Symmetric polynomials, generalized Jacobi-Trudi identities and \tau-functions

Abstract

An element [\Phi] 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 {\phi} for C(x). The Pl\"ucker coordinates S^\phi_{\lambda,n}(x_1,..., x_n) of \Phi, labelled by partitions \lambda, 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 [\Phi], 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^\phi_{\lambda,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_{\lambda}S_{\lambda,n}^\phi([x]) S^\theta_{\lambda,n} ({\bf t}) associated to any pair of polynomial bases (\phi, \theta), which are shown to be 2D Toda lattice \tau-functions. A number of applications are given, including classical group character expansions, matrix model partition functions and generators for random processes.