Reduction theorems for Clifford classes

Abstract

Clifford Theory concerns the representations over a field F of a finite group G and one of the normal subgroups N of G. It is often viewed as a series of reduction theorems. Clifford classes are certain equivalence classes of G/N-algebras over F. They have been used to describe the Schur indices of the irreducible characters of certain families of classical groups. In the present paper, we show how Clifford classes can also be used to explain and to reflect the reduction theorems of classical Clifford theory. We show that certain products of characters correspond to certain products of Clifford classes. We show that induction, restriction, inflation, and extension of the base field can all be defined naturally for Clifford classes. We prove that the properties of these elementary operations on Clifford classes imply all of the standard Clifford-theoretic reductions in the case when the field is ℂ. They also provide important information in the case when the field F is an arbitrary field of characteristic zero.