Generalizations of 𝑄-systems and orthogonal polynomials from representation theory

Abstract

We briefly describe what tau-functions in integrable systems are. We then define a collection of tau-functions given as matrix elements for the action of GL 2 on two-component Fermionic Fock space. These tau-functions are solutions to a discrete integrable system called a Q-system.We can prove that our tau-functions satisfy Q-system relations by applying the famous "Desnanot-Jacobi identity" or by using "connection matrices", the latter of which gives rise to orthogonal polynomials. In this paper, we will provide the background information required for computing these tau-functions and obtaining the connection matrices and will then use the connection matrices to derive our difference relations and to find orthogonal polynomials.We generalize the above by considering tau-functions that are matrix elements for the action of GL 3 on three-component Fermionic Fock space, and discuss the new system of discrete equations that they satisfy. We will show how to use the connection matrices in this case to obtain "multiple orthogonal polynomials of type II".2010 Mathematics Subject Classification. 17B80. Thanks to Rinat Kedem and Philippe Di Francesco for their helpful comments.

τ-Functions, Birkhoff Factorizations and Difference Equations

“…In this appendix we collect some results on one-component fermions. In other words, we are dealing with the fermionic Fock space F = F (1) , based on H = H (1) . The results in this Appendix should be known, for instance Lemma E.2 can be found (without proof) in [4], but we could not find references with complete proofs of the facts we need.…”

“…This is not an accident, but is a basic property of vertex algebras, see [13,14,18], referred as rationality of vertex operators. Indeed, the one-component fermionic Fock space F (1) is an example of a vertex algebra, and the fermionic fields are vertex operators for this vertex algebra structure. Another basic property of vertex algebras is called commutativity; roughly speaking it says that if we permute the vertex operators in a matrix element of a product of vertex operators the answer is again an expansion of the same rational function, but in a different region, up to a sign.…”

“…When we look at the special case where E(z) = 0, we obtain τ -functions that are determinants of block Hankel matrices, related to bi-orthogonal polynomials, and 4-band Toda lattices (see, e.g., [5]). See [1] for a preliminary report on this.…”