A generalized character related to the local structure and representation theory of a finite group

TL;DR

This paper introduces a generalized character related to the local structure of finite groups, conjectures its positivity, and verifies this in specific cases like PSL(2,q) and SL(2,q), impacting the understanding of their representation theory.

Contribution

It proposes a new generalized character for finite groups, conjectures its nature as a projective module character, and confirms this in key cases such as PSL(2,q) and SL(2,q).

Findings

Conjecture holds for all primes p when G is PSL(2,q) or SL(2,q).

Generalized character vanishes on p-singular elements and counts p-elements in centralizers.

Implications for the representation and character theory of finite groups.

Abstract

We consider the generalized character $\Psi_{1,p,G}$ of a finite group $G$ which vanishes on all $p$-singular elements of $G$ and whose value at each $p$-regular $y \in G$ is the number of $p$-elements of $C_{G}(y)$. We conjecture that this is always a character, and may be afforded by a projective $RG$-module, where $R$ is an appropriate complete discrete valuation ring whose residue field has characteristic $p$. We examine a number of case where this is the case, and consider consequences for the representation theory and character theory of $G$ when this conjecture is known to hold. In particular, we prove, among other things, that the conjecture is valid for all primes $p$ in the case that $G \cong {\rm PSL}(2,q)$ or ${\rm SL}(2,q)$ for every prime power $q$.