Asymptotics of the number of involutions in finite classical groups

TL;DR

This paper determines the asymptotic proportion of involutions in various finite classical groups, providing new identities and a comprehensive analysis across different group types and characteristics.

Contribution

It introduces new sum=product identities and offers a detailed asymptotic enumeration of involutions in classical groups, extending previous work to multiple group types and characteristics.

Findings

Asymptotic proportion of involutions in GL(n,q) computed

Results extended to unitary, symplectic, and orthogonal groups

New sum=product identities developed for enumeration

Abstract

Answering a question of Geoff Robinson, we compute the large n limiting proportion of i(n,q)/q^[n^2/2], where i(n,q) denotes the number of involutions in GL(n,q). We give similar results for the finite unitary, symplectic, and orthogonal groups, in both odd and even characteristic. At the heart of this work are certain new "sum=product" identities. Our self-contained treatment of the enumeration of involutions in even characteristic symplectic and orthogonal groups may also be of interest.