John Harnad: Schur Function expansions of Tau functions

Tau functions are a basic ingredient in the modern theory of integrable systems. They were originally introduced by Hirota, in his studies of particular classes of integrable systems, such as the Toda lattice and the sine-Gordon equation, simply as a clever "device" that allowed such nonlinear systems to be recast as bilinear systems for an associated auxiliary object, for which explicit solutions, such as multi-solitons, could be given in an algebraic or exponential form. Subsequently, this was incorporated into the very powerful inverse spectral method (based upon the multiplicative matrix Riemann-Hilbert problem) that is currently viewed as lying at the root of the modern theory. 
   In classical systems, the tau function is viewed as a generating function for the complete set of commuting flows that comprise the integrable system and its invariants. Sato found a remarkable geometrical interpretation of the tau function and the Hirota bilinear relations as defining the Plucker cooridinates, and Plucker relations chacterizing the embedding of a suitably defined infinite dimensional Grassmann manifold into a corresponding projective space, with the flows  simply determined as a standard lift, to this space of the action of an infinite abelian group on an associated Hilbert space. 
   Thus, elements of the Grassmannian serve to parametrize all possible solutions of integrable hierarchies like the KP (Kadomtsev-Petvishvili) as initial values in a universal phase space under a standard abelian group action.
   The basic "building blocks" for such tau functions are the Schur functions, labelled by integer partitions, which, besides their role as a basis for the symmetric polynomials, are also group characters (for Lie groups, such as U(n)) and character generators (for finite groups, such as S_n)
   All tau functions may be expanded within this basis, with the coefficients in such an expansion identified as the Plucker coordinates of the initial  value.
   The Schur function expansions themselves play a very central role in the applications of the theory of tau functions, which include such diverse things as: matrix model integrals; generating functions for random processes of "exclusion" type; generating functions for topological and algebraic invariants such as Gromov-Witten invariants; Hurwitz numbers, enumerative geometry invariants, etc.
  The construction of such tau functions using, in particular, the very naturally associated fermionic operatorial methods, as well as their character expansions, interpretation and applications will form the content of this survey talk.