# Frobenius-Schur indicators of characters in blocks with cyclic defect

## TL;DR

This paper studies Frobenius-Schur indicators of characters in blocks with cyclic defect groups, showing uniform indicators for exceptional characters and relating indicators of non-exceptional characters to group elements.

## Contribution

It establishes that all exceptional characters in such blocks share the same Frobenius-Schur indicator and provides a method to compute it, also relating indicators of non-exceptional characters to group structure.

## Key findings

- Exceptional characters have identical Frobenius-Schur indicators.
- The common indicator can be computed via the canonical character.
- The number of characters with indicator -1 exceeds or equals the count of certain real elements.

## Abstract

Let $p$ be an odd prime and let $B$ be a $p$-block of a finite group which has cyclic defect groups. We show that all exceptional characters in $B$ have the same Frobenius-Schur indicators. Moreover the common indicator can be computed, using the canonical character of $B$. We also investigate the Frobenius-Schur indicators of the non-exceptional characters in $B$. For a finite group which has cyclic Sylow $p$-subgroups, we show that the number of irreducible characters with Frobenius-Schur indicator $-1$ is greater than or equal to the number of conjugacy classes of weakly real $p$-elements in $G$.