Asymptotics of commuting probabilities in reductive algebraic groups

TL;DR

This paper investigates the asymptotic behavior of commuting probabilities in reductive algebraic groups, revealing their relation to the maximal dimension of Abelian subgroups and providing explicit formulas for finite groups over finite fields.

Contribution

It establishes the asymptotic formulas for commuting probabilities in reductive groups and finite reductive groups, connecting these probabilities to the structure of Abelian subgroups.

Findings

For large d, cp_d(G) ~ alpha/n where alpha is the maximal Abelian subgroup dimension.

For finite groups over F_q, cp_{d+1}(G(F_q)) ~ q^{(alpha - n)d}.

Provides several examples illustrating the asymptotic behavior.

Abstract

Let $G$ be an algebraic group. For $d\geq 1$, we define the commuting probabilities $cp_d(G) = \frac{dim(\mathfrak C_d(G))}{dim(G^d)}$, where $\mathfrak C_d(G)$ is the variety of commuting $d$-tuples in $G$. We prove that for a reductive group $G$ when $d$ is large, $cp_d(G)\sim \frac{\alpha}{n}$ where $n=\dim(G)$, and $\alpha$ is the maximal dimension of an Abelian subgroup of $G$. For a finite reductive group $G$ defined over the field $\mathbb F_q$, we show that $cp_{d+1}(G(\mathbb F_q))\sim q^{(\alpha-n)d}$, and give several examples.