Asymptotics of commuting probabilities in reductive algebraic groups

Abstract

Let $G$ be an algebraic group. For $d\ge 1$, we define the commuting probabilities $c{p}_d(G)=\frac{dim({\mathfrak{C}}_d(G))}{dim(G^d)}$, 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, $c{p}_d(G)\sim \frac{\alpha }{n}$ 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 $c{p}_{d+1}(G({\mathbb{F}}_q))\sim q^{(\alpha -n)d}$, and give several examples.

References in corpus (2)

• Commuting probability and simultaneous conjugacy classes of commuting tuples in a group
• Commuting probability in algebraic groups