Reflection factorizations of Singer cycles
Combinatorics
2015-10-15 v2 Representation Theory
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.
Keywords
Cite
@article{arxiv.1308.1468,
title = {Reflection factorizations of Singer cycles},
author = {Joel Brewster Lewis and Victor Reiner and Dennis Stanton},
journal= {arXiv preprint arXiv:1308.1468},
year = {2015}
}
Comments
Historical references added; final version to appear in J. Algebraic Combinatorics