The Imprimitive Faithful Complex Characters of the Schur Covers of the Symmetric and Alternating Groups

TL;DR

This paper characterizes imprimitive faithful complex characters of Schur covers of symmetric and alternating groups, identifying their minimal block stabilizers and monomial characters using combinatorics and character theory.

Contribution

It provides a complete classification of imprimitive faithful characters and their stabilizers for Schur covers of these groups, a novel extension of character theory.

Findings

Identified all imprimitive faithful characters of Schur covers.

Determined minimal block stabilizers for each imprimitive character.

Classified monomial faithful characters of the Schur covers.

Abstract

Using combinatorics and character theory, we determine the imprimitive faithful complex characters, i.e., the irreducible faithful complex characters which are induced from proper subgroups, of the Schur covers of the symmetric and alternating groups. Furthermore, for every imprimitive character we establish all its minimal block stabilizers. As a corollary, we also determine the monomial faithful characters of the Schur covers.