International Workshop on Physics and Combinatorics

August 23 -27, 1999

Graduate School of Mathematics, Nagoya University, Nagoya, Japan

Proceedings, full contents is here
List of participants


K. Aomoto, A.N. Kirillov, T. Nakanishi, A. Tsuchiya, H. Umemura

Invited Speakers:

  • K. Aomoto (Nagoya University)
  • E. Date (Osaka University)
  • J.-F. van Diejen (Chile University, Chile)
  • N. Kurokawa (Tokyo Institute of Technology)
  • S.C. Milne (Ohio State University, U.S.A.)
  • T. Miwa (RIMS, Kyoto University)
  • A. Schilling (Amsterdam University, Netherlands)
  • P. Terwillinger (University of Wisconsin, Madison, U.S.A.)
  • H. Umemura (Nagoya University)
  • M. Yoshida (Kyushu University)


    K. Hikami (University of Tokyo)
    N. Kawanaka (Osaka University)
    A.N. Kirillov (Nagoya University and Steklov Institute, Russia)
    A. Kuniba (University of Tokyo)
    K. Mimachi (Kyushu University)
    H. Murakami (Waseda University)
    T. Nakanishi (Nagoya University)
    M. Nishizawa (Waseda University)
    M. Noumi (Kobe University)
    T. Tokihiro (University of Tokyo)
    M. Taneda (Kumamoto)
    H. Terao (Tokyo Metropolitan University)
    T. Terasoma (University of Tokyo)
    K. Ueno (Waseda University)
    H.-F. Yamada (Hokkaido University)
    Y. Yamada (Kobe University)


    August 23, Monday
    August 24, Tuesday
    August 25, Wednesday
    August 26, Thursday
    August 27, Friday


    (all talks in Main Lecture Hall 509 (except for Monday afternoon in Room 111), refreshments in Room 522, computer facilities Room 109)

    Monday, August 23
    10:30-11:30 Hiroshi Umemura, Painleve equations in the past 100 years, abstract
    11:30-13:30 lunch break
    13:30-14:30 Masaaki Yoshida, Uniformizations of point-configuration spaces, abstract
    14:45-15:45 Anne Schilling, Level-restricted rigged configurations, abstract
    16:00-17:00 Atsuo Kuniba, Soliton cellular automata associated with finite crystals, abstract

    Tuesday, August 24
    9:30-10:30 Stephen C. Milne, Transformations of U(n+1) multiple basic hypergeometric series, abstract
    10:45-11:45 Etsuro Date (joint with S.S. Roan), On quotients of the Onsager alagebra, abstract
    11:45-13:30 lunch break
    13:30-14:30 Masatoshi Noumi and Yasuhiko Yamada, A realization of Weyl groups and combinatorics of special polynomials, abstract
    14:45-15:45 Noriaki Kawanaka, A q-Cauchy identity for Schur functions and complex reflection groups, abstract
    16:00-17:00 Michimoto Nishizawa (joint with Kimio Ueno), Integral solutions of hypergeometric q-difference systems with |q|=1, abstract
    17:15-18:15 Hitoshi Murakami (joint with Jun Mirakami), The colored Jones polynomials and the simplicial volume of a knot, abstract

    Wednesday, August 25
    9:30-10:30 Stephen C. Milne (joint with Verne E. Leininger), Some new infinite families of eta function identities, abstract
    10:45-11:45 Paul Terwillinger (joint with Tatsuro Ito and Kenichiro Tanabe) A q-analog of the Onsager algebra, abstract
    11:45-13:30 lunch break
    13:30-14:30 Jan Felipe van Diejen, Combinatorial formulas for q-spherical functions associated to rank-one (quantum-) symmetric spaces, abstract
    14:45-15:45 Tetsuji Tokihiro, Box and ball systems as ultradiscrete soliton equations, abstract
    16:00-17:00 Tomohide Terasoma, Selberg integral and multiple zeta values, abstract
    17:15-18:15 Makoto Taneda, Representation of Umemura Polynomials, abstract

    Thursday, August 26
    9:30-10:30 Stephen C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions, abstract
    10:45-11:45 Nobushige Kurokawa, Multiple sine functions I. Basic properties, abstract
    11:45-13:30 lunch break
    13:30-14:30 Kazuhiko Aomoto, Wu's equations and Quasi Hypergemetric Functions, abstract
    14:45-15:45 Tomoki Nakanishi (joint with Atsuo Kuniba), String solutions of the Bethe ansatz equation in the q=0 limit and combinatorial character formula for affine quantum group, abstract
    16:00-17:00 Katsuhisa Mimachi, A duality of the Macdonald-Koornwinder polynomials and its application to the integral representations, abstract
    17:15-18:15 Kazuhiro Hikami, Discrete Integrable Systems and Quantum Dilogarithm Function, abstract

    Friday, August 27
    9:30-10:30 Tetsuji Miwa, Combinatorics of $\widehat{sl}_2$ Spaces of Coinvariants, abstract
    10:45-11:45 Nobushige Kurokawa, Multiple sine functions II. Applications, abstract
    11:45-13:30 lunch break
    13:30-14:30 Hiroaki Terao, Flat connections arising from a family of arrangements, abstract
    14:45-15:45 Hiro-Fumi Yamada, Schur's Q-functions and affine Lie algebras, abstract
    16:00-17:00 Anatol Kirillov, Ubiquity of Kostka-Foulkes polynomials, abstract


    Kazuhiko Aomoto, August 26, 13:30-14:30
    Wu's equations and Quasi Hypergeometric Functions
    Abstract: Among the class of quasi hypergeometric functions the ones obtained by solving Wu's equations (which was discovered by Kazumoto Iguchi) have particularly simple properties.
    I shall discuss the properties of transformation formulas, singularities and monodromy for them.

    Etsuro Date (joint with S.S. Roan), August 24, 10:45-11:45
    On quotients of the Onsager alagebra
    Abstract: We call an ideal of a Lie algebra closed if the quotient by it does not contain central elements. After classifying closed ideals of the Onsager algebra, we study the structure of quotients by closed ideals. For that purpose we use a realization of the Onsager algebra in a completion of ${\bf C}[t]\otimes sl_2$.

    Jan Felipe van Diejen, August 25, 13:30-14:30
    Combinatorial formulas for (q-)spherical functions associated to rank-one (quantum-) symmetric spaces
    Abstract: This talk reports on joint work with A.N. Kirillov regarding the study of rank-one zonal spherical functions and their q-deformations. We present combinatorial representations for the zonal spherical funcions on odd-dimensional spheres and hyperboloids, as well as q-deformations thereof. The expansion coefficients in the combinatorial formulas are interpreted as characters of irreducible representations of the general linear group. To derive the combinatorial formulas we use the inverse scattering theory of reflectionless Jacobi operators.

    Kazuhiro Hikami, August 26, 17:15-18:15
    Discrete Integrable Systems and Quantum Dilogarithm Function
    Abstract: We study a quantization of discrete analogue of both the KdV-type equation and the Toda field theory. We shall construct the generating functions for the conserved operators by use of the quantum dilogarithm function.

    Noriaki Kawanaka, August 26, 14:45-15:45
    A q-Cauchy identity for Schur functions and complex reflection groups
    Abstract: We consider a q-analogue of the Frobenius-Schur indices for the irreducible characters of imprimitive finite complex reflection groups. This leads to a q-analogue of the classical Cauchy identity for Schur functions. Related (partly conjectural) q-identities involving Hall-Littlewood, or Macdonald symmetric functions will also be presented.

    Anatol Kirillov, August 27, 16:00-17:00
    Ubiquity of Kostka-Foulkes polynomials
    Abstract: We will describe appearences of Kostka-Foulkes polynomials in the theory of finite abelian p-groups, algebraic geometry, the representation theory of the general linear and symmetric groups and statistical physics.

    Atsuo Kuniba, August 23, 16:00-17:00
    Soliton cellular automata associated with finite crystals
    Abstract: I report on a joint work with G. Hatayama and T. Takagi. We introduce a class of cellular automata associated with irreducible finite dimensional representations of quantum affine algebras $U'_q(\hat{\geh}_n)$. They exhibit soliton behavior whose scattering is described conjecturally by the combinatorial R-matrices of the smaller algebra $U'_q(\hat{\geh}_{n-1})$. For the $A^{(2)}_2$ example, we derive piecewise linear evolution equations for N-solitons. They admit a tropical variable change into Toda-type equations which are very similar to the Y and T-systems for transfer matrices. We construct an explict solution in some sector, which leads to the N-soliton solution of the automaton through the ultra-discretization.

    Nobushige Kurokawa, August 26, 27, 10:45-11:45
    Multiple Sine Functions, I. Basic Properties, II. Applications
    Abstract: In part I, I recall two ways to construct multiple sine functions. The key properties are periodicity and algebraic differential equations. Special values are very interesting. In part II, these properties are applied to investigate special values of zeta functions containing zeta(3), and to determine the gamma factors of Selberg zeta functions. Some q-analogues will be reported also.

    Stephen C. Milne, August 24, 9:30-10:30
    Transformations of U(n+1) multiple basic hypergeometric series
    Abstract: The purpose of this talk is to survey the transformation theory of U(n+1) multiple basic hypergeometric series starting with the U(n+1) terminating very-well-poised $_6\phi_5$ summation theorems. These series were strongly motivated by L. C. Biedenharn and J. D. Louck and coworkers mathematical physics research involving angular momentum theory and the unitary groups U(n+1), or equivalently A_n. They are directly related to the corresponding Macdonald identities. This U(n+1) or A_n theory has also been extended to the root systems C_n and D_n. There are now many applications of A_n and/or subsequently C_n and D_n multiple basic hypergeometric series. These include the following topics: A. N. Kirillov - quantum groups; R. Gustafson - multidimensional beta and/or Barnes integral evaluations; C. Krattenthaler and I. Gessel - plane partition enumeration; S. Milne - analytic number theory (sums of squares); S. Milne, G. Lilly, G. Bhatnagar, C. Krattenthaler, and M. Schlosser - multidimensional matrix inversions; S. Milne and V. Leininger - new infinite families of eta function identities; Y. Kajihara and M. Noumi - applications to raising operators for Macdonald polynomials. As an introduction to this area we discuss some of the main results and techniques from the following outline of the development of U(n+1) basic hypergeometric series.
    The U(n+1) terminating very-well-poised $_6\phi_5$ summation theorems extend Rogers' classical one-variable work and are central to our theory. They may be proved using q-difference equations arising from the Lagrange interpolation formula and partial fraction expansions. The U(n+1) $_6\phi_5$ summation theorems may in turn be specialized to obtain the U(n+1) extension of Andrews' matrix formulation of the Bailey Transform. The U(n+1) Bailey transform is then applied to the U(n+1) $_6\phi_5$ summation theorems to derive the U(n+1) terminating, balanced $_3\phi_2$ summation theorems, whose special cases include U(n+1) q-Gauss summation theorems, q-Chu-Vandermonde theorems and U(n+1) q-binomial theorems. An analytic continuation argument applied to a U(n+1) q-binomial theorem yields the U(n+1) extension of Ramanujan's $_1\psi_1$ sum, and Gustafson's U(n+1) $_6\psi_6$ summation turns out to be the next higher dimensional version of the U(n+1) $_1\psi_1$ sum. For example, the two-dimensional U(3) $_1\psi_1$ summation is equivalent to Bailey's classical one-dimensional $_6\psi_6$ summation. We also obtain U(n+1) extensions of the Jacobi triple product identity. The Bailey transform coupled with the U(n+1) balanced $_3\phi _2$ summation theorems yields the U(n+1) extension of Andrews' explicit formulation of the Bailey Lemma, which upon iteration gives several U(n+1) generalizations of Watson's q-analogue of Whipple's transformation formula. Special and limiting cases include the non-terminating U(n+1) $_6\phi_5$ summation, the U(n+1) extension of the terminating balanced $_8\phi_ 7$ summation theorem, and the U(n+1) Rogers-Ramanujan-Schur identities. A classical interchange of summation argument leads to the U(n+1) $_{10}\phi_9$ transformation formulas. Important limiting cases include the U(n+1) generalization of Bailey's non-terminating extension of Watson's transformation. This in turn leads to the non-terminating U(n+1) extension of Bailey's balanced $_8\phi_ 7$ summation theorem. The classical case of all this work, corresponding to $A_1$ or equivalently U(2), contains a substantial amount of the theory and application of one-variable basic hypergeometric series.

    Stephen C. Milne (joint with Verne E. Leininger), August 25, 9:30-10:30
    Some new infinite families of eta function identities
    Abstract: Ever since Euler proved his expansion for $\prod_{i=1}^\infty (1-q^i)\equiv (q;q)_\infty$ mathematicians have been looking for other identities of this form. In 1829, Jacobi utilized his triple product identity to derive an elegant expansion for $(q;q)_\infty^3$. Since then, expansions have been found for $(q;q)_\infty^c$ for many values of c. These included several infinite families of expansions and a few exceptional cases. In 1892 F. Klein and R. Fricke gave a result for c=8. This was rediscovered by S. Ramanujan in 1916. L. Winquist in 1969 proved a result for c=10 but stated that this was first found by J. Rushforth, then independently discovered by A. Atkin. L. Winquist also noted that A. Atkin had formulae for c=14 and c=26. The existence of these identities had been suggested in 1955 by M. Newman. In 1972 F. Dyson gave his famous formula for c=24 and stated that formulae corresponding to c= 3,8,10,14,15,21,24,26,28,35,36,... had been found, but noted that these ad hoc results had been unified by I. Macdonald. In his landmark 1972 paper, Macdonald related most of these expansions for $(q;q)_\infty^c$ to affine root systems. (This connection with Lie Algebras has been the main focus of work since I. Macdonald.) A few notable exceptions remained: c=2 found by Hecke and Rogers, c=4 by Ramanujan, and c=26 by Atkin.
    In this talk we discuss our derivation of new, more symmetrical expansions for $(q;q)_\infty^{n^2+2n}$ by means of our multivariable generalization of Andrews' variation of the standard proof of Jacobi's $(q;q)_\infty^3$ result. We also present examples of our general expansion for $(q;q)_\infty^c$ where c=3,8,15,24. Our proof relies upon Milne's new U(n) multivariable extension of the Jacobi triple product identity. This result is deduced from a U(n) multiple basic hypergeometric series generalization of Watson's very-well-poised $\_8\phi_7$ transformation. The derivation of our $(q;q)_\infty^{n^2+2n}$ expansion also utilizes partial derivatives and dihedral group symmetries to write the sum over regions in n-space. We note that our expansions for $(q;q)_\infty^{n^2+2n}$ are equivalent to Macdonald's A_n family of eta-function identities. In addition, we utilize various summation and transformation formulas for U(n+1), equivalently A_n, multiple basic hypergeometric series to derive new infinite families of expansions for $(q;q)_\infty^{n^2+2}$ and $(q;q)_\infty^{n^2}$, similar products of these, and the corresponding powers of the $\eta$-function. (Recall that the eta function is defined by $\eta(q):= q^{1/24}(q;q)_\infty$.) These additional infinite families of expansions extend the list in Appendix I of Macdonald's 1972 paper. All of this work is motivated by Milne's U(n+1) multiple basic hypergeometric series treatment of the Macdonald identities for $A_{\ell}^{(1)}$.

    Stephen C. Milne, August 26, 9:30-10:30
    Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions
    Abstarct: In this talk we give several infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's (1829) 4 and 8 squares identities to $4n^2$ or $4n(n+1)$ squares, respectively, without using cusp forms. The Schur function form of these infinite families of identities are analogous to the $\eta$-function identities of Macdonald. We also utilize a special case of our methods to give a proof of the two Kac-Wakimoto conjectured identities involving representing a positive integer by sums of $4n^2$ or $4n(n+1)$ triangular numbers, respectively. These results, depending on new expansions for powers of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular C-fractions, Hankel or Tur\'anian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson's $C_{\ell}$ nonterminating $_6\phi_5$ summation theorem, and Andrews' basic hypergeometric series proof of Jacobi's 2, 4, 6, and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for $n^2$ or n(n+1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto (1994), and many others.

    Katsuhisa Mimachi, August 26, 16:00-17:00
    A duality of the Macdonald-Koornwinder polynomials and its application to the integral representations
    Abstract: A formula representing a duality of the Macdonald-Koornwinder polynomials will be presented. The formula induces several integral representations of the Macdonald-Koornwinder polynomials, of Heckman-Opdam's Jacobi polynomials of type $BC_m,$ etc.

    Tetsuji Miwa, August 27, 9:30-10:30
    Combinatorics of $\widehat{sl}_2$ Spaces of Coinvariants
    Abstract: I will talk some results obtained in collaboration with B. Feigin, R. Kedem, S. Loktev and E. Mukhin on the $\widehat{sl}_2$ coinvariants.

    Hitoshi Murakami (joint with Jun Murakami), August 24, 17:15-18:15
    The colored Jones polynomials and the simplicial volume of a knot
    Abstract: We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect non-trivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev's quantum dilogarithm invariants for links. Therefore Kashaev's conjecture can be restricted as follows: The colored Jones polynomials determine the hyperbolic volume for a hyperbolic knot. Modifying this, we propose a stronger conjecture: The colored Jones polynomials determine the simplicial volume for any knot. If our conjrcture is true, then we can prove that a knot is trivial if and only if and only if all of its Vassiliev invariants are trivial.

    Tomoki Nakanishi (joint with Atsuo Kuniba), August 26, 14:45-15:45
    String solutions of the Bethe ansatz equation in the q=0 limit and combinatorial character formula for affine quantum group
    Abstract: We study a class of meromorphic solutions (string solutions) of the XXZ Bethe ansatz equation. We show the string solutions are obtained by solving the linear congruence equation (string equation). The counting of the off-diagonal solutions of the string equation leads us to an interesting combinatorial character formula for finite dimensional modules of affine quantum group. The formula is related to the character formula by Kirillov and Reshetikhin.

    Michimoto Nishizawa (joint with Kimio Ueno), August 24, 16:00-17:00
    Integral solutions of hypergeometric q-difference systems with |q|=1
    Abstract: By using the double sine function, we construct two kinds of integral solutions of hypergeometric q-difference system with |q|=1 . One is the Barnes type and the other is the Euler type. We consider some properties of these solutions.

    Masatoshi Noumi (joint with Yasuhiko Yamada), August 24, 13:30-14:30
    A realization of Weyl groups and combinatorics of special polynomials
    Abstract: We will discuss a realization of Weyl groups as groups of birational transformations and combinatorial properties of certain cocycles of Weyl groups. These cocycles give a generalization of special polynomials arising from the Painleve equations, to arbitrary root systems.

    Anne Schilling, August 23, 14:45-15:45
    Level-restricted rigged configurations
    Abstract: Recently, in collaboration with A.N. Kirillov and M. Shimozono, we established a statistic-preserving bijection between Littlewood-Richardson tableaux and rigged configurations. In this talk we review this bijection and its main properties, and then show that it respects level-restriction. More precisely, there is a natural way to impose a level-restriction on the set of Littlewood-Richardson tableaux. The image of this restricted set under the bijection can be described explicitly. As a corollary, fermionic formulas for the level-restricted Kostka-Foulkes polynomials and for branching functions of affine type A cosets can be derived/proved.

    Makoto Taneda, August 25, 17:15-18:15
    Representation of Umemura Polynomials
    Abstract: Umemura presented the special polynomials associated with solutions of the sixth Painleve equation. Noumi, Okada and Okamoto gave the conjectural representation of these special polynomials using an interesting combinatorial interpretation. We will show that these conjectures are true.

    Hiroaki Terao, August 27, 13:30-14:30
    Flat connections arising from a family of arrangements
    Abstract:A combinatorially stable family of arrangements determines a flat connection on the moduli space of the family. When the family is universal, the connection is completely determined by the combinatorial type of the arrangements. We study the exlplicit form of the connection matrices presented with respect to $\beta$ nbc-bases.

    Tomohide Terasoma, August 25, 16:00-17:00
    Twisted Ehrhart polynomials
    Abstract: Let $f(x_1, \dots , x_n) \in \bold C [x_1, \dots ,x_n]$ be a non-degenerate Laurant polynomail with respect to its Newton polytope $\Delta$. For a sufficiently generic rational number $\alpha_0 ,\dots , \alpha_n \in \frac{1}{d}\bold Z$, the twisted cohomology group $H^n(\bold C^{\times}-V(f), \bold C(\eta))$ with the coefficient in the local system $\bold (\eta)$ of the multivalued function $\eta = f^{-\alpha_0}x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ has a natural $\bold Q(\zeta_d)$-Hodge structure: $H^n(\bold C^{\times}-V(f), \bold C(\eta)) = \oplus_{i=1}^n H^{i,j}.$ It is called a hypergeometric Hodge structure. The generating function $p(t) = \sum_{i=0}^n (\dim H^{i,j}) t^i$ is called the Hodge polynomial of $H^n(\bold C^{\times}-V(f), \bold C(\eta))$ and it is expressed in terms of twisted Ehrhart polynomial of the Newton polytope $\Delta$. It is know that there exists many non-tirival identity for hypergeomtric Hodge structure which is a generalization of Gauss multiplication formula. This identity arise from algebraic correspondeces. In this lecture, we will discuss about the identity of twisted Ehrhart polynomial arising from these generalized Gauss multiplication correspondences.

    Paul Terwillinger (joint with Tatsuro Ito and Kenichiro Tanabe), August 25, 10:45-11:45
    A q-analog of the Onsager algebra
    Abstract: The Onsager algebra is a certain infinite dimensional complex Lie algebra that has been used to study the two dimensional Ising and chiral Potts model. Perk and Davies showed the Onsager algebra has a presentation involving two generators $A$, $A^*$ satisfying the relations

    $[A,[A,[A,A^*]]] = 16[A,A^*]$
    $[A^*,[A^*,[A^*,A]]] = 16[A,A^*]$
    where [ , ] denotes the Lie bracket. The above relations are known as the Dolan-Grady relations. Later Roan showed the Onsager algebra is isomorphic to a certain subalgebra of the affine Lie algebra $A_1^{(1)}$. In this talk, we define an algebra T which can be viewed as a q-analog of the Onsager algebra. Let $\fld$ denote any field, and let $\beta, \gamma, \gamma^*,\varrho, \varrho^*$ denote scalars in $\fld$. We define $T=T( \beta, \gamma, \gamma^*, \varrho, \varrho^*)$ to be the associative $\fld$-algebra with identity generated by two symbols $A$, $A^*$ subject to the relations
    $0 = [A,A^2A^*-\beta AA^*A + A^*A^2 -\gamma (AA^*+A^*A)- \varrho A^*],$
    $0 = [A^*,A^{*2}A-\beta A^*AA^* + AA^{*2} -\gamma^* (A^*A+AA^*)- \varrho^* A],$
    where [r,s] means rs-sr. If one sets $\beta = 2$, $\gamma = 0$, $\gamma^*=0$, $\varrho=16$, $\varrho^*=16$, in the above relations, one gets essentially the Dolan-Grady relations. To understand the algebra T, we consider the finite dimensional irreducible T-modules on which $A$ and $A^*$ act as semi-simple linear transformations. One type of T-module of this sort is given by what we call a Leonard pair. Let V denote a vector space over $\fld$ that has finite positive dimension. By a Leonard pair on V, we mean an ordered pair $A,A^*$ consisting of linear transformations from V to V that satisfy both conditions (i) there exists a basis for V with respect to which the matrix representing $A$ is diagonal, and the matrix representing $A^*$ is irreducible tridiagonal,(ii) there exists a basis for V with respect to which the matrix representing $A^*$ is diagonal, and the matrix representing $A$ is irreducible tridiagonal. (A tridiagonal matrix is said to be irreducible whenever all entries immediately above and below the main diagonal are nonzero). The Leonard pairs have been completely classified by Terwilliger, and they turn out to be closely related to the q-Racah polynomials. It is an intriguing fact that there exist finite dimensional irreducible T-modules on which $A$ and $A^*$ act as semi-simple linear transformations, but are not Leonard pairs. It is an open problem to classify these modules, and the main goal of the present talk is to explain what is known so far along this line.

    Tetsuji Tokihiro, August 25, 14:45-15:45
    Box and ball systems as ultradiscrete soliton equations
    Abstract: A soliton cellular automaton, which represents a movement of finite number of balls in any array of boxes, is investigated. Its dynamics is described by two kinds of ultra-discrete equations obtained from the degenerating formula of KP hierarchy. The combinatorial rules for soliton interactions and factorization property of the scattering matrices (Yang-Baxter relation) are proved by means of inverse ultra-discretization. The interpretation with the aid of combinatorial R-matrix lattice model is also presented.

    Hiroshi Umemura, August 23, 10:30-11:30
    Painleve equations in the past 100 years
    Abstract: We review developments in the research of the Painleve equations. The Painleve equations are a hundred years old. They were almost forgoten during several decades. Today they revive and we can admet that they are more interesting mathematical objects than expected in the early years of the century. For example, recent discovery of their relation with the combinatorics reveals a new feature of the Painleve equations.

    Hiro-Fumi Yamada, August 27, 14:45-15:45
    Schur's Q-functions and affine Lie algebras
    Abstract: This talk deals with the weight vectors of the basic representations of the twisted affine Lie algebra of type D^(2)_{l+1}. The weight vectors are expressed in terms of Schur's Q-functions. The up and down motion along a string of the fundamental imaginary root is described as a combinatorial game on an abacus. As a consequence one can determine the homogeneous polynomial solutions to the reduced BKP hierarchies as the maximal weight vectors. The results obtained will be applied to the case of A^(1)_1. The embedding of A^(1)_1 into D^(2)_4 gives rise to a realization of the basic representation of A^(1)_1. The isomorphism between the "standard" and this "non-standard" realizations of the basic representation induces a "funny" identity of Schur's S-functions and Q-functions indexed by certain partitions. In fact this identity reflects a nature of decomposition matrices of the modular spin representations of the symmetric groups with characteristic 2.

    Masaaki Youshida, August 23, 13:30-14:30
    Uniformizations of point-configuration spaces
    Abstract: Some examples of (real and complex) point-configuration spaces (number of points less then 6) will be maniacly studied.

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