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

TL;DR

This paper develops new methods using plethysms and specializations of Schur functions to define and analyze formal characters of certain groups, extending classical results and providing vertex operator realizations.

Contribution

It introduces novel plethysm-based series and formal characters for groups H_, extending known character theories and offering explicit vertex operator representations.

Findings

Defined and evaluated Schur functions s_[X] and plethysms for complex 

Constructed inverse series M_ and L_ for arbitrary partitions 

Provided vertex operator realizations for the formal characters

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_\lambda^{(\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^\perp dual to L_\sigma, and then more explicitly in exponential form. Finally the replicated form of such vertex operators are written down.