Representation Theory Seminar
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
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.
Place
July 20 (Thu), 10:30--12:00, 2023
Room
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).
Place
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
We investigate partition functions defined from integrals over the handsaw quiver varieties of type $A_{1}$ via wall-crossing phenomena. We consider vortex partition functions defined by two types of cohomology classes, and get functional equations for each of them. This gives proofs to formulas suggested by physical computations. In particular, we obtain geometric interpretation of formulas for multiple hypergeometric functions including rational limit of the Kajihara transformations formula. We also introduce analogous computations for integrals over Grassmannian manifolds, and more general settings for framed quiver representations.
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/4509)
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/en/event/seminar/4505)
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/en/event/seminar/4475)
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
Yoshinori Namikawa (RIMS)
Title
Universal coverings of nilpotent orbits and birational geometry
Abstract
The normalization of a nilpotent orbit closure of a complex semisimple Lie algebra is a symplectic variety. Its symplectic resolution or Q-factorial terminalization has been extensively studied. In this lecture, we take a symplectic variety associated with the universal covering of a nilpotent orbit and consider similar problems. When the Lie algebra is classical, we will give an explicit algorithm for constructing a Q-factorial terminalization of such a symplectic variety. Moreover, we can give an explicit formula how many different Q-factorial terminalizations it has.
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/en/event/seminar/4413)
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/en/event/seminar/4395)
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/en/event/seminar/4399)
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/en/event/seminar/4382 )
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/en/event/seminar/4366 )
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/en/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/en/event/seminar/4345 )
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/en/event/seminar/4331 )
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/en/event/seminar/4328 )
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
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
[CANCEL] 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 Ikuo Satake (Kagawa University)
Title
An approach to the invariant theory for the elliptic Weyl groups
Abstract
We have 3 kinds of generators of the invariant ring for the elliptic Weyl groups: (1) the fundamental characters of the affine Lie algebras, (2) the (generalized weak) Jacobi forms, (3) the flat invariants for the Frobenius structure. For the D4 case, we could construct (2) by (1) by a determinant of a matrix whose entries are fundamental characters. By these Jacobi forms, we could give the explicit description of the Frobenius structure, since the Frobenius structure has the modular invariance. Then we could construct (3) by (2). So we could construct (3) by (1) (arxiv:1708.03875). Afterwards we find a characterization of (3) by using the behavior on the fixed points of the modified Coxeter transformation for the elliptic root system. Combining these results, we give some new properties of the fundamental characters and the Weyl denominator for the affine Lie algebra of type D4.
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
Ikuo Satake (Kagawa University)
Title
see the Japanese page
Abstract
see the Japanese page
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
See the Japanese page
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
Taiki Shibata (Okayama University of Science)
Title
Quasireductive supergroups and their representations
Abstract
An algebraic supergroup is a group-valued functor on the category of commutative superalgebras represented by a finitely generated commutative Hopf superalgebra. It has been known that representations of algebraic supergroups can be applied in non-super (modular) representation theory. In 2011, V. Serganova introduced the notion of quasireductive supergroups as a super version of the notion of split reductive groups. This is an interesting and forms an important class of algebraic supergroups including Chevalley supergroups (introduced by R. Fioresi and F. Gavarini) and queer supergroups $Q(n)$ (whose Lie superalgebra is a queer superalgebra $q(n)$). She constructed irreducible representations of quasireductive supergroups over an algebraically closed field of characteristic zero, in terms of their Lie superalgebras. In this talk, I will explain a Hopf-algebraic approach to the study of quasireductive supergroups $G$, and give a category equivalence between rational $G$-supermodules and "locally finite" $hy(G)$-supermodules, where $hy(G)$ is a certain cocommutative Hopf superalgebra, called the super-hyperalgebra of $G$ (due to M. Takeuchi). As an application, we get (1) a generalization of Serganova's result to the case when the base field is arbitrary, and (2) a super-analogue of Kempf's cohomology vanishing theorem when $G$ has a "good" parabolic super-subgroup.
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
see the Japanese page
Abstract
see the Japanese page
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
Room 507, Maskawa Building for Eduation and Research
Speaker
Akishi IKEDA (IPMU)
Title
Double graded Ginzburg Calabi-Yau algebras, q-deformations of root lattices and q-stability conditions on their derived categories
Abstract
In this talk, first we introduce a double graded version of the Ginzburg's Calabi-Yau algebra for a quiver Q and show that the derived category of dg-modules over this algebra gives a categorification of the q-deformed root lattice associated with the quiver Q. It is also shown that the action of Seidel-Thomas spherical twists on the q-deformed root lattice factors the Hecke algebra of Q. In the case that Q is of type affine ADE, this construction describes the derived category of C^*-equivariant coherent sheaves on the corresponding Kleinian singularity and also relates to the graded preprojective algebra. Next we introduce a q-stability condition on this derived category which is a Bridgeland stability condition with the additional axiom. We state the conjecture in the case Q is of type ADE (theorem for type A) that central charges of q-stability conditions give the horizontal section of the Cherednik's KZ type connection (whose monodromy is described by the Hecke algebra) on the Cartan subalgebra.
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
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
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
June 1 (Fri), 13:00--14:30, 2018
Room
Room 402, RIMS
Speaker
Yusuke Ohkubo (Tokyo)
Title
Singular vectors of DIM algebra and generalized Macdonald functions arising from AGT conjecture
Abstract
The Ding-Iohara-Miki algebra (DIM algebra) is a Hopf algebra regarded as q-deformation of the W-infinity algebra. In the free field representation of the DIM algebra, a certain algebra can be obtained and plays an essential role in a q-deformed version of the AGT correspondence. In this talk, I will explain the representation of this algebra, AGT correspondence and properties of generalized Macdonald functions (used in AGT correspondence and also called AFLT basis or fixed point basis). In particular, I will describe the coincidence between singular vectors of that algebra and generalized Macdonald functions with some specific N-tuples of Young diagrams. This coincidence can be regarded as a sort of generalization of the one between ordinary Macdonald functions and singular vectors of the deformed W-algebra.
Date
June 1 (Fri), 15:00--16:00, 2018
Room
Room 402, RIMS
Speaker
Tianshu Liu (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
June 1 (Fri), 16:30--18:00, 2018
Room
Room 402, RIMS
Speaker
David Ridout (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 (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 (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)
Title
Parabolic analogue of periodic Kazhdan-Lusztig polynomials
Abstract
We construct a parabolic analogue of so-called periodic modules, which are modules of Hecke algebra associated with an affine Weyl group. These modules have a basis similar to Kazhdan-Lusztig basis. Our construction enables us to see the relation between periodic KL-polynomials and parabolic ones.
Date
9月25日 September 25(Fri), 16:30--18:00, 2015
Room
RIMS Room 402
Speaker
Simon Wood氏 (Australian National University)
Title
Classifying simple modules at admissible levels symmetric polynomials
Abstract
From infinite dimensional Lie algebras such as the Virasoro algebra or affine Lie (super)algebras one can construct universal vertex operator algebras. These vertex operator algebras are simple at generic central charges or levels and only contain proper ideals at so called admissible levels. The simple quotient vertex operator algebras at these admissible levels are called minimal model algebras. In this talk I will present free field realisations of the universal vertex operator algebras and show how they allow one to elegantly classify the simple modules over the simple quotient vertex operator algebras by using a deep connection to symmetric polynomials.
Date
8月7日 August 7(Fri), 16:30--18:00, 2015
Room
RIMS Room 402
Speaker
Oren Ben-Bassat 氏 (Oxford)
Title
Banach Algebraic Geometry
Abstract
I will present a 'categorical' way of doing analytic geometry in which
analytic geometry is seen as a precise analogue of algebraic geometry.
Our approach works for both complex analytic geometry and p-adic analytic
geometry in a uniform way. I will focus on the idea of an 'open set' as
used in these various areas of math and how it is characterised
categorically. In order to do this, we need to study algebras and their
modules in the category of Banach spaces. The categorical
characterization that we need uses homological algebra in these
'quasi-abelian'
categories which is work of Schneiders and Prosmans. In fact, we work
with the
larger category of Ind-Banach spaces for reasons I will explain. This
gives us
a way to establish foundations of analytic geometry and to compare with
the standard notions such as the theory of affinoid algebras,
Grosse-Klonne's theory of dagger algebras (over-convergent functions),
the theory of
Stein domains and others. I will explain how this extends to
a formulation of derived analytic geometry following the relative
algebraic geometry approach of Toen, Vaquie and Vezzosi.
This is joint work with Federico Bambozzi (Regensburg) and Kobi
Kremnizer (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)
Title
Sheaves of asymptotic chiral differential operators on symplectic resolutions
Abstract
In this seminar, we discuss sheaves of (h-adic) vertex algebras on symplectic manifolds, which give quantization of vertex Poisson algebras of their Jet bundles. On each formal coordinate, these sheaves are isomorphic to the vertex algebra of a formal beta-gamma system and we can determine the Lie algebra of derivations. Using Harish-Chandra extensions, we consider the classification of such sheaves. Such sheaves include localization of affine W-algebras which were constructed by Arakawa, Malikov and the speaker. Moreover, they include quantization of Jet bundles of hypertoric varieties and Nakajima quiver varieties. We also discuss construction of such quantization by semi-infinite reduction.
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)
Title
Calabi-Yau圏の安定性条件の空間とフロベニウス多様体
Abstract
三角圏に対してBridgelandは安定性条件の概念を導入し, 安定性条件全体の成す空間は複素多様体になることを示した. この空間はフロベニウス多様体と関連があることが期待されていたが, 最近, A型箙のGinzburgのdg代数の導来圏上の安定性条件の空間と A型特異点に付随するフロベニウス多様体の関係性が明らかになった. 本講演ではその結果をきっかけとして, Calabi-Yau圏の安定性条件の空間とフロベニウス多様体の間に 期待される関係性について説明する. また, 安定性条件の空間の中心電荷とフロベニウス多様体の 周期の間の関係性についての予想を述べる.
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)
Title
Positive Casimir and Central Characters of Split Real Quantum Groups
Abstract
The notion of the positive representations was introduced in a joint work with Igor Frenkel as a new research program devoted to the representation theory of split real quantum groups. Explicit construction of the these irreducible representations have been made corresponding to classical Lie type. In this talk, I will discuss the action of the generalized Casimir operators, which is important to understand the tensor product decomposition of these representations. These operators are shown to admit positive eigenvalues, and that their image defines a semi-algebraic region bounded by real points of the discriminant variety.
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)
Title
アフィン・スーパー・リー環の表現と モック・テータ函数
Abstract
アブストラクト： アフィン・スーパー・リー環の指標のモジュラー性質が どのようなものかを調べるのは難しい問題であった。S. Zwegers は 学位論文（２００３年）において，アフィン・スーパー・リー環 $\widehat sl(2|1)$ の レベル１の表現 $L(\Lambda_0)$ のスーパー指標に非正則な補正項を付加する ことにより実解析的なモジュラー函数が得られることを示した （S.P. Zwegers: Mock Theta Functions, ArXiv:0807.4834）。 Zwegers の方法を適用することによって，すべての basic classical アフィン・スーパー・リー環の maximally atypical 表現について， それらの（スーパー）指標を実解析的なモジュラー函数に拡張し， そのモジュラー変換行列を計算することが出来る。このセミナーでは これについて V.G. Kac との共同研究で得られた最近の成果を解説する。
Date
2月6日 Feb 6(Fri) 16:30--18:00, 2015
Room
RIMS Room 402
Speaker
Hironori Oya氏 (University of Tokyo)
Title
Representations of quantized function algebras the transition matrices from Canonical bases to PBW bases
Abstract
Let $G$ be a connected simply connected simple complex
algebraic group of type $ADE$ and $\mathfrak{g}$ the corresponding
simple Lie algebra. In this talk, I will explain our new algebraic proof
of the positivity of the transition matrices from the canonical basis to
the PBW bases of $U_q(\mathfrak{n}^+)$. Here, $U_q(\mathfrak{n}^+)$
denotes the positive part of the quantized enveloping algebra
$U_q(\mathfrak{g})$.
We use the relation between $U_q(\mathfrak{n}^+)$ and the specific
irreducible representations of the quantized function algebra
$\mathbb{Q}_q[G]$. This relation has recently been pointed out by
Kuniba, Okado and Yamada (SIGMA. 9 (2013)). Firstly, we study it taking
into account the right $U_q(\mathfrak{g})$-algebra structure of
$\mathbb{Q}_q[G]$. Next, we calculate the transition matrices from the
canonical basis to the PBW bases using the result obtained in the first
step.
I mention also some remarks which have recently been perceived.
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)
Title
Toroidal and affine Yangian algebras, and their commutative subalgebras
Abstract
We will recall the construction of certain families of representations of the toroidal algebras of $sl_n$, due to [Feigin-Jimbo-Miwa-Mukhin]. We explain how to adapt those to the setting of the affine Yangians. We will generalize the result of [Gautam-Toledano Laredo] to the toroidal setting. Using an alternative realization of Fock representations, due to [Saito], we recover a functional realization of certain commutative subalgebras in the toroidal/affine quantum algebras.
Attention
Date and Place are different from the usual.
Date
1月27日 Jan 27(Tue) 16:30--18:00, 2015
Room
Room 110, Building No.3
Speaker
Anthony Henderson氏 (University of Sydney)
Title
Geometric Satake, Springer correndence, and small representations
Abstract
Let $G$ be a connected reductive group and $W$ its Weyl group.
Consider the
functor $\Phi$ from representations of $G$ to representations of $W$ defined by
taking the zero weight space. This functor contains important
information, but is hard to describe in general. Note that when $G = GLn$,
the restriction of $\Phi$ to the subcategory of representations whose weights
$(a_{1},\cdots ,a_{n})$ satisfy $a_{1}+\cdots +a_{n}=0$ and
$a_{i} \ge -1$ is
essentially the famous Schur
functor. In particular, this restriction is of the form
$Hom_{GL_{n}}(E, - )$ where $E$ is a tilting module that carries a
commuting $S_{n}$-action.
For general $G$, the analogous subcategory to consider is that of small
representations, and the restriction of $\Phi$ to this subcategory was
studied by Broer and Reeder in the complex case. However, there is no
representation analogous to $E$ in other types. In joint work with Pramod
Achar (Louisiana State University) and Simon Riche (Universit\'e Blaise
Pascal - Clermont-Ferrand II), we describe the restriction of
$\Phi$ geometrically, in terms of the perverse sheaves on the affine
Grassmannian of the complex dual group $G^{\lor}$ that correspond to small
representations under geometric Satake; this makes sense for any ground
field. As we show, the correct substitute for Eis the Springer sheaf on
the nilpotent cone of $G^{\lor}$, with its $W$-action that gives rise to the
Springer correspondence.
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)
Title
量子アフィン展開環のレベル・ゼロ表現に対するギャラリー模型
Abstract
Gaussent-LittelmannのLakshmibai-Seshadriギャラリー模型を自然に 拡張することに よって、量子アフィン展開環の端ウェイト加群、及び基本加群（のテンソル積）の結 晶基底に対する実現が得られることについてお話しする。Gaussent-Littelmannの研 究目的はLakshmibai-Seshadriパス模型とアフィンGrassmann多様体 （Mirkovic-Vilonenサイクル）との間の関係を記述することであったが、 我々の設定では半無限Lakshmibai-Seshadriパス模型とFeigin-Frenkelの 半無限旗多様体との間の関係が観察されることについても述べる。
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)
Title
Coulomb branches of 3d N=4 gauge theories and the affine Grassmannian
Abstract
We propose a mathematically rigorous definition of Coulomb branch of a 3d N=4 SUSY gauge theory, as an affine algebraic variety, based on the homology group of a variant of the affine Grassmannian. In particular, coordinate rings of various hyper-Kaehler manifolds, such as instanton moduli spaces on ALE spaces, nilpotent orbits, etc, are conjecturally given by such a construction.
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)
Title
Local Fourier transform and blowing up
Abstract
We study linear ordinary differential equations with ramified
irregular singularies with the help of the theory of singularities of
plane curve germs.
Especially we shall see analogies between
- Komatsu-Malgrange irregularities of ODEs and intersection numbers and
Milnor numbers of curves,
- Local Fourier transform of ODEs and blow up of curves,
- Stokes structures of ODEs and iterated torus knots of curves.
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)
Title
1) COMMUTATORS IN ALGEBRAIC GROUPS
2) K-THEORY OF EXCEPTIONAL GROUPS
Abstract
1)
As we teach our students, in an abstract group an element of the
commutator subgroup is not necessarily a commutator.
However, the famous Ore conjecture, recently completely settled
by Ellers---Gordeev and Liebeck---O'Brien---Shalev--Tiep,
asserts that any element of a finite simple group, or, more generally,
of an adjoint elementary Chevalley group over a field,
is a single commutator.
On the other hand, from the work of van der Kallen, Dennis and
Vaserstein it was known that nothing like that can possibly hold
in general, for commutators in classical groups over rings. Actually,
these groups do not even have bounded width with respect to
commutators.
Using new versions of localisation methods, Stepanov, partly
in cooperation with myself, Hazrat and Zhang, succeeded in showing
that there is finiteness on the other end. Namely,
it turned our that commutators have bounded width with respect to
elementary generators.
Morally, these amazing results show that in algebraic groups over
rings there are very few commutators. The only reason,
why it appears that there are many commutators in the groups
of points over zero-dimensional rings (such as fields or local rings)
is that in these cases there exist very short expressions of arbitrary
elements in terms of elementary generators.
Also, I plan to discuss some further applications of our methods,
such as multiple commutator formulae, etc.,
as well as some further related asymptotic problems.
2)
Let $\Phi$ be a reduced irreducible root system,
$R$ be a commutative ring with $1$.
We study the following three closely related groups,
associated to $(\Phi,R)$.
* The (simply-connected) Chevalley group $G(\Phi,R)$.
* The (simply-connected) elementary Chevalley group $E(\Phi,R)$.
* The Steinberg group $\St(\Phi,R)$.
We set $K_1(\Phi,R)=G(\Phi,R)/E(\Phi,R)$ and denote by
$K_2(\Phi,R)$ the kernel of the natural projection
$\St(\Phi,R)\map E(\Phi,R)$.
For the classical groups, the initial groundbreaking contributions
to the study of these groups were made by Bass, Steinberg, Milnor
in the early 1960-ies, followed by the monumental works by Bak,
Suslin, Dennis, Vaserstein, van der Kallen, and many others.
But for exceptional groups, apart from the very important work of
Matsumoto, Stein, and their followers, in particular
Plotkin and myself, very little was known until recently.
We are mainly interested in the four large exceptional groups of
types $\E_6$, $\E_7$, $\E_8$ and $\F_4$, but actually many of the
outstanding problems first stated some 50 years ago still remain open
even for classical groups, apart from the linear case.
I plan to discuss recent progress towards solution of these probelms,
including
* Nilpotent structure of relative $K_1$
(Bak--Vavilov--Hazrat, and recent generalisations due to
Hazrat--Vavilov--Zhang and Stepanov),
* Centrality of $\K_2$, where the first major progress in 30 years
(after the solution of linear case by van der Kallen and Tulenbaev)
was recently achieved by Lavrenov, who solved the symplectic case,
* Stability for $K_1$ and $K_2$, where Sinchuk has succeeded to
improve stability results obtained by Stein and Plotkin for
exceptional embeddings
(as also recent versions of stability results for classical groups
by Bak--Petrov--Tang).
I will also survey some of the background and history, some of the
methods used, and relevance of these results in other branches of
the algebraic group theory.
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)
Title
Elliptic stable bases and applications
Abstract
This will be a report on a joint work in progress with Mina Aganagic.
Our goal is to produce an elliptic generalization of the stable
envelopes in K-theory (which will be briefly reviewed).
Elliptic stable envelopes depend on
an additional parameter $z$ in the complexification of Pic(X).
Elliptic stable envelopes limit to K-theoretic stable envelopes
with slope $s$ as the elliptic curve degenerates and
the Kahler/dynamical parameter $z$ goes
to infinity so that the ratio $z/\tau$ has a finite limit $s$.
Application awaiting such elliptic generalization include:
(1) geometric construction of elliptic R-matrices,
(2) monodromy of the K-theoretic quantum difference equation,
(3) precise correspondence of boundary conditions in dual 3-dimensional
susy gauge theories, and others.
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)
Title
Elliptic Quantum Groups, Drinfeld Coproduct and Deformed W-algebras
Abstract
We first discuss a quantum Z-algebra structure of the elliptic algebra U_{q,p}(g) associated with an untwisted affine Lie algebra g, and show that the irreducibility of the level-k representation of the U_{q,p}(g)-module is governed by the corresponding Z-algebra module. The level-1 examples for g=A_l^{(1)}, B_l^{(1)}, D_l^{(1)} show that the irreducible U_{q,p}(g)-modules are decomposed as a direct sum of the irreducible W-algebra modules. We secondly introduce the Drinfeld coproduct to U_{q,p}(g) and discuss the intertwining operators (vertex operators) with respect to this new coproduct. Constructing the vertex operators for the level-1 U_{q,p}(g)-modules with g=A_l^{(1)}, B_l^{(1)}, D_l^{(1)} explicitly, we show that these vertex operators factor the generating functions of the known deformed W-algebras associated with A_l^{(1)}, D_l^{(1)}, and further obtain a conjectural expression for the B_l^{(1)} case corresponding to a deformation of Fateev-Lukyanov's WB_l-algebra.
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)
Title
Symmetric quiver Hecke algebras and R-matirce of quantum affine algebras
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。
