Irreducible characters of finite simple groups constant at the p-singular elements

TL;DR

This paper investigates irreducible characters of finite simple groups that are constant on p-singular elements, extending the concept of p-defect 0 characters and classifying such characters across various group types.

Contribution

It generalizes the notion of p-defect 0 characters to those constant on p-singular elements and classifies these characters for multiple classes of finite simple groups.

Findings

All such characters of non-zero defect are determined for alternating, symmetric, and sporadic groups.

For groups of BN-pair rank > 2, only the Steinberg and trivial characters are constant on non-identity unipotent elements.

Characters with degrees differing by 1 from the Steinberg degree are identified.

Abstract

In representation theory of finite groups an important role is played by irreducible characters of p-defect 0, for a prime p dividing the group order. These are exactly those vanishing at the p-singular elements. In this paper we generalize this notion investigating the irreducible characters that are constant at the p-singular elements. We determine all such characters of non-zero defect for alternating, symmetric and sporadic simple groups. We also classify the irreducible characters of quasi-simple groups of Lie type that are constant on the non-identity unipotent elements. In particular, we show that for groups of BN-pair rank greater than 2 the Steinberg and the trivial characters are the only characters in question. Additionally, we determine all irreducible characters whose degrees differ by 1 from the degree of the Steinberg character.