n-Schur Functions and Determinants on an Infinite Grassmannian

Abstract

A set of functions is defined which is indexed by a positive integer $n$ and partitions of integers. The case $n=1$ reproduces the standard Schur polynomials. These functions are seen to arise naturally as a determinant of an action on the frame bundle of an infinite grassmannian. This fact is well known in the case of the Schur polynomials ($n=1$) and has been used to decompose the $τ$-functions of the KP hierarchy as a sum. In the same way, the new functions introduced here ($n>1$) are used to expand quotients of $τ$-functions as a sum with Plucker coordinates as coefficients.