Symmetric polynomials, generalized Jacobi-Trudi identities and τ-functions

Abstract

An element [Φ]∈GrnH+,F of the Grassmannian of n-dimensional subspaces of the Hardy space H+=H2, extended over the field F = C(x1, …, xn), may be associated to any polynomial basis ϕ = {ϕ0, ϕ1, ⋯ } for C(x). The Plücker coordinates Sλ,nϕ(x1,…,xn) of [Φ], labeled by partitions λ, provide an analog of Jacobi’s bi-alternant formula, defining a generalization of Schur polynomials. Applying the recursion relations satisfied by the polynomial system ϕ to the analog {hi(0)} of the complete symmetric functions generates a doubly infinite matrix hi(j) of symmetric polynomials that determine an element [H]∈Grn(H+,F). This is shown to coincide with [Φ], implying a set of generalized Jacobi identities, extending a result obtained by Sergeev and Veselov [Moscow Math. J. 14, 161–168 (2014)] for the case of orthogonal polynomials. The symmetric polynomials Sλ,nϕ(x1,…,xn) are shown to be KP (Kadomtsev-Petviashvili) τ-functions in terms of the power sums [x] of the xa’s, viewed as KP flow variables. A fermionic operator representation is derived for these, as well as for the infinite sums ∑λSλ,nϕ([x])Sλ,nθ(t) associated to any pair of polynomial bases (ϕ, θ), which are shown to be 2D Toda lattice τ-functions. A number of applications are given, including classical group character expansions, matrix model partition functions, and generators for random processes.