The character degree product and the conjugacy length product for finite general linear groups

TL;DR

This paper verifies Harada's conjecture that the product of irreducible character degrees divides the product of conjugacy class lengths for finite general linear and unitary groups.

Contribution

It provides the first proof of Harada's conjecture for these important classes of finite groups.

Findings

Confirmed the divisibility conjecture for finite general linear groups.

Extended the verification to finite unitary groups.

Contributed to understanding the relationship between character degrees and conjugacy class sizes.

Abstract

Let $G$ be a finite group. K. Harada conjectured that the product of degrees of all irreducible characters of $G$ divides the product of lengths of all conjugacy classes of $G$. We verify this conjecture for finite general linear groups and finite unitary groups.