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

Abstract

An element [Φ] of the Grassmannian of n-dimensional subspaces of the Hardy space H^2, extended over the field C(x_1,..., x_n), may be associated to any polynomial basis ϕ for C(x). The Plücker coordinates S^ϕ_{λ,n}(x_1,..., x_n) of Φ, labelled 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 to the analog of the complete symmetric functions generates a doubly infinite matrix of symmetric polynomials that determine an element [H] of the Grassmannian. This is shown to coincide with [Φ], implying a set of {\it quantum Jacobi-Trudi identities} that generalize a result obtained by Sergeev and Veselov for the case of orthogonal polynomials. The symmetric polynomials S^ϕ_{λ,n}(x_1,..., x_n) are shown to be KP (Kadomtsev-Petviashvili) tau-functions in terms of the monomial sums [x] in the parameters x_a, viewed as KP flow variables. A fermionic operator representation is derived for these, as well as for the infinite sums \sum_λS_{λ,n}^ϕ([x]) S^θ_{λ,n} ({\bf 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.

32 pages. References added. Character expansions for classical groups added. Title modified. Examples 4.1 and 4.2 revised

References in corpus (3)

• Classical tau-function for quantum spin chains
• Fermionic construction of partition function for multi-matrix models and multi-component TL hierarchy
• Finite Dimensional KP τ-functions I. Finite Grassmannians

Cited by in corpus (6)

• Bilinear expansions of lattices of KP $\tau$-functions in BKP $\tau$-functions: a fermionic approach
• Polynomial KP and BKP $\tau$-functions and correlators
• ABJM Matrix Model and 2D Toda Lattice Hierarchy
• The generalized Giambelli formula and polynomial KP and CKP tau-functions
• Tau functions, infinite Grassmannians and lattice recurrences
• Notes about KP/BKP correspondence