ALGEBRAIC COMBINATORICS

no. 2

A note on non-reduced reflection factorizations of Coxeter elements
Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 465-469.

We extend a result of Lewis and Reiner from finite Coxeter groups to Coxeter groups of finite rank by showing that two reflection factorizations of a Coxeter element lie in the same Hurwitz orbit if and only if they share the same multiset of conjugacy classes.

Received: 2019-01-20
Revised: 2019-06-24
Accepted: 2019-09-23
Published online: 2020-04-01
DOI: https://doi.org/10.5802/alco.99
Classification: 05E15,  05E18,  20F55
Keywords: Coxeter groups, Hurwitz action, reflection factorizations, Coxeter element 
@article{ALCO_2020__3_2_465_0,
     author = {Wegener, Patrick and Yahiatene, Sophiane},
     title = {A note on non-reduced reflection factorizations of Coxeter elements},
     journal = {Algebraic Combinatorics},
     pages = {465--469},
     publisher = {MathOA foundation},
     volume = {3},
     number = {2},
     year = {2020},
     doi = {10.5802/alco.99},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2020__3_2_465_0/}
}
Wegener, Patrick; Yahiatene, Sophiane. A note on non-reduced reflection factorizations of Coxeter elements. Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 465-469. doi : 10.5802/alco.99. https://alco.centre-mersenne.org/item/ALCO_2020__3_2_465_0/

[1] Baumeister, Barbara; Dyer, Matthew; Stump, Christian; Wegener, Patrick A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements, Proc. Am. Math. Soc., Ser. B, Volume 1 (2014), pp. 149-154 | Article | MR 3294251 | Zbl 1343.20041

[2] Bessis, David The dual braid monoid, Ann. Sci. Éc. Norm. Supér. (4), Volume 36 (2003) no. 5, pp. 647-683 | Article | Numdam | MR 2032983 | Zbl 1064.20039

[3] Deligne, Pierre Letter to E. Looijenga (1974) http://homepage.rub.de/christian.stump/Deligne_Looijenga_Letter_09-03-1974.pdf

[4] Dyer, Matthew Reflection subgroups of Coxeter systems, J. Algebra, Volume 135 (1990) no. 1, pp. 57-73 | Article | MR 1076077 | Zbl 0712.20026

[5] Dyer, Matthew On the “Bruhat graph” of a Coxeter system, Compos. Math., Volume 78 (1991) no. 2, pp. 185-191 | Numdam | MR 1104786 | Zbl 0784.20019

[6] Humphreys, James E. Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Volume 29, Cambridge University Press, Cambridge, 1990, xii+204 pages | Article | MR 1066460 | Zbl 0725.20028

[7] Igusa, Kiyoshi; Schiffler, Ralf Exceptional sequences and clusters, J. Algebra, Volume 323 (2010) no. 8, pp. 2183-2202 | Article | MR 2596373 | Zbl 1239.16019

[8] Lewis, Joel Brewster; Reiner, Victor Circuits and Hurwitz action in finite root systems, New York J. Math., Volume 22 (2016), pp. 1457-1486 | MR 3603073 | Zbl 1392.20033

[9] Wegener, Patrick Hurwitz action in Coxeter groups and elliptic Weyl groups (2017) https://pub.uni-bielefeld.de/record/2913106 (Ph. D. Thesis)