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 conjugacy-class lengths divides the product of irreducible character degrees for finite groups, specifically proving it for $\mathrm{GL}_n(q)$ and $\mathrm{GU}_n(q)$. It develops a detailed partition-parameterization framework for conjugacy classes and irreducible characters, analyzes $q$-adic and $q'$-parts via $\Omega$-valuations, and reduces the problem to combinatorial bounds on integer partitions through the key quantity $N(n)$. The authors establish sharp upper bounds on $N(n)$ and combine these with a case-split analysis to show $v_q(h(\mathrm{GL}_n(q))) \ge 0$ for all $n,q$, with a parallel argument for unitary groups. Consequently, Harada's conjecture holds for both $\mathrm{GL}_n(q)$ and $\mathrm{GU}_n(q)$, and in many cases $|G'|$ also divides $h(G)$ (verified for small instances by computation). These results deepen understanding of the relationship between class sizes and character degrees in finite classical groups and advance the broader study of Harada-type divisibility phenomena.

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.

Theorems & Definitions (40)

• Conjecture 1.1: Ha
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• Lemma 3.1
• proof
• Lemma 3.2