Geodesic
Reflection factorizations of Singer cycles
Journal of Algebraic Combinatorics, Tome 40 (2014) no. 3, pp. 663-691.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

The number of shortest factorizations into reflections for a Singer cycle in $\mathrm{GL}_n(\mathbb F_q)$ 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. 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.
Classification : 05E10, 05E15, 05C70, 20B30
Keywords: Singer cycle, reflection, transvection, factorization 
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     author = {Lewis, J.B. and Reiner, V. and Stanton, D.},
     title = {Reflection factorizations of {Singer} cycles},
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Lewis, J.B.; Reiner, V.; Stanton, D. Reflection factorizations of Singer cycles. Journal of Algebraic Combinatorics, Tome 40 (2014) no. 3, pp. 663-691. http://geodesic.mathdoc.fr/item/JAC_2014__40_3_a9/