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

2025/05/06 by Geoffrey R. Robinson, Robinson, Geoffrey R.
Mathematics · Computer Science · #Finite Group Theory Research #Rings, Modules, and Algebras #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.2505.03976

Abstract

We consider the generalized character Ψ_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 ∈ 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 ≅ \rm PSL(2,q) or \rm SL(2,q) for every prime power q.

Citations

• Projective modules and involutions

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