the notion of stabilities.

Date

6月1日(Fri)(June 1) 16:30--18:00, 2012

Room

Rims 402, Kyoto University

Speaker

和田 堅太郎氏(信州)

Title

cyclotomic q-Schur 代数の Drinfeld 型の表示について

Abstract

　 cyclotomic q-Schur 代数は，Ariki-Koike 代数の quasi-hereditary cover の１つである。講演では，cyclotomic q-Schur 代数 （加算無限個の)生成元とその間の関係式を与え，（可能な限り) その表現論へ の応用をお話ししたいと思います。

Date

4月20日(Fri)(April 20) 16:30--18:00, 2012

Room

Rims 402, Kyoto University

Speaker

Masaki Kashiwara氏 (RIMS)

Title

Parameters of quiver Hecke algebras

Abstract

　 Varagnolo-Vasserot and Rouquier proved that, in a symmetric generalized Cartan matrix case, the simple modules over the quiver Hecke algebra with a special parameter correspond to the upper global basis.

In this talk I will show that the simple modules over the quiver Hecke algebras with a generic parameter also correspond to the upper global basis in a symmetric generalized Cartan matrix case.

Date

4月13日(Fri)(April 13) 16:30--18:00, 2012

Room

Rims 402, Kyoto University

Speaker

Sarah Scherotzke氏 (Univ. Bonn)

Title

Linear recurrence relations for cluster variables

Abstract

　 In a recent paper by Asssem Reitenaure and Smith, frieze sequences were associated to acyclic quiver. They are a natural generalization of the Coxeter-Convey frieze pattern. Using categorification of cluster algebras, we show that frieze sequences associated to acyclic quivers satisfy linear recurrence relations if and only if the quiver is an affine quiver.

Date

2月10日(Fri)(Februrary 10) 16:30--18:00, 2012

Room

Rims 402, Kyoto University

Speaker

小寺諒介氏(東大・数理)

Title

Self-extensions and prime factorizations for simple $U_q(L\mathfrak{sl}_2)$-modules

Abstract

　 In the category of finite-dimensional modules over a quantum loop algebra, it often occurs that a simple module is factorized into a tensor product of smaller simple modules. Hence the notion of prime simple module (a simple module which does not admit a nontrivial factorization) was introduced and prime factorizations of simple modules have been studied. For the quantum loop algebra of $\mathfrak{sl}_2$, prime simple modules exactly coincide with the evaluation modules.

Chari-Moura-Young proposed in a recent paper arXiv:1112.6376 a study of prime simple modules from a homological point of view. They conjectured that a simple module is prime if and only if it has the one-dimensional self-extension group, and proved it in the $\mathfrak{sl}_2$ case.

In this talk, I will review Chari-Moura-Young's paper and explain that my previous result on nonself-extensions can be applied to refining their result. It establishes a relation between the dimension of the self-extension group and the number of factors in the prime factorization for a simple $U_q(L\mathfrak{sl}_2)$-module.

Date

2月3日(Fri)(Februrary 3) 16:30--18:00, 2012

Room

Rims 402, Kyoto University

Speaker

疋田辰之氏(京大・理)

Title

A型アファインシュプリンガーファイバーとdiagonal coinvariantの組み合わせ論

Abstract

　 Diagonal coinvariant ringのbigraded Frobenius seriesに関してHaglund, Haiman, Loehr, Remmel, Ulyanovはそれを記述する組み合わせ論的公式を予想した。講演ではこの公式がA 型のアファインシュプリンガーファイバーのホモロジーを用いることで幾何的に 現れることを説明したい。

Date

12月16日(Fri)(December 16) 15:00--18:00, 2011

Room

Rims 402, Kyoto University

Speaker

木村嘉之氏(京大・数研)

Title

次数付き箙多様体と量子クラスター代数

Abstract

　 クラスター代数の正値性予想とは、任意のクラスター変数の任意の 種におけるローラン展開(クラスター展開)に関する正 値性に関する予想である。 今回、非輪状の箙を種として含むようなクラスター代数における正 値性予想が解決された。 中島啓氏によるbipartite quiverに付随する(量子) クラスター代数のモノイダル圏論化の証明の手法に従い、 「非輪状型(acyclic quiver)に適合した」次数付き(アフィン) 箙多様体上の同変偏屈層の量子表現環による量子クラ スター代数の実現により証明される。 講演では、(クラスター代数のモノイダル圏論化の中心的な問 題である)(量子)クラスター単項式が``双対標準基底'' に含まれることを説明したい。 本研究は、パリ第７大学のFan Qin氏との共同研究に基づく。

Date

12月2日(金)(December 2) 16:30--18:00, 2011

Room

Rims 402, Kyoto University

Speaker

中島啓氏(京大・数研)

Title

Coproduct on Yangian

Abstract

　 Consider the Yangian $Y$ associated with an affine Lie algebra, which is not of type $A^{(1)}_1$ nor $A^{(2)}_2$. We define a coproduct $\Delta$, which takes value in a certain completion of $Y\otimes Y$. This is a work in progress, with Nicolas Guay.

Date

11月18日(金)(November 18) 16:30--18:00, 2011

Room

Rims 402, Kyoto University

Speaker

加藤周氏(京大・理)

Title

グリーン関数のホモロジー論的側面について

Abstract

　 グリーン関数は簡約群の冪単指標の一般化として得られる直交関数系であり、二 つの複素鏡映群の既約指標の組を添字にもつ。この話では一般の複素鏡映群(と 良い付加データ)に対してコストカ系と呼ぶ(一般には存在するかどうか分からな いが、一旦存在するとよい性質を満たす)加群の族を導入し、(標数が良い時に) 簡約群の冪単指標に付随するグリーン関数は常にコストカ系として実現される事 を説明する。この事は特に任意の簡約群に付随するグリーン関数が直既約加群の 次数付き指標という解釈を許す事を意味する。この解釈を用いると小ワイル群を BC型として等しくする任意の簡約群の冪単指標に付随するグリーン関数達がどの ように互いに移りあうか等も見る事ができる。

Date

11月11日(金)(November 11) 16:30--18:00, 2011

Room

Rims 402, Kyoto University

Speaker

阿部紀行氏(北大・創成研究機構)

Title

対称空間のコンパクト化によるJacquet加群の実現

Abstract

　 Beilinson-Bernstein対応は，実半単純Lie群の表現の旗多様 体上における幾何学 的な実現を与えるが，一方で対称空間上での幾何学的な実現も与え る．三枝洋一 氏との共同研究により，対称空間の境界に向けてうまく極限をとる と，Jacqeut 加群が実現されることがわかったので，それについて話をする．

Attention

いつもとは時間・場所が異なりますのでご注意ください。

Date

10月31日(月), 11月1日(火),2日(水),4日(金)
(October 31, November 1, 2, 4 (4days)) 13:15--14:15, 2011

Room

Room 475 of Research Bldg. No.2 , Kyoto University
http://www.kyoto-u.ac.jp/en/access/campus/main.htm

Speaker

Ivan Losev氏 (Northeastern)

Title

Finite W-algebras

Abstract

　 Finite W-algebras are associative algebras that can be thought as generalizations of universal enveloping algebras of semisimple Lie algebras. Each W-algebra is constructed from a pair of a semisimple Lie algebra and its nilpotent orbit. These algebras first appeared in the work of Kostant in the late 70's in a special case. In the whole generality they were defined by Premet in the beginning of 2000's.

In my lectures I am going to emphasize connections between W-algebras and universal enveloping algebras. I will start by giving two definitions of W-algebras, one due to Premet and one due to myself. Then I will introduce various functors between the representation categories for W-algebras on one side and semisimple Lie algebras on the other side. Using this functors I will explain an interplay between primitive ideals in the universal enveloping algebras and irreducible finite dimensional modules for W-algebras.

Date

10月28日(金) (October 28) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

大島芳樹氏(東大・数理)

Title

コホモロジカル誘導の局所化

Abstract

　 コホモロジカル誘導は(g,K)-加群に対して代数的に定義さ れ、半単純リー群の離 散系列表現、主系列表現(のHarish-Chandra加 群)、 Zuckerman加群などを生成する。 Borel部分代数の1次元表現からの誘導の場合、誘導された表 現は旗多様体上のD 加群を用いて実現できることが、Hecht, Milicic, Schmid, Wolfにより示されている。 講演では、より一般の表現からの誘導についてこの結果を拡張する ことを考える。

Date

10月21日(金) (October 21) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

市野篤史氏(京大・理)

Title

形式次数とテータ対応

Abstract

　 局所テータ対応とは, (p進体上の)古典群の(複素数係数)表現から, 別の古典群 の表現を, ある種の分岐則を用いて構成する方法である.
この構成の下での, 表現の解析的不変量の振る舞いについて述べ, これを表現の 分類(局所Langlands対応)を用いて, 数論的に解釈する. この講演は, Wee Teck Gan氏との共同研究に基づく.

Date

10月14日(金) (October 14) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

Leonid Rybnikov氏　(HSE)

Title

Quantization of Quasimaps' Spaces (joint work with M. Finkelberg)

Abstract

　 Quasimaps' space $Z_d$ (also known as Drinfeld's Zastava space) is a remarkable compactification of the space of based degree d maps from the projective line to the flag variety of type A. The space $Z_d$ has a natural Poisson structure, which goes back to Atiyah and Hitchin. We describe the Quasimaps' space as some quiver variety, and define the Atiyah-Hitchin Poisson structure in quiver terms. This gives a natural way to quantize this Poisson structure. The quantization of the coordinate ring of the Quasimaps' space turns to be some natural subquotient of the Yangian of type A. I will also discuss some generalization of this result to the BCD types.

Date

7月22日(金) (July 22) 15:00--16:30, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

土岡俊介 (Shunsuke Tsuchioka)氏　(IPMU)

Title

Quiver Hecke superalgebras

Abstract

　 We introduce two families of superalgebras $R_n$ and $RC_n$ which are weakly Morita superequivalent each other. The quiver Hecke superalgebra $R_n$ is a generalization of the Khovanov-Lauda-Rouquier algebras. We show that, after suitable specialization and completion, the quiver Hecke-Clifford superalgebra $RC_n$ is isomorphic to the affine Hecke-Clifford superalgebras and its rational degeneration.
This is a joint work with Seok-Jin Kang and Masaki Kashiwara.

Date

7月8日(金) (July 8) 15:00--16:30, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

荒川 知幸氏 (RIMS)

Title

Localization of affine W-algebras at the critical level

Abstract

　 We localize the simple affine W-algebras at the critical level on the infinite jet schemes of Slodowy varieties, by introduction a chiral analogue of the Kashiwara-Rouquier deformation quantization algebra.

This is a joint work with Toshiro Kuwabara and Fyodor Malikov.

Attention

今回はいつもより開始時間が早めになり、
講演時間も長くなっていますのでご注意ください

Date

7月1日(金) (July 1) 13:30--15:00, 15:15--16:45, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

山川大亮氏（神戸大）

Title

Fourier-Laplace変換とKatz-Deligne-Arinkinの定理 その2

Abstract

　 6/17に同セミナーにて行った講演では， 射影直線上の有理型接続に対するFourier-Laplace変換を ある条件下で接続の係数行列に対する変換として初等的に書き下し， それを利用したKatz-Deligne-Arinkinの定理（genericな場合） の別証明を駆け足で紹介した．
今回はもう少し踏み込んで，Fourier-Laplace変換が誘導する 接続の（ナイーヴな）モジュライ空間の間のシンプレクティック同型写像や， モジュライ空間と箙多様体との関係等，幾何学的側面も紹介したい． なお，前回の講演の内容は仮定せず話をする．

Date

6月24日(金) (June 24) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

笹木集夢氏（東海大）

Title

An application of the classification of visible linear actions to nilpotent orbits

Abstract

　 小林俊行氏によって導入された複素多様体における可視的作用という概念は， 無重複表現の統一理論において重要な役割を果たすことが近年明らかになってき ている． 作用が線型な場合は可視的作用の分類は与えられ， それはKacやBenson-Ratcliff, Leahyによるmultiplicity-free作用の分類に一致する． さらに最近，線型な可視的作用の分類を用いることで， 複素リー環の冪零軌道における(線型でない)作用が 可視的であることと冪零軌道がsphericalであることが同値であることが分かった．

本講演では，線型な可視的作用の研究結果について概説した後， この結果を冪零軌道における可視的作用の研究に応用する様子を解説する予定で ある．

Attention

!今回6/17は2コマ講演でいつもより開始時間が早めですのでご注意ください!
第1報から時間が変更になりました. また,1コマ目はいつもと部屋が異なります.

Date

6月17日(金) (June 17) 11:30--13:00, 2011

Room

Room 204 of RIMS, Kyoto University

Speaker

山川大亮氏（神戸大）

Title

Fourier-Laplace変換とKatz-Deligne-Arinkinの定理

Abstract

　 Katz-Deligne-Arinkinの定理は，rigidと呼ばれる性質を満たす射影直線上の有 理型接続を， 座標変換・階数1の有理型接続によるテンソル積・Fourier-Laplace変換，の3つ の操作を有限回繰り返す事によって， 必ず階数1の有理型接続にする事ができると主張する． これはもともとKatzによって有理型接続が確定特異点のみを持つ場合に示され （Katzの定理）， 後にDeligneとArinkinによって不確定特異点の場合に拡張された．

この講演では，Dettweiler-ReiterによるKatzの定理の別証明が， 自然な形で不確定特異点の場合（ただしgenericな仮定を課す）に拡張される事 を紹介する．
特に，Dettweiler-Reiterの議論では明示されていないFourier-Laplace変換の具 体的な記述に焦点を当てる．

Date

6月17日(金) (June 17) 14:45--16:15, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

廣恵一希氏（東京大）

Title

線形常微分方程式とルート系

Abstract

　 直既約な箙の表現とルートとの対応を示すKacの定理は箙の表現論では基本的な 定理だが， Fuchs型の線形常微分方程式においてこのKacの定理の不思議なアナロジーが知ら れている． すなわち微分方程式に対してあるルート系とルート格子の元が決まり，方程式の 既約性とルートの条件が対応する． さらに微分方程式のある種のモジュライ空間の次元がルートの長さによって決定 されるというのである． 本講演ではこの対応をFuchs型でない場合にも拡張することを目標として今まで に得られている結果を報告する．

Date

6月10日(金) (June 10) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

鈴木咲衣氏 （京大数理研）

Title

On the universal sl_2 invariant of bottom tangles

Abstract

　 Jones多項式の発見を機に量子トポロジーと呼ばれる分野が誕生し，現在までに急速 な発展を遂げてきた． まず単純リー環gの量子群とその表現を用いて定義される絡み目の「量子g不変 量」，次に量子群のみを用いて定義され，表現に関して量子g不変量に普遍性を 持つ「普遍量子g不変量」，さらにはリー環の関係式を用いて定義され，量子群 に関して量子g不変量に普遍性をもつ「Kontsevich不変量」の構成である．そし てその３つの段階の不変量に対応して，絡み目に沿った手術の理論を経由して３ 次元多様体の不変量が構成される．
この講演では「底タングル」を用いた普遍量子sl_2不変量の研究の枠組みを説明し， 講演者の結果を簡単に紹介する．

Attention

!今回5/27は2コマ講演でいつもより開始時間が早めですのでご注意ください!

Date

5月27日(金) (May 27) 14:30--16:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

榎本直也氏（京大理）

Title

曲面の写像類群に付随するJohnson余核のSp-加群構造について（佐藤隆夫 氏（東京理科大）との共同研究）

Abstract

　 境界を1つ持つ種数gの向き付けられたコンパクトリーマン面$\Sigma_{g,1}$の写像類 群$M_{g,1}$は、 $H_1(\Sigma_{g,1},Z)$に自明に作用するTorelli部分群とよばれる部分群を持ち、そ の商はSp(2g,Z)と同型になる。 Torelli部分群のJohnson filtrationの次数商を自由Lie代数の微分代数へ移すJohnson準同型は、 Torelli部分群の構造を調べるための重要な道具のひとつであり，言わばTorelli群の 近似物を記述していると考えられる。 その後、森田茂之氏によってJohnson準同型の像がある次数付き部分Lie代数$\mf{h}_ {g,1}$に埋め込まれることが示され、 その余核の次数k-部分（k：奇数）にSp-既約表現[k]が含まれることがわかった。こ れは森田障害と呼ばれている。 本講演では、自由群の自己同型群におけるJohnson準同型とその余核のGL（あるいはS p）-構造をもとに、 写像類群のJohnson余核に現れる既約成分のあるクラスについて紹介し、 具体的に、Sp-既約表現[1^k]が次数k-部分（$k \equiv 1 (mod 4）$に重複度1で現れることを述べたい。

Date

5月27日(金) (May 27) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

高尾尚武氏（京大数理研）

Title

Johnson準同型と外Galois表現

Abstract

　 Johnson準同型の定義は一般の双曲型Riemann面に対して一般化されます。 その余核の次数k-部分（$k \equiv 2 (mod 4）$には有理数体上の絶対Galois群が「現れる」ことが、 織田孝幸氏によって予想され、現在ほぼ解決されています。 Deligne-伊原予想の解決を合わせると、より精確な定量的な評価も可能になりま した。
講演では、織田予想を中心に、Johnson準同型の余核をめぐる数論側からの進展につ いて紹介します。

Date

5月20日(金) (May 20) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

飯島和人氏 (名古屋大学)

Title

A COMPARISON OF q-DECOMPOSITION NUMBERS IN THE q-DEFORMED FOCK SPACES OF HIGHER LEVELS

Abstract

　 The q-deformed Fock spaces of higher levels were introduced by Jimbo-Misra-Miwa-Okado. The q-decomposition matrix is a transition matrix from the standard basis to the canonical basis defined by Uglov in the q-deformed Fock space. In this talk, we show that parts of q-decomposition matrices of level $\ell$ coincides with that of level $\ell$ － 1 under certain conditions of multicharge.

(This talk will be given in Japanese, but the slides in English.)

Date

5月13日(金) (May 13) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

Anatol Kirillov (RIMS)氏**

Title

Saga of Dunkl elements.

Abstract

　 Dunkl operators has been introduced in the middle of 80's by Charles Dunkl to solve certain problems in the theory of orthogonal polynomials. Later it was observed a close connection of Dunkl operators with the theory integrable systems, as well as construction of different kinds of generalizations. Connection of (truncated) Dunkl operators with the coinvariant algebra of a finite Coxeter group has been observed by C. Dunkl and clarified by Y. Bazlov.
In my talk I introduce a certain quadratic algebra and a distinguish set of mutually commuting elements in it (Dunkl elements). It appears that different kind of Dunkl operators (rational, trigonometric, elliptic, multiplicative,...) are images of the Dunkl elements in the corresponding representation of the quadratic algebra in question. The main goal of my talk is to relate the algebra generated by Dunkl elements with generalized cohomology theories of complete flag varieties of type A . Applications to other fields of Mathematics will be presented.

My talk partly is based on joint works with T.Maeno.

Date

4月22日(金) (April 22) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

直井克之(Katsuyuki Naoi)氏 (東大数理)

Title

Weyl module, Demazure moduleとfundamental representationのテンソル積のcrystal basisとの関係について

Abstract

　 Weyl moduleは生成元と関係式によって定義されるcurrent algebra（単純リー代 数と多項式環のテンソル積で定義される無限次元リー代数）の有限次元表現である。 一方fundamental representationはcrystal basisを持つ重要なquantum affine algebraの有限次元表現である。
これらは一見それほど関係がなさそうであるが、Demazure加群およびその crystalにおける対応物(Demazure crystal)を用いることで二つの対象の間に非 常に強い関係があることを示すことができる。今回の公演ではこの結果について 紹介する。
また、上で述べた結果とX=M予想との関係についても紹介する。ここでX=M予想と は、1-dimensional sumと呼ばれる有限crystalのテンソル積から定義される多項 式とfermionic formulaと呼ばれる多項式が一致する、という予想である。

Date

4月15日(金) (April 15) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

有家雄介(Yusuke Arike)氏(大阪大)

Title

Pseudo-trace functions for orbifold models associated with symplectic fermions

Abstract

　 頂点作用素代数Vに付随する楕円曲線上の一点関数は, Vの元に上半平面上の正則 関数を対応させるある性質をもつ写像です. 頂点作用素代数VがC_2有限かつ有理 的(加群の圏が半単純性であること)であるとき, 一点関数の空間はVの単純加群 上のtrace functionと呼ばれるもので張られることがZhuにより示されています. またtrace functionの真空ベクトルでの値は加群の指標と一致します.

有理性を仮定しない頂点作用素代数に付随する一点関数の空間は, pseudo-trace functionと呼ばれるもので生成されることが示されています.しかしpseudo- trace functionの定義は高次のZhu代数と呼ばれる結合代数を用いるもので, 具 体的な例に対して一点関数を構成することは非常に困難です.

本講演ではまず, Zhu代数を用いずにpseudo-trace functionを定義する方法を解 説し,次にsymplectic fermionic頂点作用素超代数のeven partとして得られる頂 点作用素代数の直既約加群上のpseudo-trace functionを構成します.さらに得ら れたpseudo-trace functionの真空ベクトルでの値を調べます. その結果として, 一点関数の真空での値で定義される一般化された指標の空間と一点関数の空間の 次元が異なる場合があることを示します.

Date

4月8日(金) (April 8) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

柏原正樹(Masaki Kashiwara)氏 (RIMS)

Title

Cyclotomic quiver Hecke algebras and categorifications of highest weight module

Date

4月1日(金) (April 1) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

加藤周(Syu Kato)氏 (Kyoto)

Title

Harish-Chandra bimodules for quantized Slodowy slices (survey)

Abstract

　 Ginzburgの同名の論文[Represent. Theory 13 (2009), 236-271]の サーベイをします。

Date

3月25日(金) (March 25), 16:30--18:00, 2011

Room

Room 204 of RIMS, Kyoto University

Speaker

和田堅太郎(Kentaro Wada)氏 (RIMS)

Title

Induction and Restriction functors for cyclotomic q-Schur algebras.

Abstract

　 異なるランクの cyclotomic q-Schur 代数の有限次元加群のなす圏の間に関する 誘導, 制限関手を定義し，その性質を調べます。特に，Weyl 加群 (既約加群) に対する誘導，制限関手の性質を調べることによって (部分的にはまだ予想の段 階ですが) higher level の Fock 空間との関係を考えます。これは，Ariki- Koike 代数に対する LLT-有木理論の quasi-hereditary cover 版の１部分とな ります。　さらに，Rouquier による quasi-hereditary cover の同値定理に よって，cyclotomic q-Schur 代数の有限次元加群のなす圏が, 有理 Cherednik 代数の圏 O と同値になる場合には，今回の誘導，制限関手は，有理 Cherednik 代数に対する Bezrukavnikov-Etingof の誘導，制限関手と (cover の同値を通 じて) 同値な関手となり，Shan, Gordon-Martino によって得られている Fock 空間の圏化の "ドミナント版" が得られることになります。
講演では，これらの概略を話してみようと思います。

Date

March 16 (Wed) 14:00--（３時間程度）

Room

Room 204 of RIMS, Kyoto University

Speaker

中島 啓氏(RIMS)

Title

Maulik-Okounkovの理論の紹介 - 応用として、AGT予想の証明

Date

February 16 (Wed), 14:45-16:15, 2011

Room

Room 204 of RIMS, Kyoto University

Speaker

Kari Vilonen (Northwestern University)

Title

Langlands duality for real groups

Abstract

　 In the case of real groups Langlands duality acquires a symmetry as both sides can be interpreted as (derived) categories of representations. We explain this duality and its proof in the case of quasi-split groups. The result was also known as the Soergel conjecture. This is joint work with R. Bezrukavnikov.

Date

Feb. 9 (Wed) 14:45--16:15, 2011

Room

Room 204 of RIMS, Kyoto University

Speaker

岡田 聡一 (Soichi Okada)氏　(名大多元)

Title

Two-parameter deformation of multivariate hook product formulae

Abstract

　 The hook product formula due to Frame, Robinson, and Thrall gives the number of standard tableaux of a given shape, which is equal to the dimension of the irreducible representation of the symmetric group. The FRS hook product formula is obtained from Gansner's multivariate hook product formula for the trace generating function of reverse plane partitions. In this talk, we give another proof and a (q,t)-deformation of Gansner's formula by using operator calculus on the ring of symmetric functions. Also we present a conjectural deformation of Peterson-Proctor's hook product formula for P-partitions on d-complete posets.

Date

Jan. 28 (Fri) 14:30--16:15, 2011

Room

Room 204 of RIMS, Kyoto University

Speaker

柳田伸太郎氏(神戸大・理)

Title

A finite analog of the AGT relation (survey)

Abstract

　 Braverman-Feigin-Finkelberg-Rybnikovの仕事 (arXiv:1008.3655)のサーベイをしま す.特にshifted YangianのGelfand-Tsetlin基底を詳しく扱います.

Date

Jan. 28 (Fri) 16:45--18:00, 2011

Room

Room 204 of RIMS, Kyoto University

Speaker

柳田伸太郎氏(神戸大・理)

Title

Ding-Iohara algebra and K-theoretic AGT conjecture

Abstract

　 K理論的AGT予想はインスタントンのモジュライ空間の同変K理論に変形Virasoro代数( ないし変形W代数)が作用することを示唆します.講演ではDing-Iohara代数という量子 アフィン環の類事物とK理論的AGT予想(及び通常のAGT予想)の関係について述べます.

Date

Jan. 19 (Wed) 14:45--16:15, 2011

Room

Room 204 of RIMS, Kyoto University

Speaker

Seok-Jin Kang (Seoul National University)

Title

Quantum queer superalgebra and crystal bases.

Abstract

　 We will give a brief survey of recent developments in the crystal basis theory for the quantum queer superalgebra $U_q(q(n))$. The odd Kashiwara operators and 'queer' tensor product rule will be introduced. We will also discuss their combinatorial realization in terms of semistandard decomposition tableaux.

Date

Jan. 12 (Wed) 15:00--16:00, 2011
Jan. 13 (Thu) 11:00--12:00,13:30--14:30, 2011

Room

Jan.12:Room 204 of RIMS, Kyoto University
Jan.13:Room 110 of RIMS, Kyoto University

Speaker

Ian M. Musson（The University of Wisconsin-Milwaukee）

Title

Lie Superalgebras and Enveloping Algebras

Abstract

　 I will give 3 lectures mainly about enveloping algebras of classical simple Lie superalgebras. The second of these will concern the center of the enveloping algebra, and the third will contain material about primitive ideals. The first lecture will contain some background material.

Date

December 10 (Fri), 16:30--18:00, 2010

Room

Room 204 of RIMS, Kyoto University

Speaker

山田泰彦氏(神戸大・理)

Title

共形場理論、モノドロミー保存変形とAGT予想

Abstract

　 ４次元超対称ゲージ理論の分配関数と２次元共形場理論の相関関数が一致するという 興味深い現象(Alday-Gaiotto-Tachikawa予想）について、モノドロミー保存変形の 量子化の観点から考察する。応用として、ゲージ理論のある分配関数が満たすと期待 される微分方程式を定式化する。

Date

December 1 (Wed), 14:45--16:15, 2010
December 8 (Wed), 14:45--16:15, 2010

Room

Room 204 of RIMS, Kyoto University

Speaker

土岡俊介氏(京大・数理研)

Title

Shifted Yangians and finite W-algebras (survey)

Abstract

　 Brundan-Kleschevの同名の論文(Adv. Math. 200 (2006), 136--195, arXiv:math/040 7012)の内容を二回に分けてサーベイします。

Date

November 22 (Mon), 16:30--18:00, 2010

Room

Room 111 of RIMS, Kyoto University

Speaker

Ben Webster (Oregon)

Title

Hypertoric (and other) categories O
(joint w/ Braden, Licata and Proudfoot)

Abstract

　 The category O defined by Bernstein, Bernstein and Gelfand has been an active area of representation theory for over 30 years now. I'll explain how this construction is a special case of a more general picture, and explain how things like Koszulity, cells, the localization theorem, and the action by shuffling and twisting functors generalize.

A particularly well-developed special case is hypertoric category O, which arises from torus invariant differential operators on a vector space. In this case, we can find an analogue of almost any theorem about the Lie theoretic category O, though sometimes with subtle and interesting changes.

Perhaps most interestingly, results on Koszul duality in this picture point the way toward a notion of duality between certain symplectic singularities, as I will explain.

Date

November 17 (Wed), 14:45--16:15, 2010

Room

Room 204 of RIMS, Kyoto University

Speaker

Scott Carnahan (IPMU)

Title

Borcherds products in monstrous moonshine

Abstract

　 During the 1980s, Koike, Norton, and Zagier independently found an infinite product expansion for the difference of two modular j-functions on a product of half planes. Borcherds showed that this product identity is the Weyl denominator formula for an infinite dimensional Lie algebra that has an action of the monster simple group by automorphisms, and used this action to prove the monstrous moonshine conjectures.

I will describe a more general construction that yields an infinite product identity and an infinite dimensional Lie algebra for each element of the monster group. The above objects then arise as the special cases assigned to the identity element. Time permitting, I will attempt to describe a connection to conformal field theory.

Date

October 27 (Wed), 14:45--16:15, 2010

Room

Room 204 of RIMS, Kyoto University

Speaker

中島啓 (RIMS)

Title

Instanton and W-algebras (review)

Abstract

　 物理学者の Alday-Gaiotto-Tachikawa の最近の研究から、R⁴ 上のインスタントンのモジュライ空間の同変交叉ホモロジーに W-代数の表現の構造が入ることが期待されている。これについて、概観する。

Date

2010年10月13日(水曜日) 14:50--16:20

Room

数理解析研究所204号室

Speaker

木村嘉之氏(京大・数理研)

Title

量子ベキ単部分群と双対標準基底（Quantum Unipotent Subgroup and dual canonical basis）

Abstract

　 (量子)クラスター代数構造は、Berenstein-Fomin-Zelevinskyらによって、双対標準 基底の乗法的性質の研究のため導入された組み合わせ的な構造である。 Weyl群の元wに付随したKac-Moody群の冪単部分群N(w)の座標環のクラスター代数構造 は、Berenstein-Fomin-Zelevinskyらにより予想され、前射影多元環(preprojective algebra)に関するGeiss-Leclerc-Schr\"{o}erらによる研究により、双対準標準基底( dual semicanonical basis)との整合性が知られている。 Geiss-Leclerc-Schr\"{o}erらによる結果の量子化として、冪単部分群N(w)の座標環 の量子変形O_q[N(w)]には量子クラスター構造が存在し、双対標準基底との整合性が 予想される(量子化予想)。

本講演では、問題の背景と量子化予想とそのいくつかの帰結について述べ、量子化予 想の準備と言えるいくつかの結果について紹介する。 これらは、Calderoの有限ADE型の結果の一般化である。

Date & Room

2010年10月6日(水) 14:45--16:15 数理解析研究所204号室
2010年10月8日(金) 16:30--18:00 理学部３号館数学教室109号室

Speaker

荒川知幸氏

Title

W-algebras and their representations

Abstract

　 本講演では（アフィン)W代数とその表現論についてのreviewを行いたいと思います。