## Representation Theory Seminar

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI, Ryo SATO

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI, Ryo SATO

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI, Ryo SATO

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI

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).

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Yuto MORIWAKI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE

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).

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE

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).

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE

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.)

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Ryo FUJITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Ryo FUJITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA,Ryo FUJITA, Hideya WATANABE

--- CANCEL ---

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI

Date

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

Room

RIMS 006

Speaker

Mamoru Ueda (RIMS Kyoto)

Title

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA, Sota ASAI

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

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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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

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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.

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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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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).

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

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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.

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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

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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).

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

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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.

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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.

Organizers 荒川知幸、加藤周、疋田辰之

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.

Organizers Tomoyuki ARAKAWA, Shu KATO, Tatsuyuki HIKITA

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.

Organizers 荒川知幸、加藤周、疋田辰之

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.

Organizers 荒川知幸、加藤周、疋田辰之

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.

Organizers 荒川知幸、加藤周、疋田辰之

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.

Organizers 中島啓、加藤周、疋田辰之

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.

Organizers 中島啓、加藤周、疋田辰之

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).

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Date

2月16日 (Fri), 15:45--16:45, 2018

Room

Room 402, RIMS

Speaker

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.

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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.

Organizers 中島啓、加藤周、疋田辰之

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.

Organizers 中島啓、加藤周、疋田辰之

Date

12月22日 (Fri), 10:30--12:00, 2017

Room

Room 402, RIMS

Speaker

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.

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Date

12月22日 (Fri), 13:00--14:30, 2017

Room

Room 402, RIMS

Speaker

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.

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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.

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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.

Organizers 中島啓、加藤周、疋田辰之

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.

Organizers 中島啓、加藤周、疋田辰之

Date

11月17日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

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.

Organizers 中島啓、加藤周、疋田辰之

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.

Organizers 中島啓、加藤周、疋田辰之

Date

10月20日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

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.

Organizers 中島啓、加藤周、疋田辰之

Date

10月6日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

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.

Organizers 中島啓、加藤周、疋田辰之

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.

Organizers 中島啓、加藤周、疋田辰之

Date

7月28日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

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.

Organizers 中島啓、加藤周、疋田辰之

Date

7月21日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

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.

Organizers 中島啓、加藤周、疋田辰之

Date

6月16日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

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.

Organizers 中島啓、加藤周、疋田辰之

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.

Organizers 中島啓、加藤周、疋田辰之

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.

Organizers 中島啓、加藤周、疋田辰之

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.

Organizers 中島啓、加藤周、疋田辰之

Date

5月12日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

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.

Organizers 中島啓、加藤周、疋田辰之

Date

4月21日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

Title

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。

Organizers 中島啓、加藤周、疋田辰之

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.

Organizers 中島啓、加藤周、疋田辰之

Date

4月7日 (Fri), 16:30--18:00, 2017

Room

Room 402, RIMS

Speaker

Title

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。

Organizers 中島啓、加藤周、疋田辰之

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.

Organizers 中島啓、加藤周、疋田辰之

Date

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

Room

Room 402, RIMS

Speaker

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.

Organizers 中島啓、加藤周、疋田辰之

Date

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

Room

Room 402, RIMS

Speaker

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.

Organizers 中島啓、加藤周、疋田辰之

Date

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

Room

Room 402, RIMS

Speaker

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.

Organizers 中島啓、加藤周、疋田辰之

Date

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

Room

Room 402, RIMS

Speaker

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.

Organizers 中島啓、加藤周、疋田辰之

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.

Organizers 中島啓、加藤周、疋田辰之

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.

Organizers 中島啓、加藤周、疋田辰之

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.

Organizers 中島啓、加藤周、疋田辰之

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.

Organizers 中島啓、加藤周、疋田辰之

Date

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

Room

RIMS 402

Speaker

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 $\mathop{GL}_n$.

Organizers 中島啓、加藤周、疋田辰之

Date

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

Room

RIMS 402

Speaker

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).

Organizers 中島啓、加藤周、疋田辰之

Date

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

Room

RIMS 402

Speaker

Abstract

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

Organizers 中島啓、加藤周、疋田辰之

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.

Organizers 中島啓、疋田辰之

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.

Organizers 中島啓、疋田辰之

Date

7月8日 July 8(Fri), 16:30--18:00, 2016

Room

RIMS Room 402

Speaker

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.

Organizers 中島啓、疋田辰之

Date

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

Room

RIMS Room 402

Speaker

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.

Organizers 中島啓、疋田辰之

Date

6月3日 June 3(Fri), 16:30--18:00, 2016

Room

RIMS Room 402

Speaker

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.

Organizers 中島啓、疋田辰之

Date

5月20日 May 20(Fri), 16:30--18:00, 2016

Room

RIMS Room 402

Speaker

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.

Organizers 中島啓、疋田辰之

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.

Organizers 中島啓、疋田辰之

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.

Organizers 中島啓、疋田辰之

Date

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

Room

RIMS Room 402

Speaker

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。

なお、本研究は Wee Teck Gan 氏との共同研究である。

Organizers 中島啓、疋田辰之

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.

Organizers 中島啓、疋田辰之

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.

Organizers 中島啓、疋田辰之

Date

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

Room

RIMS Room 402

Speaker

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).

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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

Title

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．

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

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

Room

RIMS Room 402

Speaker

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

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

Room

RIMS Room 402

Speaker

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．

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

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

Room

RIMS Room 402

Speaker

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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).

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

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

Room

RIMS Room 402

Speaker

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

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

Room

RIMS Room 402

Speaker

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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

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。

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

Attention

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

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。

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

1月9日 Jan 9(Fri) 16:30--18:00, 2015

Room

RIMS 402

Speaker

Title

Abstract

Organizers 荒川知幸、加藤周、柳田伸太郎

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

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

11月21日 Nov 21(Fri) 17:00--18:30, 2014

Room

Room 402, RIMS

Speaker

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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．

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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).

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

5月16日 May 16(Fri) 16:30--18:00, 2014

Room

Rims 402, Kyoto University

Speaker

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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).

Organizers 荒川知幸、加藤周、柳田伸太郎

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

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

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

Room

Rims 402, Kyoto University

Speaker

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

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

Room

Rims 402, Kyoto University

Speaker

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。

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

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

Room

Rims 402, Kyoto University

Speaker

Title

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．

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

7月19日 July 19(Fri) 16:30--18:00, 2013

Room

Rims 402, Kyoto University

Speaker

Title

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

Organizers 荒川知幸、加藤周、柳田伸太郎

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

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。

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

5月31日 May 31(Fri) 16:30--18:00, 2013

Room

Rims 402, Kyoto University

Speaker

Title

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。

Organizers 荒川知幸、加藤周、柳田伸太郎

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

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．

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

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

Room

Rims 402, Kyoto University

Speaker

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

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

Room

Rims 402, Kyoto University

Speaker

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

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).

Organizers 荒川知幸、加藤周、柳田伸太郎

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.

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

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

Room

Rims 402, Kyoto University

Speaker

Title

Symmetric quiver Hecke algebras and R-matirce of quantum affine algebras

Organizers 荒川知幸、加藤周、柳田伸太郎

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).

Organizers 荒川知幸、加藤周、柳田伸太郎

Date

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

Room

Rims 402, Kyoto University

Speaker

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).

Organizers 荒川知幸、加藤周、柳田伸太郎

Attention

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.

Organizers 荒川知幸、加藤周

Date

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

Room

Rims 402, Kyoto University

Speaker

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。

Organizers 荒川知幸、加藤周

Date

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

Room

Rims 402, Kyoto University

Speaker

Title

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）。

Organizers 荒川知幸、加藤周

Date

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

Room

Rims 402, Kyoto University

Speaker

Title

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．

Organizers 荒川知幸、加藤周

Date

6月8日(Fri)(June 8) 16:30--18:00, 2012

Room

Rims 402, Kyoto University

Speaker

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.

Organizers 荒川知幸、加藤周

Date

6月1日(Fri)(June 1) 16:30--18:00, 2012

Room

Rims 402, Kyoto University

Speaker

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。

Organizers 荒川知幸、加藤周

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.

Organizers 荒川知幸、加藤周

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.

Organizers 荒川知幸、加藤周

Date

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

Room

Rims 402, Kyoto University

Speaker

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.

Organizers 荒川知幸、加藤周

Date

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

Room

Rims 402, Kyoto University

Speaker

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。

Organizers 荒川知幸、加藤周

Date

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

Room

Rims 402, Kyoto University

Speaker

Title

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。

Organizers 荒川知幸、加藤周

Date

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

Room

Rims 402, Kyoto University

Speaker

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.

Organizers 荒川知幸、加藤周

Date

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

Room

Rims 402, Kyoto University

Speaker

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。

Organizers 荒川知幸、加藤周

Date

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

Room

Rims 402, Kyoto University

Speaker

Title

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．

Organizers 荒川知幸、加藤周

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.

Organizers 荒川知幸、加藤周

Date

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

Room

Room 402 of RIMS, Kyoto University

Speaker

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。

Organizers 荒川知幸、加藤周

Date

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

Room

Room 402 of RIMS, Kyoto University

Speaker

Title

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.

Organizers 荒川知幸、加藤周

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.

Organizers 荒川知幸、加藤周

Date

7月22日(金) (July 22) 15:00--16:30, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

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.

Organizers 荒川知幸、加藤周

Date

7月8日(金) (July 8) 15:00--16:30, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

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.

Organizers 荒川知幸、加藤周

Attention

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

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．

Organizers 荒川知幸、加藤周、榎本直也、木村嘉之

Date

6月24日(金) (June 24) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

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．

Organizers 荒川知幸、加藤周、榎本直也、木村嘉之

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!

Date

6月17日(金) (June 17) 11:30--13:00, 2011

Room

Room 204 of RIMS, Kyoto University

Speaker

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．

Date

6月17日(金) (June 17) 14:45--16:15, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

Title

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．

Organizers 荒川知幸、加藤周、榎本直也、木村嘉之

Date

6月10日(金) (June 10) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

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．

Organizers 荒川知幸、加藤周、榎本直也、木村嘉之

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

Title

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

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。

Organizers 荒川知幸、加藤周、榎本直也、木村嘉之

Date

5月20日(金) (May 20) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

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.)

Organizers 荒川知幸、加藤周、榎本直也、木村嘉之

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.

Organizers 荒川知幸、加藤 周、榎本直也

Date

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

Room

Room 402 of RIMS, Kyoto University

Speaker

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。

Organizers 荒川知幸、加藤 周

Date

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

Room

Room 402 of RIMS, Kyoto University

Speaker

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.

Organizers 荒川知幸、加藤 周

Date

4月8日(金) (April 8) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

Title

Cyclotomic quiver Hecke algebras and categorifications of highest weight module

Organizers 荒川知幸、加藤 周

Date

4月1日(金) (April 1) 16:30--18:00, 2011

Room

Room 402 of RIMS, Kyoto University

Speaker

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。

Organizers 荒川知幸、加藤 周

Date

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

Room

Room 204 of RIMS, Kyoto University

Speaker

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。

Organizers 荒川知幸、加藤 周

Date

March 16 (Wed) 14:00--（３時間程度）

Room

Room 204 of RIMS, Kyoto University

Speaker

Title

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

Organizers 荒川知幸、加藤 周

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.

Organizers 荒川知幸、加藤 周

Date

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

Room

Room 204 of RIMS, Kyoto University

Speaker

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.

Organizers 荒川知幸、加藤 周

Date

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

Room

Room 204 of RIMS, Kyoto University

Speaker

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.

Organizers 荒川知幸、加藤 周

Date

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

Room

Room 204 of RIMS, Kyoto University

Speaker

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.

Organizers 荒川知幸、加藤 周

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.

Organizers 荒川知幸、加藤 周

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.

Organizers 荒川知幸、加藤 周

Date

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

Room

Room 204 of RIMS, Kyoto University

Speaker

Title

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。

Organizers 荒川知幸、加藤 周

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

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。

Organizers 荒川知幸、加藤 周

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.

Organizers 荒川知幸、加藤 周

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.

Organizers 荒川知幸、加藤 周

Date

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

Room

Room 204 of RIMS, Kyoto University

Speaker

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。

Organizers 荒川知幸、加藤 周

Date

2010年10月13日(水曜日) 14:50--16:20

Room

Speaker

Title

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)。

Organizers 荒川知幸、加藤 周

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

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。

Organizer 加藤 周

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