RIMS
Research Inspiration from
Mathematical Sciences
Research Institute for Mathematical Sciences
Kyoto University

International Workshop on Integrable Models, Combinatorics and Representation Theory

August 12-16, 2001


Speakers | Program | Abstracts | RIMS | Kansai Seminar House | Weather in Kyoto | Welcome to Kyoto | Japan Travel Guide
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O r g a n i z e r s:

M. Kashiwara, A.N. Kirillov, T. Miwa
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Invited Speakers:

A. Braverman (Harvard University, USA), abstract
P. Caldero (University Claude Bernard Lyon I, France), abstract
H. Derksen (University of Michigan, Ann Arbor, USA), abstract
J.-F. van Diejen (University of Talca, Chile), abstract
C. Dong (University of California at Santa Cruz, USA), abstract
B. Feigin (Landau Inst. for Theoretical Physics, Russia, and RIMS, Kyoto University), abstract
E. Frenkel (University of California at Berkeley, USA), abstract
M. Jinzenji (Hokkaido University), abstract
A. Kirillov (RIMS, Kyoto University, and Steklov Institute, St.Petersburg, Russia), abstract
A. Malkin (Yale University, USA), abstract
H. Miyachi (Science University of Tokyo), abstract
S. Naito (University of Tsukuba), abstract
A. Nakayashiki (Kyushu University), abstract
M. Noumi (Kobe University), abstract
T. Otofuji (Nihon University), abstract
Y. Saito (University of Tokyo), abstract
M. Semenov-Tian-Shansky (University of Bourgogne, France, and Steklov Institute, St.Petersburg, Russia), abstract
A. Takahashi (RIMS, Kyoto University), abstract
C. Teleman (Cambridge University, United Kingdom), abstract
H. Terao (Tokyo Metropolitan University), abstract
M. Wakimoto (Kyushu University), abstract
M. Zabrocki (York University, Canada), abstract

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P r o g r a m

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The Workshop is to be held at Kansai Seminar House where accommodation and meals (according to the program below) for all speakers and most participants will be provided.

time August 12
(Sun)
August 13
(Mon)
August 14
(Tue)
August 15
(Wed)
August 16
(Thu)
8:00 - 9:00 Breakfast Breakfast Breakfast Breakfast
9:30 - 10:30

Miyachi Naito Otofuji Dong
10:45 - 11:45 Braverman Frenkel Semenov-
Tian-Shansky
Zabrocki
12:15 - 13:00 Registration Lunch Lunch Lunch Lunch
13:15 - 14:15 Wakimoto Feigin Discussions Discussions Noumi
14:30 - 15:30 Caldero Discussions Discussions Discussions van Diejen
15:45 - 16:45 Terao Discussions Discussions Discussions Kirillov
17:00 - 18:00 Nakayashiki Discussions Discussions Discussions Discussions
18:00 - 18:45 Dinner Dinner Dinner Dinner Dinner
19:00 - 20:00 Reception Malkin Teleman Jinzenji Bonfire
20:15 - 21:15 Reception Saito Derksen Takahashi Bonfire


Daily Program

August 12
Sunday
12:30 - 13:00 Registration

13:15 - 14:15 Minoru Wakimoto (Kyushu University):
N=2 superconformal modules with half-modular properties


14:30 - 15:30 Philippe Caldero (University Claude Bernard Lyon I, France):
Adapted algebras for the Berenstein-Zelevinsky conjecture


15:45 - 16:45 Hiroaki Terao (Tokyo Metropolitan University):
Multiderivations of Coxeter arrangements


17:00 - 18:00 Atsushi Nakayashiki (Kyushu University):
On the space of local operators of SU(2) invariant Thirring model
18:00 - 19:00 Dinner
19:00 - 21:00 Reception

August 13
Monday
8:00 - 9:00 Breakfast
9:30 - 10:30 Hyohe Miyachi (Science University of Tokyo):
On some crystallized decomposition numbers


10:45 - 11:45 Alexander Braverman (Harvard University, USA):
Drinfeld's compactifications and periodic Kazhdan-Lusztig polynomials
12:15 - 13:00 Lunch
13:15 - 14:15 Boris Feigin (Landau Inst. for Theoretical Physics, Russia, and RIMS, Kyoto University):
to be announced
14:30 - 18:00 Discussions
18:00 - 18:45 Dinner
19:00 - 20:00 Anton Malkin (Yale University, USA):
Induction of singular quiver varieties


20:15 - 21:15 Yoshihisa Saito (University of Tokyo):
On elliptic Lie algebras
August 14
Tuesday
8:00 - 9:00 Breakfast
9:30 - 10:30 Satoshi Naito (University of Tsukuba):
Twining characters, Path models, and crystal bases


10:45 - 11:45 Edward Frenkel (University of California at Berkeley, USA):
Representations of p-adic and quantum affine groups
12:15 - 13:00 Lunch
13:00 - 18:00 Discussions
18:00 - 18:45 Dinner
19:00 - 20:00 Constantin Teleman (Cambridge University, United Kingdom):
The Strong Macdonald conjecture and geometric applications


20:15 - 21:15 Harm Derksen (University of Michigan, Ann Arbor, USA):
Quiver representations and Littlewood-Richardson coefficients
August 15
Wednesday
8:00 - 9:00 Breakfast
9:30 - 10:30 Takashi Otofuji (Nihon University):
Quantum cohomology algebra of infinite-dimensional flag manifolds


10:45 - 11:45 Michael Semenov-Tian-Shansky (University of Bourgogne, France, and Steklov Mathematical Institute, St.Petersburg, Russia):
q-deformed quantum Toda lattice: a hint to representation theory of noncompact quantum groups
12:15 - 13:00 Lunch
13:00 - 18:00 Discussions
18:00 - 18:45 Dinner
19:00 - 20:00 Masao Jinzenji (Hokkaido University):
Gauss-Manin system and the virtual structure constants


20:15 - 21:15 Atsushi Takahashi (RIMS, Kyoto University):
BPS invariants on Calabi-Yau 3-fold
August 16
Thursday
8:00 - 9:00 Breakfast
9:30 - 10:30 Chongying Dong (University of California at Santa Cruz, USA):
Vertex operator algebras and reductive Lie algebras


10:45 - 11:45 Michael Zabrocki (York University, Canada):
q-analogs in Hopf algebra structures
12:15 - 13:00 Lunch
13:15 - 14:15 Masatoshi Noumi (Kobe University):
q-Painleve equations arising from a discrete version of the modified KP hierarchy


14:30 - 15:30 Jan Felipe van Diejen (University of Talca, Chile):
Modular hypergeometric sums


15:45 - 16:45 Anatol Kirillov (RIMS, Kyoto University, and Steklov Mathematical Institute, St.Petersburg, Russia):
Generalized saturation conjecture
16:45 - 18:00 Discussions
18:00 - 19:00 Dinner
19:00 - 22:00 Bonfire

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A b s t r a c t s

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Alexander Braverman (Harvard University, USA)
Drinfeld's compactifications and periodic Kazhdan-Lusztig polynomials
Abstract: The purpose of this talk is to introduce certain compactifications of the moduli space of parabolic bundles on a smooth projective curve over an algebraically closed field (these compactifications were originally defined by Drinfeld). I will explain how one can think about Drinfeld's compactifications as certain global models for the (at the moment) non-existing "partial semi-infinite flag manifolds". In particular, I will explain that one can realize the "periodic polynomials" introduced recently by Lusztig in terms of stalks of some natural perverse sheaves on Drinfeld's compactifications.
This is a joint work with M.Finkelberg, D.Gaitsgory and I.Mirkovic.

Philippe Caldero (University Claude Bernard Lyon I, France)
Adapted algebras for the Berenstein-Zelevinsky conjecture
Abstract: Let $G$ be a simply connected semi-simple complex Lie group and fix a maximal unipotent subgroup $U^-$ of $G$. Let $q$ be an indeterminate and let's denote by $\B^*$ the dual canonical basis of the quantized algebra $\C_q[U^-]$ of regular functions on $U^-$. A.Berenstein and A.Zelevinsky conjecture that two elements of $\B^*$ $q$-commute if and only if they are multiplicative, i.e. their product is an element of $\B^*$ up to a power of $q$. For all reduced decomposition ${\tilde w_0}$ of the longest element of the Weyl group of $\g$, we associate a subalgebra $A_{\tilde w_0}$, called adapted algebra, of $\C_q[U^-]$ such that
1) $A_{\tilde w_0}$ is a $q$-polynomial algebra which equals $\C_q[U^-]$ up to localization,
2) $A_{\tilde w_0}$ is spanned by a subset of $\B^*$,
3) the Berenstein-Zelevinsky conjecture is true on $A_{\tilde w_0}$.

Harm Derksen (University of Michigan, Ann Arbor, USA)
Quiver Representations and Littlewood-Richardson Coefficients
Abstract: I will discuss quiver representations and a theorem about generators of rings of semiinvariants for quivers. Applied to the triple flag quiver, the theorem implies the result of Knutson and Tao about the saturation of Littlewood-Richardson coefficients. Using quiver representations, one can also describe the faces (of arbitrary dimension) of the cone of nonzero Littlewood-Richardson coefficients.

Jan-Felipe van Diejen (University of Talca, Chile)
Modular hypergeometric sums
Abstract:
Recenlty Frenkel and Turaev introduced a modular hypergeometric generalization of the celebrated very-well-poised balanced basic hypergeometric ${}_8\Phi_8$ sum. In this talk various multidimensional generalizations of the Frenkel-Turaev sum are discussed.

Chongying Dong (University of California at Santa Cruz, USA)
Vertex operator algebras and reductive Lie algebras
Abstract: A VOA is of CFT type if the negative weight space is zero and weight zero subsapce is one-dimensional. The weight one subspace in this case forms a Lie algebra. In this talk we present three results concerning VOAs of CFT type:
1. The weight one subspace is a reductive Lie algebra.
2. The rank of the reductive Lie algebra is less than or equal to the central charge of the VOA.
3. Apply the results in 1 and 2 to the classification of holomorphic VOAs of small central charges.

Edward Frenkel (University of California at Berkeley, USA)
Representations of p-adic and quantum affine groups
Abstract: It is well-known that the category of unipotent representations of p-adic GL_n over the field of complex numbers is equivalent to the category of finite-dimensional representations of the corresponding affine Hecke algebra, which is in turn equivalent to a subcategory of the category of finite-dimensional representations of the quantum affine algebra of gl(N) for large N, via Jimbo's generalization of the Schur functor (with q=p). Recently, Vigneras has considered unipotent representations of p-adic GL_n over a field of characteristic l>0, prime to p. This category turns out to be related to the category of finite-dimensional representations of quantum affine gl(N), where q is a root of unity of order equal to the order of p mod l. In this talk I will discuss this correspondence, and also review recent results (obtained jointly with E.Mukhin) on the Hopf algebra structure of the Grothendieck ring of finite-dimensional representations of quantum affine gl(N), as N goes to infinity, both when q is generic and a root of unity.

Masao Jinzenji (Hokkaido University)
Gauss-Manin System and the Virtual Structure Constants
Abstract: In this talk, we discuss some application of ODE to enumerative problems on rational curves in projective hypersurfaces. In particular, we clarify the correspondence between the virtual structure constants and Givental's differential equation.

Anatol N. Kirillov (RIMS, Kyoto University, and Steklov Institute, St.Petersburg, Russia)
Generalized saturation conjecture
Abstract: Saturation Conjecture, now theorem by A.Knutson and T.Tao, 1998, was a final step in the proof of longstanding Horn Conjecture about the spectrum of sum of two hermitian matrices. Since the original proof by A.Knutson and T.Tao, several new proofs and generalizations have been given by H.Derksen and J.Weyman, A.Schofield, and others.
In my talk I will explain a new proof of the saturation theorem which is based on the theory of rigged configurations, parabolic Kostka polynomials, and use some ideas from Bethe's ansatz and Corner transfer matrix theory. I will also state several conjectures which are further generalizations of the saturation theorem and Fulton's conjecture.

Anton Malkin (Yale University, USA)
Induction of singular quiver varieties
Abstract: I'll describe an induction procedure for singular quiver varieties of ADE type which allows one to give a geometric construction of the tensor category of finite-dimensional representations of a reductive algebraic group over C. More elementary, one obtains a generalization of Hall polynomials to ADE case. The proof uses geometric crystals associated to quiver varieties.

Hyohe Miyachi (Science University of Tokyo)
On some crystallized decomposition numbers
Abstract: I'd like to talk about some recent results on crystallized decomposition numbers defined by Leclerc-Thibon. We can get some closed formulas for certain vectors of the canonical bases of the Fock space representation of $U_v(\hat{gl}_n)$. (This result is a joint work with B. Leclerc.) On the other hand, the origin of these formulas comes from $\ell$-modular representations of $GL_m(F_q)$. (This result is derived from a joint work with A. Hida.) I'd like to explain their connections, too.

Satoshi Naito (University of Tsukuba)
Twining characters, Path models, and crystal bases
Abstract: We talk about a combinatorial approach to a twining character formula for a Demazure module over a Kac-Moody algebra (this is a joint work with Dr. D. Sagaki). In this approach, we use the path model of a Demazure module and the global crystal base of a quantum Demazure module, in order to describe the basis elements of the module fixed by a diagram automorphism in terms of the orbit Lie algebra.

Atsushi Nakayashiki (Kyushu University)
On the space of local operators of SU(2) invariant Thirring model
Abstract: I will present a huge family of local operators of SU(2) invariant Thirring model in the form of form factors. In this case form factors of local operators are characterized as the infinite sets of solutions of sl_2 qKZ equation at level zero satisfying certain conditions. Up to now there are two important method to construct form factors, Smirnov's and Lukyanov's. These two constructions are very different. We generalize the results of Smirnov and discuss the relation of it with the latter construction.
This is a joint work with Y. Takeyama.

Masatoshi Noumi (Kobe University)
q-Painleve equations arising from a discrete version of the modified KP hierarchy
Abstract: I will discuss a class of q-Painleve equations with affine Weyl group symmetry which arise from a q-version of the modified KP hierarchy.
This talk is based on a joint work with K.Kajiwara and Y.Yamada.

Yoshihisa Saito (University of Tokyo)
On elliptic Lie algebras
Abstract: I will introduce a new class of infinite dimensional Lie algebras so-called Elliptic Lie algebras. This algebra was originally defined by K.Saito and D.Yoshii. I will discuss some class of representations and compute their characters.
This talk is based on a joint work with H.Awata, A.Kato, Y.Shimizu and A.Tsuchiya.

Michael Semenov-Tian-Shansky (University of Bourgogne, France, and Steklov Mathematical Institute, St.Petersburg, Russia)
q-deformed quantum Toda lattice: a hint to representation theory of noncompact quantum groups
Abstract: The q-deformed quantum Toda lattice may be studied by a combination of the methods of representation theory and of the Quantum Inverse Scattering Method. The key phenomenon which is under scrutiny is the role of the modular duality concept (first discovered by L.Faddeev) in the representation theory of noncompact semisimple quantum groups. Explicit formulae for the Whittaker vectors are presented in terms of the double sine functions and the wave functions of the N-particle q-deformed open Toda chain are expressed by means of a multiple integral of the Mellin-Barnes type.

Atsushi Takahashi (RIMS, Kyoto University)
BPS invariants on Calabi-Yau 3-fold
Abstract: I shall explain a mathematical definition of ``BPS invarants" of Calabi-Yau 3-fold based on the joint work with S.Hosono and M.-H.Saito. In particular, for any projective morphism $f:X\to Y$ of normal projective varieties, it is shown that there exists a natural $sl_2\times sl_2$ action on the intersection cohomology group of $X$. Some evidences for the Gopakumar-Vafa formula, the equivalence between BPS and Gromov-Witten invariants, will be given. Relation with BPS algebra will also be discussed.

Constantin Teleman (Cambridge University, United Kingdom)
The Strong Macdonald conjecture and geometric applications
Abstract: We shall give a quick sketch of the proof of the "Strong Macdonald Constant Term conjecture" of Feigin and Hanlon, and discuss one of its geometric applications, to wit, the computation of the coherent sheaf cohomology of the cotangent bundle of the moduli of holomorphic G-bundles over a Riemann surface, and the cohomology of the sheaf of differential operators of the same The proof of the conjecture is joint work with Fishel and Grojnowski; the application is joint work with E. Frenkel.
The work is based on ideas of Hitchin, Feigin and Beilinson.

Hiroaki Terao (Tokyo Metropolitan University)
Multiderivations of Coxeter arrangements
Abstract: Let $V$ be an $\ell$-dimensional Euclidean space. Let $G \subset O(V)$ be a finite irreducible orthogonal reflection group. Let ${\cal A}$ be the corresponding Coxeter arrangement. Let $S$ be the algebra of polynomial functions on $V.$ For $H \in {\cal A}$ choose $\alpha_H \in V^*$ such that $H = {\rm ker}(\alpha_H).$ For each nonnegative integer $m$, define the derivation module $D^{(m)}({\cal A}) = \{ \theta \in {\rm Der}_S ~|~\theta(\alpha_H) \in S \alpha^m_H\}$. The module is known to be a free $S$-module of rank $\ell$ by K. Saito (1975) for $m=1$ and L. Solomon-H. Terao (1998) for $m=2$. The main result of this paper is that this is the case for all $m$. Moreover we explicitly construct a basis for $D^{(m)} (\cal A)$. Their degrees are all equal to $mh/2$ (when $m$ is even) or are equal to $((m-1)h/2) + m_i (1 \leq i \leq \ell)$ (when $m$ is odd). Here $m_1 \leq \cdots \leq m_{\ell}$ are the exponents of $G$ and $h= m_{\ell} + 1$ is the Coxeter number. The construction heavily uses the primitive derivation $D$ which plays a central role in the theory of flat generators by K. Saito (or equivalently the Frobenius manifold structure for the orbit space of $G$.) Some new results concerning the primitive derivation $D$ are obtained in the course of proof of the main result.

Takashi Otofuji (Nihon University)
Quantum cohomology algebra of infinite-dimensional flag manifolds
Abstract: I will discuss a generalization of Givental-Kim's result on the quantum cohomology algebra of finite-dimensional flag manifolds and the relation with Toda lattices to the infinite-dimensional case. I will discuss also a trial for a quantum Schubert calculus of those spaces.
This is a joint work with Martin Guest.

Minoru Wakimoto (Kyushu University)
N=2 superconformal modules with half-modular properties
Abstract: In the construction of W-algebras associated to affine superalgebras via the quantized Drinfeld-Sokolov reduction, it turns out that the W-algebra of sl(2|1)^ is the direct sum of the centerless Virasoro algebra and the N=2 superconformal algebra. This picture takes, of course, care of the usual N=2 discrete series representations which are obtained from sl(2|1)^-modules of boundary levels, and moreover, using admissible sl(2|1)^-modules in general, reveals series of representations of the N=2 superconformal algebra whose characters are "half" of modular functions in the sense that, for each N=2 superconformal module belonging to this series, there exists another module such that the sum of characters of these two is a modular function. It is well known that the central charges of discrete series representations of the N=2 superconformal algebra are c(m)=3m/(m+2) where m are non-negative integers. The central charges of these half-modular series of representations are just equal to c(m) with m being rational numbers such that m+2 >0.

Michael Zabrocki (York University, Canada)
q-analogs in Hopf algebra structures
Abstract: Using only operations that exist in any graded Hopf algebra, we introduce a formula for taking a q-analog of homomorphisms of the Hopf algebra. This quantization seems to be important in the theory of symmetric functions as it gives rise to Hall-Littlewood and Macdonald polynomials from formulas for better known symmetric functions. By changing the Hopf algebra, we will show that this q-analog also gives rise to other q-coefficients using the same sort of algebraic construction.


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