On Héthelyi–Külshammer’s conjecture for principal blocks

Nguyen, Ngoc Hung; Schaeffer Fry, A. A.

doi:10.2140/ant.2023.17.1127

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

On Héthelyi–Külshammer's conjecture for principal blocks

Nguyen Ngoc Hung and A. A. Schaeffer Fry

Vol. 17 (2023), No. 6, 1127–1151

DOI: 10.2140/ant.2023.17.1127

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 Héthelyi and Külshammer (2000) for principal blocks and provides an affirmative answer to Brauer’s problem 21 (1963) for principal blocks of bounded defect. Our proof relies on recent works of Maróti (2016) and Malle and Maróti (2016) on bounding the conjugacy class number and the number of $p^{'}$ -degree irreducible characters of finite groups, earlier works of Broué, Malle and Michel (1993) and Cabanes and Enguehard (2004) 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.

Keywords

finite groups, principal blocks, characters, Héthelyi–Külshammer conjecture, Alperin–McKay conjecture

Mathematical Subject Classification

Primary: 20C15, 20C20, 20C33, 20D06

Milestones

Received: 22 April 2021

Revised: 16 January 2022

Accepted: 6 July 2022

Published: 26 May 2023

Authors

Nguyen Ngoc Hung
Department of Mathematics Buchtel College of Arts and Sciences The University of Akron Akron, OH United States

A. A. Schaeffer Fry
Department of Mathematics and Statistics Metropolitan State University of Denver Denver, CO United States

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 17, No. 6, 2023