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

Specializations of Schur functions are exploited to define and evaluate the Schur functions s(lambda)[alpha X] and plethysms s(lambda)[as(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)((pi)), 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-sigma(perpendicular to) dual to L-sigma, and then more explicitly in the exponential form. Finally the replicated form of such vertex operators are written down.