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Reflection factorizations of Singer cycles

Abstract : The number of shortest factorizations into reflections for a Singer cycle in $GL_n(\mathbb{F}_q)$ 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.
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https://hal.inria.fr/hal-01207611
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J.B. Lewis, V. Reiner, D. Stanton. Reflection factorizations of Singer cycles. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.297-308. ⟨hal-01207611⟩

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