Derangements in affine classical groups and Cohen-Lenstra heuristics

TL;DR

The paper reframes the problem of derangements in finite affine classical groups in terms of symplectic and orthogonal Cohen-Lenstra type distributions on integer partitions with parity constraints. By linking Anzanello's conjectures to these distributions, the authors use terminating $_2\phi_1$ hypergeometric identities and $q$-series techniques to prove three conjectured $q$-polynomial identities. The approach provides a conceptual bridge between group-theoretic derangement proportions and Cohen-Lenstra heuristics, and yields explicit hypergeometric-transform-based proofs for all three conjectures. This work thus integrates finite-field group theory with probabilistic partition models, highlighting a deeper structure behind derangement phenomena in affine classical groups.

Abstract

We observe that Anzanello's work on the proportion of derangements in affine classical groups over finite fields is related to symplectic and orthogonal Cohen-Lenstra type distributions on integer partitions. This leads to a proof of three q-polynomial identities conjectured by Anzanello, which were crucial for her work.

Theorems & Definitions (20)

• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 2.5
• Lemma 3.1
• proof
• Definition 3.2
• Definition 3.3
• Proposition 3.4