A uniform upper bound for the character degree sums and Gelfand-Graev-like characters for finite simple groups

TL;DR

This paper classifies finite simple groups based on the sum of their irreducible character degrees relative to Sylow p-subgroup indices, highlighting cases involving Lie type groups and specific examples.

Contribution

It provides a complete classification of pairs (G,p) where the character degree sum exceeds the Sylow p-subgroup index, including all Lie type groups in characteristic p and select examples.

Findings

All Lie type groups in defining characteristic p are included.

Explicit classification of pairs (G,p) with the degree sum exceeding the Sylow p-index.

Identification of special cases outside the Lie type groups.

Abstract

Let G be a finite non-abelian simple group and let p be a prime. We classify all pairs (G,p) such that the sum of the complex irreducible character degrees of G is greater than the index of a Sylow p-subgroup of G. Our classification includes all groups of Lie type in defining characteristic p (because every Gelfand-Graev character of G is multiplicity free and has degree equal to the above index), and a handful of well-described examples.