Universal Characters and an Extension of the KP Hierarchy

Abstract

The universal character is a polynomial attached to a pair of partitions and is a generalization of the Schur polynomial. In this paper, we define vertex operators which play roles of raising operators for the universal character. By means of the vertex operators, we obtain a series of non-linear partial differential equations of infinite order, called the UC hierarchy; we regard it as an extension of the KP hierarchy. We investigate also solutions of the UC hierarchy; the totality of the space of solutions forms a direct product of two infinite-dimensional Grassmann manifolds, and its infinitesimal transformations are described in terms of the Lie algebra .