Some Remarks on Commuting Probability

TL;DR

The paper addresses estimating commutator probabilities in finite groups by leveraging representation theory. It introduces non-negative virtual characters and an associated random variable $\xi_{g,\phi}$ to bound $c(g)$ using partial spectral information, culminating in a key formula $\mathbb{E}(\xi_{g,\phi})=\sum_i (a_i/n_i^2)\Re(\chi_i(g))$ and bounds $c(g)\le \mathbb{E}(\xi_{g,\phi})/\phi(1)$ for non-negative $\phi$. This framework yields concrete corollaries, including bounds derived from the regular representation and specific irreducibles, and connects to classical results such as Frobenius's formula and Gustafson's $5/8$ theorem with examples from $S_n$ and $A_5$. The approach provides a flexible, character-based method to bound commuting probabilities with limited spectral data, offering insights into the structure of finite groups via their representations.

Abstract

We introduce a weighted sum of irreducible character ratios as an estimator for commutator probabilities. The estimator yields Frobenius formula when applied to a regular representation

Theorems & Definitions (15)

• Remark 1
• Remark 2
• Lemma 1
• Theorem 1
• Lemma 2
• Lemma 3
• Lemma 4
• Corollary 1
• Corollary 2
• Corollary 3