Counting factorizations of Singer cycles in linear and unitary groups

Abstract

We count factorizations of Singer cycles as products of reflections in the families of special and general unitary and linear groups over a finite field. In the case of minimum-length factorizations, the resulting answer is a striking product formula resembling the count for minimum-length factorizations of Coxeter elements into reflections in complex reflections groups. Moreover, for minimum length, the answers for the unitary and linear groups exhibit the phenomenon of Ennola duality, where the number of factorizations in a unitary group over the field $\mathbb{F}_q$ is given by replacing `$q$' with `$-q$' in the corresponding answer for a linear group. We use the character theory of these groups to make this count, and in particular we employ the Deligne--Lusztig theory of characters for finite reductive groups.