Difference Hierarchies for nT τ-functions

Abstract

We introduce hierarchies of difference equations (referred to as nT -systems) associated to the action of a (centrally extended, completed) infinite matrix group GL (n) ∞ on n-component fermionic Fock space. The solutions are given by matrix elements (τ -functions) for this action. We show that the τfunctions of type nT satisfy bilinear equations of length 3, 4, . . . , n + 1. The 2T -system is, after a change of variables, the usual 3 term T -system of type A.Restriction from GL (n)∞ to a subgroup isomorphic to the loop group LGLn, defines nQ-systems, studied earlier in [1] by the present authors for n = 2, 3.

τ-Functions, Birkhoff Factorizations and Difference Equations

“…, which follows from [2] by the change of variables, k → k + 1, → + 1, α → α − 2. The first term then becomes…”

“…In this paper, we discuss the construction of the nQ-systems for n = 2, 3. See [2] for more general hierarchies.…”

“…We have an action of the central extension, GL 2 , of the loop group GL 2 = GL 2 (C((z))) on twocomponent fermionic Fock space, F (2) , the semi-infinite wedge spaced based in C 2 ⊗ C z, z −1 , see, e.g., [31] for n-component fermions, and [19] for the construction of central extensions of Lie algebras and corresponding groups. Some of this material is reviewed in Appendix A.…”

“…In order to define our τ -functions, we need to define fermionic translation operators which we denote Q i , i = 0, 1. In (A.9), we will carefully define these operators and their action on F (2) , in terms of wedging and contracting operators,…”

“…for α, β ∈ F (2) . Here, e k a = e a z k , where e a denotes the standard basis vectors of C 2 , e 0 = 1 0 ,…”