Reflection factorizations of Singer cycles

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.

Historical references added; final version to appear in J. Algebraic Combinatorics

References in corpus (1)

• Character varieties with Zariski closures of GL_n-conjugacy classes at punctures

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• Quantum Hurwitz numbers and Macdonald polynomials
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• GL_n(F_q)-analogues of factorization problems in the symmetric group
• Intersection theorems for finite general linear groups
• Universality classes for purification in nonunitary quantum processes
• Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element
• Klein's ten planar dessins of degree 11, and beyond
• Cycle type factorizations in ${\mathrm{GL}}_n{\mathbb{F}}_q$