On H\'{e}thelyi-K\"{u}lshammer's conjecture for principal blocks

TL;DR

This paper proves a lower bound on the number of irreducible characters in the principal p-block of a finite group, confirming a longstanding conjecture and advancing understanding of block theory in modular representation theory.

Contribution

It confirms Hétélyi and Külshammer's conjecture for principal blocks, providing a key bound and connecting various recent results in the field.

Findings

Number of irreducible characters ≥ 2√(p-1) in principal p-blocks

Confirms Hétélyi-Külshammer conjecture for principal blocks

Provides an affirmative answer to Brauer's Problem 21 for bounded defect

Abstract

We prove that the number of irreducible ordinary characters in the principal $p$-block of a finite group $G$ of order divisible by $p$ is always at least $2\sqrt{p-1}$. This confirms a conjecture of H\'{e}thelyi and K\"{u}lshammer for principal blocks and provides an affirmative answer to Brauer's Problem 21 for principal blocks of bounded defect. Our proof relies on recent works of Mar\'{o}ti and Malle-Mar\'{o}ti on bounding the conjugacy class number and the number of $p'$-degree irreducible characters of finite groups, earlier works of Brou\'{e}-Malle-Michel and Cabanes-Enguehard on the distribution of characters into unipotent blocks and $e$-Harish-Chandra series of finite reductive groups, and known cases of the Alperin-McKay conjecture.