# zbMATH — the first resource for mathematics

A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements. (English) Zbl 1343.20041
Summary: In this note, we provide a short and self-contained proof that the braid group on $$n$$ strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank $$n$$ into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of $$W$$ also lie in this parabolic subgroup.

##### MSC:
 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20F36 Braid groups; Artin groups
Full Text:
##### References:
 [1] Bessis, David, The dual braid monoid, Ann. Sci. \'Ecole Norm. Sup. (4), 36, 5, 647-683 (2003) · Zbl 1064.20039 [2] [De74] P. Deligne, letter to E. Looijenga, 9/3/1974. available at \urlhttp://homepage.univie.ac.at/christian.stump/ Deligne_Looijenga_Letter_09-03-1974.pdf. [3] Digne, F., Pr\'esentations duales des groupes de tresses de type affine $$\widetilde A$$, Comment. Math. Helv., 81, 1, 23-47 (2006) · Zbl 1143.20020 [4] Dyer, Matthew J., On minimal lengths of expressions of Coxeter group elements as products of reflections, Proc. Amer. Math. Soc., 129, 9, 2591-2595 (electronic) (2001) · Zbl 0980.20028 [5] Dyer, Matthew, Reflection subgroups of Coxeter systems, J. Algebra, 135, 1, 57-73 (1990) · Zbl 0712.20026 [6] Humphreys, James E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, xii+204 pp. (1990), Cambridge University Press, Cambridge · Zbl 0725.20028 [7] Igusa, Kiyoshi, Exceptional sequences, braid groups and clusters. Groups, algebras and applications, Contemp. Math. 537, 227-240 (2011), Amer. Math. Soc., Providence, RI · Zbl 1241.16015 [8] Igusa, Kiyoshi; Schiffler, Ralf, Exceptional sequences and clusters, J. Algebra, 323, 8, 2183-2202 (2010) · Zbl 1239.16019 [9] [HK13] A. Hubery, H. Krause, A categorification of non-crossing partitions, arXiv:1310.1907 (2013), 30 pages · Zbl 1381.16006 [10] Franzsen, W. N.; Howlett, R. B.; M{\"u}hlherr, B., Reflections in abstract Coxeter groups, Comment. Math. Helv., 81, 3, 665-697 (2006) · Zbl 1101.20024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.