  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.
