Deligne-Lusztig theoretic derivation for Weyl groups of the number of reflection factorizations of a Coxeter element

Michel, Jean

doi:10.1090/proc12753

Deligne-Lusztig theoretic derivation for Weyl groups of the number of reflection factorizations of a Coxeter element
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Proc. Amer. Math. Soc. 144 (2016), 937-941 Request permission

Abstract:

Chapuy and Stump have given a nice generating series for the number of factorizations of a Coxeter element as a product of reflections. Their method is to evaluate case by case a character-theoretic expression. The goal of this note is to give a uniform evaluation of their character-theoretic expression in the case of Weyl groups, by using combinatorial properties of Deligne-Lusztig representations.

References

N. Bourbaki, “Groupes et algèbres de Lie”, Chap. 5, Hermann 1969.
Michel Broué and Jean Michel, Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées, Finite reductive groups (Luminy, 1994) Progr. Math., vol. 141, Birkhäuser Boston, Boston, MA, 1997, pp. 73–139 (French). MR 1429870
Michel Broué, Gunter Malle, and Jean Michel, Split spetses for primitive reflection groups, Astérisque 359 (2014), vi+146 (English, with English and French summaries). With an erratum to [MR1712862]. MR 3221618
Guillaume Chapuy and Christian Stump, Counting factorizations of Coxeter elements into products of reflections, J. Lond. Math. Soc. (2) 90 (2014), no. 3, 919–939. MR 3291807, DOI 10.1112/jlms/jdu059
G. Lusztig, Coxeter orbits and eigenspaces of Frobenius, Invent. Math. 38 (1976/77), no. 2, 101–159. MR 453885, DOI 10.1007/BF01408569
George Lusztig, Representations of finite Chevalley groups, CBMS Regional Conference Series in Mathematics, vol. 39, American Mathematical Society, Providence, R.I., 1978. Expository lectures from the CBMS Regional Conference held at Madison, Wis., August 8–12, 1977. MR 518617, DOI 10.1090/cbms/039
George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472, DOI 10.1515/9781400881772
Gunter Malle, On the rationality and fake degrees of characters of cyclotomic algebras, J. Math. Sci. Univ. Tokyo 6 (1999), no. 4, 647–677. MR 1742597
Jean Michel, The development version of the CHEVIE package of GAP3, J. Algebra 435 (2015), 308–336. MR 3343221, DOI 10.1016/j.jalgebra.2015.03.031