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 sλ[αX] and plethysms sλ[αsν(X))] for any α—integer, real or complex. Plethysms are then used to define pairs of mutually inverse infinite series of Schur functions, Mπ and Lπ, specified by arbitrary partitions π. These are used in turn to define and provide generating functions for formal characters, s(π)λ, of certain groups Hπ, 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⊥σ dual to Lσ, and then more explicitly in the exponential form. Finally the replicated form of such vertex operators are written down.The characters of the orthogonal and symplectic groups have been found by Schur [34] and Weyl [35] respectively. The method used is transcendental, and depends on integration over the group manifold. These characters, however, may be obtained by purely algebraic methods,…. This algebraic method would seem to offer a better prospect of successful application to other restricted groups than the method of group integration.Littlewood D E 1944 Phil. Trans. R. Soc. London, Ser. A 239 (809) 392