Jessica Anzanello: On the proportion of derangements in affine classical groups

On the proportion of derangements in affine classical groups Jessica Anzanello (Milano Bicocca)

Abstract:: Let $G \leq \operatorname{Sym}(\Omega)$ be a finite transitive permutation group. An element $g \in G$ is a derangement if it has no fixed points on $\Omega$.

The study of derangements has a long and rich history, but exact formulas for their number are quite rare, as they are often difficult to obtain. A natural family to consider in this context is that of affine groups. In 2017, Spiga gave a remarkably simple and explicit expression for the proportion of derangements in the affine general linear group $\operatorname{AGL}_{n}(q)$.

In this talk, I will discuss new results that provide exact formulas for the proportion of derangements in the affine classical groups $\operatorname{AU}_{n} (q)$, $\operatorname{ASp}_{2n}(q)$ and $\operatorname{AO}_{n}^{\epsilon}(q)$, $\epsilon \in \{+,-,\circ\}$. Our approach builds on the cycle index generating functions for the finite classical groups, leading to intriguing connections with integer partition theory.