# International Workshop on Physics and Combinatorics

## Organizers:

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)

## Speakers:

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)

## Schedule

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

# Schedule

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

# Abstracts

$[A,[A,[A,A^*]]] = 16[A,A^*]$
$[A^*,[A^*,[A^*,A]]] = 16[A,A^*]$
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.