Reflection factorizations of Singer cycles

TL;DR

This paper provides explicit formulas for counting shortest and longer factorizations of Singer cycles into reflections in GL_n(F_q), using character theory, and discusses open problems related to these factorizations.

Contribution

It introduces a character-theoretic approach to count reflection factorizations of Singer cycles, including formulas for various lengths and conjugacy classes, which was not previously known.

Findings

Number of shortest factorizations into reflections is (q^n-1)^(n - 1)

Formulas for counting factorizations of any length are provided

Open problems related to reflection factorizations are discussed

Abstract

The number of shortest factorizations into reflections for a Singer cycle in GL_n(F_q) is shown to be (q^n-1)^(n - 1). Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also given. The method is a standard character-theory technique, requiring the compilation of irreducible character values for Singer cycles, semisimple reflections, and transvections. The results suggest several open problems and questions, which are discussed at the end.