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 $AU_m(q)$, $ASp_{2m}(q)$, $AO_{2m+1}(q)$ 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 $λ=(λ_1, \dots, λ_m)$ into $m$ parts, with $λ_1\ge \dots \ge λ_m$, such that either $λ_1=1$ or $λ_{k-1}>λ_k=k$ 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.

Figures (6)

• Figure 1: Ferrers diagram of the partition $(6,5,4,2,2)$
• Figure 2: Example:$\lambda=(8,7,7,4,4,3,3,1,1)$ has Durfee square 4, $\pi_1(\lambda)=(4,3,3)$, $\pi_2(\lambda)=(4,3,3,1,1)$ and it satisfies $\lambda_3>\lambda_4=4$.
• Figure 3: Graphical representation of $\Phi(a,b)$
• Figure 4:
• Figure 5: Example of a symplectic signed partition: here, the $+$ corresponds to the part of size 8 and the $-$ corresponds to the parts of size 4 and 2

Theorems & Definitions (47)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Fulman and Stanton, FulmanStanton25
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 3.1
• proof
• Theorem 3.2: SpigaAGL