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

J. Harnad, Eunghyun Lee

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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.

Original languageEnglish
Article number091411
JournalJournal of Mathematical Physics
Volume59
Issue number9
DOIs
Publication statusPublished - Sep 1 2018

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ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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