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

An element [Phi] is an element of Gr(n) (H+, F) of the Grassmannian of n-dimensional subspaces of the Hardy space H+ = H-2, extended over the field F = C(x(1),...,x(n)), may be associated to any polynomial basis phi = {phi(0), phi(1), ... } for C(x). The Pliicker coordinates S-lambda,n(phi)(x(1),...,x(n)) of [Phi], labeled by partitions lambda, provide an analog of Jacobi's bialternant formula, defining a generalization of Schur polynomials. Applying the recursion relations satisfied by the polynomial system phi to the analog {h(i)((0) )of the complete symmetric functions generates a doubly infinite matrix h(i)((j)) of symmetric polynomials that determine an element [H] is an element of Gr(n) (H+,F). This is shown to coincide with [Phi], 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 S-lambda,n(phi)(x(1),...,x(n))are shown to be KP (Kadomtsev-Petviashvili) tau-functions in terms of the power sums [x] of the x(a)'s, viewed as KP flow variables. A fermionic operator representation is derived for these, as well as for the infinite sums Sigma S-lambda(lambda,n)phi([x]) S-lambda,S-n theta(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. Published by AIP Publishing.