Geodesic
Reflection factorizations of Singer cycles
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (2014).

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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. 
@article{DMTCS_2014_special_265_a26,
     author = {Lewis, J.B. and Reiner, V. and Stanton, D.},
     title = {Reflection factorizations of {Singer} cycles},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)},
     year = {2014},
     doi = {10.46298/dmtcs.2401},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2401/}
}
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Lewis, J.B.; Reiner, V.; Stanton, D. Reflection factorizations of Singer cycles. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (2014). doi : 10.46298/dmtcs.2401. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2401/

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