Plethysms, replicated Schur functions and series, with applications to vertex operators

Abstract

Specializations of Schur functions are exploited to define and evaluate the Schur functions $s_\lambda [\alpha X]$ and plethysms $s_\lambda [\alpha s_\nu(X))]$ for any $\alpha$-integer, real or complex. Plethysms are then used to define pairs of mutually inverse infinite series of Schur functions, $M_\pi$ and $L_\pi$, specified by arbitrary partitions $\pi$. These are used in turn to define and provide generating functions for formal characters, $s^{(\pi)}_\lambda$, of certain groups $H_\pi$, thereby extending known results for orthogonal and symplectic group characters. Each of these formal characters is then given a vertex operator realization, first in terms of the series $M=M_{(0)}$ and various $L^\bot_\sigma$ dual to $L_\sigma$, and then more explicitly in exponential form. Finally the replicated form of such vertex operators are written down.