Real Subpairs and Frobenius-Schur Indicators of Characters in 2-Blocks

TL;DR

This paper investigates the structure of real 2-blocks in finite groups, introducing defect couples to determine real subpairs, and applies these concepts to analyze Frobenius-Schur indicators and module vertices in blocks with dihedral defect groups.

Contribution

It introduces the concept of defect couples (D,E) to characterize real subpairs and explores their influence on Frobenius-Schur indicators and module vertices in real 2-blocks.

Findings

Defect couples (D,E) determine real subpairs in 2-blocks.

Enumeration of Frobenius-Schur indicators for characters in blocks with dihedral defect groups.

Vertices of involution modules are explicitly determined in these blocks.

Abstract

Let B be a real 2-block of a finite group G. Then B has a real defect class. Let g be an element of such a class. A defect couple of B is (D,E), where E is a Sylow 2-subgroup of the extended centralizer C^*(g) of g, and D is the intersection of E with the centralizer C(g). It is known that (D,E) is uniquely determined up to G-conjugacy. We show that (D,E) determines which B-subpairs are real. We also outline how (D,E) influences the vertices of components of the G-permutation module corresponding to the conjugation action of G on its involutions. We apply these methods to enumerate the Frobenius-Schur indicators of the irreducible characters in a real block that has a dihedral defect group. We also determine the vertices of the components of the involution module in such a block.