On the proportion of derangements in affine classical groups

Abstract

We derive exact formulas for the proportions of derangements and of derangements of $p$-power order in the affine classical groups $A{U}_m(q)$, $AS{p}_{2m}(q)$, $A{O}_{2m+1}(q)$ and $A{O}_{2m}^{\pm }(q)$, 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 ${\lambda }_1\ge \cdots \ge {\lambda }_m$, 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.

Revised version. All previous conjectures are now stated as theorems, following Fulman and Stanton's proof of the three q-polynomial identities conjectured in the first version of this paper (see arXiv:2510.16277). This version also treats the even characteristic case for affine symplectic and orthogonal groups