The Schur-Clifford subgroup of the Brauer-Clifford group

TL;DR

This paper introduces a Schur-Clifford subgroup within the Brauer-Clifford group, paralleling the Schur subgroup in the Brauer group, and explores its properties and behavior in Clifford theory of finite groups.

Contribution

It defines the Schur-Clifford subgroup, proves it forms a subgroup, and analyzes its compatibility with restriction and corestriction maps.

Findings

The Schur-Clifford subgroup is a subgroup of the Brauer-Clifford group.

It precisely captures classes relevant to Clifford theory.

The subgroup behaves well under restriction and corestriction maps.

Abstract

We define a Schur-Clifford subgroup of Turull's Brauer-Clifford group, similar to the Schur subgroup of the Brauer group. The Schur-Clifford subgroup contains exactly the equivalence classes coming from the intended application to Clifford theory of finite groups. We show that the Schur-Clifford subgroup is indeed a subgroup of the Brauer-Clifford group, as are certain naturally defined subsets. We also show that this Schur-Clifford subgroup behaves well with respect to restriction and corestriction maps between Brauer-Clifford groups.