On the proportion of derangements in affine classical groups

TL;DR

This paper provides exact formulas for the proportions of derangements in affine classical groups, involving new generating functions and verified $q$-polynomial identities, advancing understanding of group element distributions.

Contribution

The paper introduces new formulas for derangement proportions in affine classical groups, utilizing novel generating functions and confirming conjectured $q$-identities.

Findings

Exact formulas for derangement proportions in affine classical groups.

A new generating function for specific integer partitions.

Verification of three $q$-polynomial identities.

Abstract

We derive exact formulas for the proportions of derangements and of derangements of $p$-power order in the affine classical groups $AU_m(q)$, $ASp_{2m}(q)$, $AO_{2m+1}(q)$ and $AO^{\pm}_{2m}(q)$, where $p$ denotes the characteristic of the defining finite field. In the unitary case, the formulas rely on a result on partitions of independent interest: we obtain a generating function for integer partitions $\lambda=(\lambda_1, \dots, \lambda_m)$ into $m$ parts, with $\lambda_1\ge \dots \ge \lambda_m$, such that either $\lambda_1=1$ or $\lambda_{k-1}>\lambda_k=k$ for some $k \in \{2, \dots,m\}$. In the symplectic and orthogonal cases, the proofs of the formulas reduce to verifying three $q$-polynomial identities conjectured by the author and later proved by Fulman and Stanton.