International Workshop on Physics and
Combinatorics
August 2126, 2000
Weather in Nagoya 
Welcome to Nagoya 
Welcome to Kyoto 
Japan Travel Guide


O r g a n i z e r s:


K. Aomoto, F. Hiroshita,
A.N. Kirillov,
R. Kobayashi, A. Tsuchiya, H. Umemura



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 (ETHZentrum, 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



S c h e d u l e



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)
 10:0011:00
A. Tsuchiya,
SeibergWitten differential as a period of rational elliptic surface,
abstract
 11:1012:10
 M. Wakimoto,
Representation theory of affine superalgebras,
abstract
 12:1013:30

lunch break

 13:3014:30
 K. Kadell,
The Macdonald and Dyson Polynomials,
abstract
 14:4015:40
 E. Bannai,
Some results on modular forms motivated by coding theory,
abstract
15:4016:00

tea break

 16:0017:00
M. Kaneko,
Multiple zeta values and polyBernoulli numbers  a survey,
abstract
 17:1018:10
 T. Takagi,
Factorization of Combinatorial R matrices and Associated Cellular
Automata,
abstract
 18:3019:30
 Summer Academic Concert,
program
9:3010:30
M. Haiman,
The n! and Macdonald positivity conjectures,
abstract
10:4511:45
M. Wakimoto,
Representation theory of affine superalgebras II,
abstract
11:4513:15

lunch break

13:1514:15
A. Lascoux, Double Demazure character formula,
abstract
14:2515:25
M. Noumi,
Weyl group actions arising from nilpotent Poisson algebras,
abstract
15:2515:45

tea break

15:4516:45
K. Hikami,
Exclusion statistics and Universal Chiral Partition Function,
abstract
16:5517:55
M. Taneda,
Special polynomials for the sixth Painleve equation and Combinatorics
abstract
9:3010:30
J. Felder,
The elliptic gamma function, SL(3,Z), and qdeformation of conformal
field theory on elliptic curves,
abstract
10:4511:45
M. Haiman,
Hilbert schemes and the proof of the n! conjecture,
abstract
11:4513:15

lunch break

13:1514:15
H. Terao, Logarithmic GaussManin connections on the
onecodimensional strata of hyperplane arrangements,
abstract
14:2515:25
T. Shoji,
Green functions associated to complex reflection groups,
abstract
15:2515:45

tea break

15:4516:45
 H. Nakajima,
Finite dimensional representations of quantum affine algebras,
abstract
16:5517:55
K. Iguchi,
A Theory of Interacting Many Body Systems with Exclusion Statistics:
Origin of Exclusion Statistics, Haldane Liquids, SutherlandWu Equations,
Grand Partition Function and LeeYang Theorem,
abstract
9:3010:30
M. Takahashi,
Simple solution of thermodynamic Bethe ansatz equations,
abstract
10:4511:45
T. Eguchi,
Quantum cohomlogy and structure of topological string theory,
abstract
11:4513:15

lunch break

13:1514:15
M. Ioffe, Multiparticle Supersymmetrical Quantum
Mechanics and representations of Permutation Group,
abstract
14:2515:25
R. Sasaki,
Quantum CalogeroMoser models: complete integrability for all the
root systems,
abstract
15:2515:45

tea break

15:4516:45
Y. Yamada,
The combinatorial Rmatrix and the canonical basis,
abstract
16:5517:55
K. Kajiwara,
Classical solutions for Painleve and Discrete Painleve equations,
abstract
18:2020:00
Reception, information
9:3010:30
T. Miwa,
Recursion relations for rigged partitions,
abstract
10:4511:45
M. Haiman, Diagonal harmonics,
abstract
11:4513:15

lunch break

13:1514:15
R. Kashaev,
Using the noncompact quantum dilogarithm for quantizing the Teichmuller
spaces of punctured surfaces,
abstract
14:2515:25
H. Murakami, Volume conjecture for threemanifolds,
abstract
15:2515:45

tea break

15:4516:45
T. Nakanishi,
Qsystem and characters of KirillovReshetikhin modules of affine quantum
groups and Yanigans,
abstract
16:5517:55
M. Nishizawa,
Generalized Holder's theorem for multiple gamma functions,
abstract
9:3010:30
M. Yoshida, A hypergeometric story,
abstract
10:4511:45
J. Kaneko,
Forrester's constant term conjecture and ChuVandermonde formula for
generalized binomial coefficients,
abstract
11:4513:15

lunch break

13:1514:15
K. Mimachi,
A representation of the IwahoriHecke algebra on the twisted homology
associated with the Selberg type integral,
abstract
14:2515:25
B. Feigin, Gordontype filtrations and corresponding
bosonfermion character formulas, abstract
15:3516:35
A. Kirillov, Elliptic disease,
abstract


A b s t r a c t s

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
selfdual 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 halfintegral 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/lintegral weights. We show that the space of modular
forms of 1/5integral 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
1/7integral 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 genustwo amplitude in the case of
P^{1} model.
Boris Feigin (Landay Institute for Theoretical
Physics, Russia, and RIMS, Kyoto University)
Gordontype filtrations and corresponding bosonfermion 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 Shubertlike
manifolds.
Giovanni Felder (ETHZentrum, Switzerland, joint
work with Alexander Varchenko)
The elliptic gamma function, SL(3,Z), and
qdeformation of conformal field theory on elliptic curves
Abstract: The elliptic gamma function is an elliptic version
of the qgamma 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 qdeformation 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 representationtheoretic 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)^{(n1)}, 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 quasiparticles which obey exclusion statistics.
We will also explain a spinon basis of the level1 WZW model.
Kazumoto Iguchi (Freelancing Physicist)
A Theory of Interacting Many Body Systems with Exclusion Statistics:
Origin of Exclusion Statistics, Haldane Liquids, SutherlandWu Equations,
Grand Partition Function and LeeYang Theorem
Abstract: I will discuss a theory of interacting many body systems
with exclusion statistics. I first show how exclusion statistics appears
from the longranged 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 SutherlandWu functional
equation and the grand partition function. Finally, I will mention
applications of the theory to some other problems such as the LeeYang
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 wellknown 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 2dim quantum and classical
systems. The method of multidimensional SUSY QM is also applied
to the investigation of SUSY Nparticle systems on a line for
the case of separable centerofmass motion. New decomposition of
the Superhamiltonian into blockdiagonal 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
S_{N}, which
correspond to the Young tableaux
(NM, 1^{M}).
The connections with known generalizations of Nparticle 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,...,t_{n}
of the Macdonald polynomials. We give constant term orthogonality relations
for l=(r) using Good's proof, for
n=2 using the qSaalschutz 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 ChuVandermonde 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
ChuVandermonde formula for generalized binomial coefficients.
Masanobu Kaneko (Kyushu University)
Multiple zeta values and polyBernoulli 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
"polyBernoulli 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 noncompact quantum dilogarithm for quantizing the Teichmuller
spaces of punctured surfaces
Abstract: The noncompact 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 noncompact 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 Cauchytype kernel
$\prod_{i+j less then n}1/(1x_iy_j)$, one recovers
the Demazure character formula, as well as doublesided crystal graphs
for type A_{n}. 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 IwahoriHecke 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 IwahoriHecke 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 threemanifolds
Abstract: I will explain how one can get the volume and the
ChernSimons invariant from an asymptotic behavior of the
WittenReshetikhinTuraev invariants of threemanifolds.
Hiraku Nakajima (Kyoto University)
Finite dimensional representations of quantum affine algebras
Abstract: We define a `tanalogue' of the qcharacter
of finite dimensional representations of quantum affine algebras (untwisted,
type ADE), introduced by FrenkelReshetkhin. When t = 1, it reproduces
the original qcharacter. There is a combinatorial alogorithm to
compute this tanalogues for all irreducible finite dimensional
representations via the theory of quiver varieties.
Tomoki Nakanishi: (Nagoya University, joint work
with A. Kuniba)
Qsystem and characters of KirillovReshetikhin modules of affine quantum
groups and Yanigans
Abstract: In 1989, Kirillov and Reshetikhin proposed the algebraic
relation (Qsystem) among the characters of special class of Yangian
modules (KR modules). Except for special
cases, it still remains a conjecture. The Qsystem is related with several
mathematicalphysical problems  such as dilogarithm formula of CFT
central charge, formal completeness of Bethe
vector, etc. In this talk, we review the recent development on Qsystem:
Under a certain convergence condition, there is a unique solution of
Qsystem 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 XXXtype and XXZtype.
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 KacMoody 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 CalogeroMoser models: complete integrability for all the root
systems
Abstract: The issues related with the integrability
of quantum CalogeroMoser 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. å_{µ,nÎ}_{R}(L^{n})_{µn}, 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 Aseries, 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
GL_{n}, Green functions are
constructed, based on a combinatorics
associated to partitions, as the transition matrix between Schur functions
and HallLittlewood functions. Schur functions are easily generalized to
the case of W.
In this talk, we construct a new type of HallLittlewood
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 GL_{n}.
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 boxball systems in terms of
crystal base theory, where the ballmoving 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 onedimensional
solvable models are generally coupled integral equations which contain
many unknown functions. For XXZ model at
D³ 1,
GaudinTakahashi 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 GaussManin connections on the onecodimensional 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)
SeibergWitten 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 "subprincipal 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 WeylKac type, theta function type and quasiparticle 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 Rmatrix and the canonical basis
Abstract: Lusztig's canonical bases
B_{s}
depends on the choice of a reduced decomposition
s of the longest element
w_{0}.
If s' is another reduced decomposition of
w_{0}, there is a bijection
R_{s}^{s'} :
B_{s}®B_{s'}.
Using the explicit combinatorial description of the bijection
R_{s}^{s'}, we derive certain
combinatorial representation of affine Weyl group of type
A_{n}^{(1)},
which turns out to be equivalent with the combinatorial Rmatrix
for symmetric tensors of
U_{q}(sl_{n}).
As a byproduct we obtain a version of inverse ultradiscretized formula of
the combinatorial Rmatrix similar with that
given by Hatayama et.al. (qalg/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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Last updated September 8, 2000