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.
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