International Workshop on Physics and Combinatorics

August 21-26, 2000


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Proceedings, full contents is here
List of participants

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O r g a n i z e r s:

K. Aomoto, F. Hiroshita, A.N. Kirillov, R. Kobayashi, A. Tsuchiya, H. Umemura
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S p e a k e r s:

E. Bannai (Kyushu University), abstract
T. Eguchi (Physical Department, University of Tokyo), abstract
B. Feigin (Landau Inst. for Theoretical Physics, Russia, and RIMS, Kyoto University), abstract
G. Felder (ETH-Zentrum, Switzerland), abstract
M. Haiman (University of California, San Diego, USA), abstract
K. Hikami (University of Tokyo), abstract
K. Iguchi (Freelancing Physicist), abstract
M. Ioffe (St. Petersburg University, Russia), abstract
K. Kadell (Arizona State University, USA), abstract
K. Kajiwara (Doshisha University), abstract
J. Kaneko (Kyushu University), abstract
M. Kaneko (Kyushu University), abstract
R. Kashaev (Steklov Institute, St.Petersburg, Russia, and Helsinki Institute of Physics), abstract
A. Kirillov (Nagoya University and Steklov Institute, St.Petersburg, Russia), abstract
A. Lascoux (CNRS, Marne la Vallee, France), abstract
K. Mimachi (Kyushu University), abstract
T. Miwa (Kyoto University), abstract
H. Murakami (Tokyo Institute of Technology), abstract
H. Nakajima (Kyoto University), abstract
T. Nakanishi (Nagoya University), abstract
M. Nishizawa (Waseda University), abstract
M. Noumi (Kobe University), abstract
R. Sasaki (Yukawa Institute for Theoretical Physics), abstract
T. Shoji (Science University of Tokyo), abstract
T. Takagi (National Defense Academy), abstract
M. Takahashi (Institute for Solid State Physics, University of Tokyo), abstract
M. Taneda (Freelancing Mathematician), abstract
H. Terao (Tokyo Metropolitan University), abstract
A. Tsuchiya (Nagoya University), abstract
M. Wakimoto (Kyushu University), abstract
Y. Yamada (Kobe University), abstract
M. Yoshida (Kyushu University), abstract

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S c h e d u l e

August 21, Monday
August 22, Tuesday
August 23, Wednesday
August 24, Thursday
August 25, Friday
August 26, Saturday

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S c h e d u l e

(all talks in Main Lecture Hall 509, computer facilities Room 555, discussion Rooms 109 & 555, refreshments in Room 555, tea in Room 109)

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Monday, August 21

10:00-11:00
A. Tsuchiya, Seiberg-Witten differential as a period of rational elliptic surface, abstract
11:10-12:10
M. Wakimoto, Representation theory of affine superalgebras, abstract
12:10-13:30
lunch break
13:30-14:30
K. Kadell, The Macdonald and Dyson Polynomials, abstract
14:40-15:40
E. Bannai, Some results on modular forms motivated by coding theory, abstract
15:40-16:00
tea break
16:00-17:00
M. Kaneko, Multiple zeta values and poly-Bernoulli numbers - a survey, abstract
17:10-18:10
T. Takagi, Factorization of Combinatorial R matrices and Associated Cellular Automata, abstract
18:30-19:30
Summer Academic Concert, program

Tuesday, August 22

9:30-10:30
M. Haiman, The n! and Macdonald positivity conjectures, abstract
10:45-11:45
M. Wakimoto, Representation theory of affine superalgebras II, abstract
11:45-13:15
lunch break
13:15-14:15
A. Lascoux, Double Demazure character formula, abstract
14:25-15:25
M. Noumi, Weyl group actions arising from nilpotent Poisson algebras, abstract
15:25-15:45
tea break
15:45-16:45
K. Hikami, Exclusion statistics and Universal Chiral Partition Function, abstract
16:55-17:55
M. Taneda, Special polynomials for the sixth Painleve equation and Combinatorics abstract

Wednesday, August 23

9:30-10:30
J. Felder, The elliptic gamma function, SL(3,Z), and q-deformation of conformal field theory on elliptic curves, abstract
10:45-11:45
M. Haiman, Hilbert schemes and the proof of the n! conjecture, abstract
11:45-13:15
lunch break
13:15-14:15
H. Terao, Logarithmic Gauss-Manin connections on the one-codimensional strata of hyperplane arrangements, abstract
14:25-15:25
T. Shoji, Green functions associated to complex reflection groups, abstract
15:25-15:45
tea break
15:45-16:45
H. Nakajima, Finite dimensional representations of quantum affine algebras, abstract
16:55-17:55
K. Iguchi, A Theory of Interacting Many Body Systems with Exclusion Statistics: Origin of Exclusion Statistics, Haldane Liquids, Sutherland-Wu Equations, Grand Partition Function and Lee-Yang Theorem, abstract

Thursday, August 24

9:30-10:30
M. Takahashi, Simple solution of thermodynamic Bethe ansatz equations, abstract
10:45-11:45
T. Eguchi, Quantum cohomlogy and structure of topological string theory, abstract
11:45-13:15
lunch break
13:15-14:15
M. Ioffe, Multiparticle Supersymmetrical Quantum Mechanics and representations of Permutation Group, abstract
14:25-15:25
R. Sasaki, Quantum Calogero-Moser models: complete integrability for all the root systems, abstract
15:25-15:45
tea break
15:45-16:45
Y. Yamada, The combinatorial R-matrix and the canonical basis, abstract
16:55-17:55
K. Kajiwara, Classical solutions for Painleve and Discrete Painleve equations, abstract
18:20-20:00
Reception, information

Friday, August 25

9:30-10:30
T. Miwa, Recursion relations for rigged partitions, abstract
10:45-11:45
M. Haiman, Diagonal harmonics, abstract
11:45-13:15
lunch break
13:15-14:15
R. Kashaev, Using the non-compact quantum dilogarithm for quantizing the Teichmuller spaces of punctured surfaces, abstract
14:25-15:25
H. Murakami, Volume conjecture for three-manifolds, abstract
15:25-15:45
tea break
15:45-16:45
T. Nakanishi, Q-system and characters of Kirillov-Reshetikhin modules of affine quantum groups and Yanigans, abstract
16:55-17:55
M. Nishizawa, Generalized Holder's theorem for multiple gamma functions, abstract

Saturday, August 26

9:30-10:30
M. Yoshida, A hypergeometric story, abstract
10:45-11:45
J. Kaneko, Forrester's constant term conjecture and Chu-Vandermonde formula for generalized binomial coefficients, abstract
11:45-13:15
lunch break
13:15-14:15
K. Mimachi, A representation of the Iwahori-Hecke algebra on the twisted homology associated with the Selberg type integral, abstract
14:25-15:25
B. Feigin, Gordon-type filtrations and corresponding boson-fermion character formulas, abstract
15:35-16:35
A. Kirillov, Elliptic disease, abstract

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A b s t r a c t s

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Eiichi Bannai (Kyushu University)
Some results on modular forms motivated by coding theory
Abstract: This talk is based on joint work with Masao Koike (Kyushu University), Akihiro Munemasa (Kyushu University) and Jiro Sekiguchi (Himeji Institute of Technology).

First we give the determination of the finite index subgroups G of the modular group SL(2,Z) for which the space of modular forms (of integral weights) for G is isomorphic to a polynomial ring. There are 17 such groups up to the conjugacy in SL(2,Z).) One example is G =G(3), and this case is related to the space of weight enumerators of ternary self-dual codes and the polynomial invariants by the unitary reflection group (No. 4) of order 24.

Then we consider a similar problem for the space of modular forms of fractional weights. We show that if the space of modular forms of half-integral weights (with respect to some multiplier system) of G is isomorphic to a polynomial ring generated by 2 modular forms of weight 1/2, then the group G must be one of the 191 subgroups up to the conjugacy in SL(2,Z). One example is G =G (4), and this case is related to the space of weight enumerators of binary Type II codes, and the polynomial invariants by the unitary reflection group (No. 8) of order 96.

Finally, we consider similar problems for the space of modular forms of 1/l-integral weights. We show that the space of modular forms of 1/5-integral weights (with respect to certain specified multiplier system) of G(5) is isomorphic to a polynomial ring generated by 2 modular forms of weight 1/5. Also, we discuss the connections with the classical work of F. Klein on the icosahedron, as well as with unitary reflection group (No. 16) of order 600.

The main theme here is that, in some cases, by considering the fractional weight modular forms, the ring of modular forms becomes simpler, and we can get better understanding of the space of modular forms of integral weights. We also mention some very recent work by T. Ibukiyama on the study of modular forms of 1/7-integral weight of G(7), which were motivated by this theme.

Tohru Eguchi (Physical Department, University of Tokyo)
Quantum cohomlogy and structure of topological string theory
Abstract: We discuss the structure of topological string theories coupled to gravity and their implications on quantum cohomology theory. We describe Virasoro conditions and topological recursion relations which determine the free energy of the topological string. We present the structure of the genus-two amplitude in the case of P1 model.

Boris Feigin (Landay Institute for Theoretical Physics, Russia, and RIMS, Kyoto University)
Gordon-type filtrations and corresponding boson-fermion character formulas
Abstract: In the talk we will discuss the algebraic manifolds which are similar to the Shubert varieties. Such manifolds naturally appear in some conformal field theories. Gordon filtrations arise when we study the space of sections of line bundles on these Shubert-like manifolds.

Giovanni Felder (ETH-Zentrum, Switzerland, joint work with Alexander Varchenko)
The elliptic gamma function, SL(3,Z), and q-deformation of conformal field theory on elliptic curves
Abstract: The elliptic gamma function is an elliptic version of the q-gamma function of Jackson which, in turn, is a trigonometric version of the classical Euler gamma function. The modular properties of the elliptic gamma function will be discussed. They are expressed in terms of a generalization of Jacobi modular forms associated to SL(3,Z). These properties are prototypes of the modular properties of conformal blocks of the q-deformation of conformal field theory on elliptic curves. I will discuss this in examples based on sl(2).

Mark Haiman (University of California, San Diego, USA)
The n! and Macdonald positivity conjectures
Abstract: Introduction to the positivity conjecture for Macdonald polynomials and its representation-theoretic interpretation via a conjecture by Garsia and me, known as the "n! conjecture". Recently I succeeded in proving these conjectures using the isospectral Hilbert scheme of points in the plane, whose connection to the n! conjecture I will explain.

Mark Haiman (University of California, San Diego, USA)
Hilbert schemes and the proof of the n! conjecture
Abstract: In this second lecture I will develop the properties of the isospectral Hilbert scheme in more detail and explain the proof of the n! conjecture.

Mark Haiman (University of California, San Diego, USA)
Diagonal harmonics
Abstract: Related to the n! conjecture is a series of conjectures on the space of harmonics for the diagonal action of the symmetric group on its doubled reflection representation. The dimension of the diagonal harmonics is conjectured to be (n+1)(n-1), and there is a series of combinatorial refinements of this. All of them pfollow from a master formula for the character of diagonal harmonics in terms of Macdonald polynomials, which can be explained and, I expect, proved, using the isospectral Hilbert scheme.

Kazuhiro Hikami (Tokyo University)
Exclusion statistics and Universal Chiral Partition Function
Abstract: We will show how to derive a partition function and to give thermodynamics of quasi-particles which obey exclusion statistics. We will also explain a spinon basis of the level-1 WZW model.

Kazumoto Iguchi (Freelancing Physicist)
A Theory of Interacting Many Body Systems with Exclusion Statistics: Origin of Exclusion Statistics, Haldane Liquids, Sutherland-Wu Equations, Grand Partition Function and Lee-Yang Theorem
Abstract: I will discuss a theory of interacting many body systems with exclusion statistics. I first show how exclusion statistics appears from the long-ranged interactions. Second, I explain the concept of Haldane liquids in higher dimensions and discuss the basic properties of the Haldane liquid in terms of the language of the Sutherland-Wu functional equation and the grand partition function. Finally, I will mention applications of the theory to some other problems such as the Lee-Yang theorem of phase transition.

Mikhail Ioffe (St.Petersburg University, Russia)
Multiparticle Supersymmetrical Quantum Mechanics and representations of Permutation Group
Abstract: A multidimensional Supersymmetrical Quantum Mechanics (SUSY QM) is proposed. Its structure for an arbitrary number of space dimensions is investigated. Supersymmetrical method leads to the multidimensional generalization of the well-known Schroedinger Factorization Method and Darboux Transformation. This approach makes it possible to find connections between the spectra and eigenfunctions of the chain of matrix quantum Hamiltonians. Some generalizations and applications of this method are considered: higher order SUSY QM, scattering problem case, Pauli equation for spin 1/2 particle, integrable 2-dim quantum and classical systems. The method of multidimensional SUSY QM is also applied to the investigation of SUSY N-particle systems on a line for the case of separable center-of-mass motion. New decomposition of the Superhamiltonian into block-diagonal form with elementary matrix components is constructed. Matrices of coefficients of these minimal blocks are shown to coincide with matrices of irreducible representations of the permutation group SN, which correspond to the Young tableaux (N-M, 1M). The connections with known generalizations of N-particle Calogero and Sutherland models are discussed briefly.

Kevin Kadell (Arizona State University, USA)
The Macdonald and Dyson Polynomials
Abstract: The Macdonald polynomials satisfy many algebraic, combinatorial and analytic properties which are related to the Schur functions, Selberg's integral and its many extensions, and certain constant term identities associated with root systems. We conjecture that there are Dyson polynomials which refine the parameters q,t to q,t,...,tn of the Macdonald polynomials. We give constant term orthogonality relations for l=(r) using Good's proof, for n=2 using the q-Saalschutz sum, and for n=3, l=(2,1) by computation. We give numerous conjectures.

Kenji Kajiwara (Doshisha University)
Classical solutions for Painleve and Discrete Painleve equations
Abstract: Classical solutions and their determinant formulas, including special polynomials, for Painleve and Discrete Painleve equations are presented. Some recent results on discrete dynamics of discrete Painleve equations will be discussed.

Jyoichi Kaneko (Kyushu University)
Forrester's constant term conjecture and Chu-Vandermonde formula for generalized binomial coefficients
Abstract: Forrester's constant term conjecture, still widely open, predicts the precise product formula of the constant term of even power of difference product multiplied by some extra factors. We give a proof of the conjecture in an extreme case where essetial use is made of the Chu-Vandermonde formula for generalized binomial coefficients.

Masanobu Kaneko (Kyushu University)
Multiple zeta values and poly-Bernoulli numbers - a survey
Abstract: Multiple zeta values" generalize rather naively the classical special values z(k) of the Riemann zeta function. They recently have attracted wide attention because of their appearance in several branches of mathematics and physics. In this talk, several basic properties, conjectures and results of these numbers will be reviewed. Also, a connection with "poly-Bernoulli numbers", which also generalize the classical Bernoulli numbers and have combinatorial side too, will be briefly touched upon.

Rinat Kashaev (Steklov Mathematical Institute, St.Petersburg, Russia, and Helsinki Institute of Physics)
Using the non-compact quantum dilogarithm for quantizing the Teichmuller spaces of punctured surfaces
Abstract: The non-compact quantum dilogarithm among other things satisfies the operator counterpart of Roger's pentagon identity on dilogarithm function. This "quantum pentagon" identity appears to be equivalent to the integral analogue of the Ramanujan summation formula. Using the non-compact quantum dilogarithm, one can construct projective representations of the mapping class groups of puctured surfaces within the quantum Teichmuller theory. The projective factor in such representaion is connected with the central charge in quantum Liouville theory.

Anatol N. Kirillov (Nagoya University and Steklov Institute, St.Petersburg, Russia)
Elliptic disease

Alain Lascoux (CNRS, Marne la Vallee, France)
Double Demazure character formula
Abstract: Using a Cauchy-type kernel $\prod_{i+j less then n}1/(1-x_iy_j)$, one recovers the Demazure character formula, as well as double-sided crystal graphs for type An. Applications are the expansion of Schubert polynomials in terms of Demazure characters, as well as vanishing properties of some generalizations of Schur functions.

Katsuhisa Mimachi (Kyushu University)
A representation of the Iwahori-Hecke algebra on the twisted homology associated with the Selberg type integral
Abstract: We consider the twisted cycle (an element of the homology with the local system coefficient) defined by the integrand of the Selberg type integral. When we choose an appropriate subspace of it, we can construct representation of the Iwahori-Hecke algebra.

Tetsuji Miwa (Kyoto University)
Recursion relations for rigged partitions
Abstract: Rigged partitions parametrize sets of symmetric functions which arise in the theory of coinvariants. We derive a recursion relation for certain sets of rigged parttions.

Hitoshi Murakami (Tokyo Institute of Technology)
Volume conjecture for three-manifolds
Abstract: I will explain how one can get the volume and the Chern-Simons invariant from an asymptotic behavior of the Witten-Reshetikhin-Turaev invariants of three-manifolds.

Hiraku Nakajima (Kyoto University)
Finite dimensional representations of quantum affine algebras
Abstract: We define a `t-analogue' of the q-character of finite dimensional representations of quantum affine algebras (untwisted, type ADE), introduced by Frenkel-Reshetkhin. When t = 1, it reproduces the original q-character. There is a combinatorial alogorithm to compute this t-analogues for all irreducible finite dimensional representations via the theory of quiver varieties.

Tomoki Nakanishi: (Nagoya University, joint work with A. Kuniba)
Q-system and characters of Kirillov-Reshetikhin modules of affine quantum groups and Yanigans
Abstract: In 1989, Kirillov and Reshetikhin proposed the algebraic relation (Q-system) among the characters of special class of Yangian modules (KR modules). Except for special cases, it still remains a conjecture. The Q-system is related with several mathematical-physical problems -- such as dilogarithm formula of CFT central charge, formal completeness of Bethe vector, etc. In this talk, we review the recent development on Q-system: Under a certain convergence condition, there is a unique solution of Q-system which is expected to be the characters of KR modules; we give the analytic and combinatorial formulae of the solution, which are related to the formal completeness of the Bethe vector of XXX-type and XXZ-type.

Michitomo Nishizawa (Waseda University)
Generalized Holder's theorem for multiple gamma functions
Abstract: We prove that Vigneras' multiple gamma function does not satisfy any algebraic differential equation over C(z) by using relations between logarithmic derivatives of these functions. Furthermore, related topics are discussed.

Masatoshi Noumi (Kobe University)
Weyl group actions arising from nilpotent Poisson algebras
Abstract: I will explain in some detail about:
1) A method to realize the Weyl group in terms of birational canonical transformations, starting from a nilpotent Poisson algebra of Kac-Moody type.
2) Tau functions and cocycles related to special polynomials.
3) Poisson manifold on which the birational action of the Weyl group is regularized.

Ryu Sasaki (Yukawa Institute of Theoretical Physics)
Quantum Calogero-Moser models: complete integrability for all the root systems
Abstract: The issues related with the integrability of quantum Calogero-Moser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems:
(i) Construction of a complete set of quantum conserved quantities in terms of a total sum of the Lax matrix L, i.e. ,nR(Ln)n, in which (R) is the representation space of the Coxeter group.
(ii) Proof of Liouville integrability.
(iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum.
(iv) Equivalence of the Lax operator and the Dunkl operator.
(v)Algebraic construction of all excited states in terms of creation operators. These are mainly generalisations of the results known for the models based on the A-series, i.e. SU(N)-type, root systems.

Toshiaki Shoji (Science University of Tokyo)
Green functions associated to complex reflection groups
Abstract: The Green functions of finite Chevalley groups are determined as the solution of a certain matrix equation. This matrix equation makes sense even for complex reflection groups such as W=G(e,1,n), and one can define Green functions associated to W. In the case of GLn, Green functions are constructed, based on a combinatorics associated to partitions, as the transition matrix between Schur functions and Hall-Littlewood functions. Schur functions are easily generalized to the case of W.

In this talk, we construct a new type of Hall-Littlewood functions which are associated to certain "symbols" instead of partitions, and show that Green functions for W can be obtained as the transition matrix between these two functions as in the case of GLn.

Taichiro Takagi (National Defense Academy, joint with A. Kuniba and G. Hatayama)
Factorization of Combinatorial R matrices and Associated Cellular Automata
Abstract: We give a description of the box-ball systems in terms of crystal base theory, where the ball-moving algorithm is given by a product of Weyl group operators.

Minoru Takahashi (Institute for Solid State Physics, University of Tokyo)
Simple solution of thermodynamic Bethe ansatz equations
Abstract: Thermodynamic Bethe ansatz equations for one-dimensional solvable models are generally coupled integral equations which contain many unknown functions. For XXZ model at |D| 1, Gaudin-Takahashi equation has infinite unknown functions. Considering properties on the complex plane it is found that equation is reuced to an integral equation which has only one unknown function.

Makoto Taneda (Freelancing Mathematician)
Special polynomials for the sixth Painleve equation and Combinatorics (joint work with A. N. Kirillov)
Abstract: We shall introduce generalized Umemura polynomials and explain a relation between these special polynomials and the sixth Painleve equations.

Hiroaki Terao (Tokyo Metropolitan University)
Logarithmic Gauss-Manin connections on the one-codimensional strata of hyperplane arrangements
Abstract: An explicit combinatorial presentation of a certain logarithmic connection matrix is given and discussed. The connection arises from a combinatorially equivalent family of arrangements of hyperplanes which have only one degeneracy. Although it is the second easiest case, next to the general position case, the matrix already has interesting features.

Akihiro Tsuchiya (Nagoya University)
Seiberg-Witten differential as a period of rational elliptic surface
Abstract: In 1994 Seiberg - Witten established the low energy effective theory of N=2 super Yang - Mills theory in 4 dimention by using so called Seiberg - Witten differential. In this talk I will show how Seiberg - Witten differential give a period mapping of rational elliptic surfaces.

Minoru Wakimoto (Kyushu University)
Representation theory of affine superalgebras
Abstract: As it is known well, the representation of superalgebras is quite different from those of usual Lie algebras. It is still quite mysterious, but is getting more and more interesting in recent years through the joint reseach with V. G. Kac.

As in the case of usual Lie algebras, an interesting class of representations of an affine superalgebra is an integrable representation, by which we mean the Weyl group invariance of its character. Then, in the case of affine superalgebras, it turns necessary to introduce the notion of "principal integrability" and "sub-principal integrability".

The simplest and most important ones among integrable representations are fundamental representations. An explicit construction of fundamental representations of sl(m,n) and osp(m,n) using bosonic and fermionic fields enables us to look at the structure of their representation space more precisely, and gives us some interesting and useful informations. For example, through its study, we can deduce some kinds of character formulas for fundamantal sl(m,n)-modules, namely Weyl-Kac type, theta function type and quasi-particle type. In particular the characters of fundamental sl(m,1)-modules are written in terms of the classical elliptic functions, which have been "forgotten" over one hundred years, and, by a detail analysis of these elliptic functions, we can compute the asymptotic behavior of characters, although they are not modular functions.

Yasuhiko Yamada (Kobe University)
The combinatorial R-matrix and the canonical basis
Abstract: Lusztig's canonical bases Bs depends on the choice of a reduced decomposition s of the longest element w0. If s' is another reduced decomposition of w0, there is a bijection Rss' : BsBs'. Using the explicit combinatorial description of the bijection Rss', we derive certain combinatorial representation of affine Weyl group of type An(1), which turns out to be equivalent with the combinatorial R-matrix for symmetric tensors of Uq(sln). As a byproduct we obtain a version of inverse ultra-discretized formula of the combinatorial R-matrix similar with that given by Hatayama et.al. (q-alg/9912209).

Masaaki Yoshida (Kyushu University)
A hypergeometric story
Abstract: About: ``Six points on the projective plane'': On such a simple object, I can tell a fascinating hypergeometric story.

--- Reception: August 24, Thursday, 18:20, Symposium, Admission fee 4,000 Yen

Everybody is cordially invited!!!


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International Workshop on Physics and Combinatorics 1999


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