Physics and Combinatorics

The following sections are included:

• PREFACE

• CONTENTS

We give a conjectural monodromy formula for quasi hypergeometric functions obtained as the solutions to Wu–Sutherland equations in case of generic exponents.

We exhibit an explicit determinantal formula for the Harish–Chandra series on the hyperboloid H^p,q = SO₀(p, q; ℝ)/SO₀(p - 1, q; ℝ) with p, q odd and larger than 1.

We consider a discretization of the quantum affine Toda field theory. We clarify the quantum group structure of this integrable lattice field theory using the lattice vertex operators. We also construct the integrals of motion explicitly in terms of the quantum dilogarithm function. Further discussed is the duality of this system.

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.

We give a method to produce Hall-Littlewood symmetric function identities from symmetric spaces over finite fields. A special case of such identities motivates to introduce a notion of q-Frobenius-Schur index of an irreducible character of a finite reflection group G. In the case G is an imprimitive complex reflection group, we calculate the indices explicitly using a new q-Cauchy identity for Schur functions.

We report about results revolving around Kostka–Foulkes and parabolic Kostka polynomials and their connections with Representation Theory and Combinatorics. It appears that the set of all parabolic Kostka polynomials forms a semigroup, which we call Liskova semigroup. We show that polynomials frequently appearing in Representation Theory and Combinatorics belong to the Liskova semigroup. Among such polynomials we study rectangular q–Catalan numbers; generalized exponents polynomials; principal specializations of the internal product of Schur functions; generalized q–Gaussian polynomials; parabolic Kostant partition function and its q–analog; certain generating functions on the set of transportation matrices. In each case we apply rigged configurations technique to obtain some interesting and new information about Kostka–Foulkes and parabolic Kostka polynomials, Kostant partition function, MacMahon, Gelfand–Tsetlin and Chan–Robbins polytopes. We describe certain connections between generalized saturation and Fulton's conjectures and parabolic Kostka polynomials; domino tableaux and rigged configurations. We study also some properties of l–restricted generalized exponents and the stable behaviour of certain Kostka–Foulkes polynomials.

The purpose of this paper is to survey some of the main results and techniques from the transformation theory of U(n + 1) multiple basic hypergeometric series associated to the root system A_n. Our approach to this theory employs partial fraction decompositions, q-difference equations, and suitable multidimensional matrix inversions. These series were strongly motivated by Biedenharn and Louck and coworkers mathematical physics research involving angular momentum theory and the unitary groups U(n + 1), or equivalently A_n. The foundation of our theory is the U(n + 1) multiple sum refinement of the terminating classical q-binomial theorem. This result contains as special, limiting, or transformed cases both the Macdonald identities for A_n, and U(n + 1) multiple sum extensions of the classical q-binomial theorem, Ramanujan's ₁ψ₁ sum, the balanced ₃ɸ₂ summation theorem, and Rogers' classical terminating very-well-poised ₆ɸ₅ summation theorem. The general terminating U(n + 1) ₆ɸ₅ summation theorems are specialized to obtain the U(n + 1) extension of Andrews' matrix formulation of the Bailey Transform. The U(n + 1) Bailey transform coupled with the U(n + 1) terminating very-well-poised ₆ɸ₅ summation theorems yields the U(n + 1) extension of Andrews' explicit formulation of the Bailey Lemma. Subsequent applications of these U(n + 1) transformations and summations have led to corresponding extensions of a substantial amount of the theory and application of one-variable basic hypergeometric series. There are now many applications by various researchers of the theory of A_n and/or subsequently C_n and D_n multiple basic hypergeometric series to quantum groups, multidimensional beta and/or Barnes integral evaluations, plane partition enumeration, infinite families of explicit sums of squares formulas, multidimensional matrix inversions, new infinite families of eta function identities, and applications to raising operators for Macdonald polynomials.

I follow Y. Yokota to explain how to obtain a tetrahedron decomposition of the complement of a hyperbolic knot and compare it with the asymptotic behavior of Kashaev's link invariant using the figure-eight knot as an example.

Integral solutions of several hypergeometric q-difference systems with |q| = 1 are constructed by using the double sine function and their contiguity relations are considered.

We propose a general method to realize an arbitrary Weyl group of Kac-Moody type as a group of birational canonical transformations, by means of a nilpotent Poisson algebra.

We give an answer to the two fundamental questions about the Painlevé equations. (i) Where do the Bäcklund transformations come from? (ii) What are the Hamiltonian functions? Our approach depends on the geometry of the projective surfaces constructed by Okamoto and reviewed in ¹⁰.

Noumi, Okada, Okamoto and Umemura proposed a generalization of Jacobi polynomials using the sixth Painlevé equation. They conjectured that their generalized Jacobi polynomials have a combinatorial representation. The purpose of this paper is to prove their conjecture.

We define an algebra on two generators which we call the Tridiagonal algebra, and we consider its irreducible modules. The algebra is defined as follows. Let 𝕂 denote a field, and let β, γ, γ*, ϱ, ϱ* denote a sequence of scalars taken from 𝕂. The corresponding Tridiagonal algebra T is the associative 𝕂-algebra with 1 generated by two symbols A, A* subject to the relations

where [r, s] means rs - sr. We call these relations the Tridiagonal relations. For β = q + q^-1, γ = γ* = 0, ϱ = ϱ* = 0, the Tridiagonal relations are the q-Serre relations where [3]_q = q + q^-1 +1. For β = 2, γ = γ* = 0, ϱ = b², ϱ* = b*², the Tridiagonal relations are the Dolan-Grady relations In the first part of this paper, we survey what is known about irreducible finite dimensional T-modules. We focus on how these modules are related to the Leonard pairs recently introduced by the present author, and the more general Tridiagonal pairs recently introduced by Ito, Tanabe, and the present author. In the second part of the paper, we construct an infinite dimensional irreducible T-module based on the Askey-Wilson polynomials. This module is on the vector space 𝕂[x] consisting of all polynomials in an indeterminant x that have coefficients in 𝕂. Denoting by A the linear transformation on 𝕂[x] which is multiplication by x, and denoting by A* an Askey-Wilson second order q-difference operator for x, we show A and A* satisfy a pair of Tridiagonal relations. Using this we give 𝕂[x] the structure of an irreducible T-module. The Askey-Wilson polynomials form a basis for this module, and these basis elements are eigenvectors for A*.