The number of shortest factorizations into reflections for a Singer cycle in GLn(Fq) is shown to be (qn-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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Acknowledgments The authors thank A. Ram and P. Diaconis for pointing them to this work of Hildebrand  used in Sect. 4.4. They also thank A. Henderson and E. Letellier for pointing them to [11,21]. This work is partially supported by NSF grants DMS-1148634 and DMS-1001933.
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- Singer cycle