Restrictions of characters in p-solvable groups

TL;DR

This paper investigates character restrictions in p-solvable groups, proving a specific sum decomposition related to subgroup indices and proposing a conjecture for more general cases, extending classical theorems and previous results.

Contribution

It proves a new sum decomposition property for character restrictions in p-solvable groups and proposes a conjecture extending Brauer--Nesbitt's theorem.

Findings

Proved a sum decomposition of character restrictions in p-solvable groups.

Extended previous results on linear constituents of character restrictions.

Conjectured a broader version applicable to more general groups.

Abstract

Let G be a p-solvable group, P a p-subgroup and chi in Irr(G) such that chi(1)_p \ge |G:P|_p. We prove that the restriction chi_P is a sum of characters induced from subgroups Q\le P such that chi(1)_p=|G:Q|_p. This generalizes previous results by Giannelli--Navarro and Giannelli--Sambale on the number of linear constituents of chi_P. Although this statement does not hold for arbitrary groups, we conjecture a weaker version which can be seen as an extension of Brauer--Nesbitt's theorem on characters of p-defect zero. It also extends a conjecture of Wilde.