Real characters in nilpotent blocks

TL;DR

This paper investigates the properties of real characters in nilpotent blocks of finite groups, establishing local determination of their count and proposing a conjecture for computing Frobenius-Schur indicators for p=2.

Contribution

It proves the local determination of real characters in nilpotent blocks and conjectures a method to compute Frobenius-Schur indicators for p=2, extending previous results.

Findings

Number of real characters is locally determined in nilpotent blocks.

Conjecture on computing Frobenius-Schur indicators for p=2.

Extension of results to blocks with one simple module.

Abstract

We prove that the number of irreducible real characters in a nilpotent block of a finite group is locally determined. We further conjecture that the Frobenius-Schur indicators of those characters can be computed for p=2 in terms of the extended defect group. We derive this from a more general conjecture on the Frobenius-Schur indicator of projective indecomposable characters of 2-blocks with one simple module. This extends results of Murray on 2-blocks with cyclic and dihedral defect groups.