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On the Hurwitz action in finite Coxeter groups. (English) Zbl 1368.20045
The authors deal with the dual approach to Coxeter groups as introduced by D. Bessis [Ann. Sci. Éc. Norm. Supér. (4) 36, No. 5, 647–683 (2003; Zbl 1064.20039)], T. Brady [Adv. Math. 161, No. 1, 20–40 (2001; Zbl 1011.20040)], and T. Brady and C. Watt [Geom. Dedicata 94, 225–250 (2002; Zbl 1053.20034)]. A dual Coxeter system is a Coxeter group together with the generating set of conjugates of all the elements of a simple system and a set of relations between these generators.
The authors provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions.
The main result that the authors obtain is the following:
Theorem 1.1. Let \((W, T)\) be a finite, dual Coxeter system of rank \(n\) and let \(w \in W\). The Hurwitz action on \(\text{Red}_T (w)\) is transitive if and only if \(w\) is a parabolic quasi-Coxeter element for \((W, T)\).

MSC:
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20F36 Braid groups; Artin groups
17B22 Root systems
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References:
[1] Armstrong D., Generalized noncrossing partitions and combinatorics of Coxeter groups, Mem. Amer. Math. Soc. 202 (2009), no. 949, 1-159. · Zbl 1191.05095
[2] Baumeister B., Dyer M., Stump C. and Wegener P., A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements, Proc. Amer. Math. Soc. Ser. B 1 (2014), 149-154. · Zbl 1343.20041
[3] Bessis D., The dual braid monoid, Ann. Sci. Éc. Norm. Supér. (4) 36 (2003), 647-683. · Zbl 1064.20039
[4] Bessis D., Finite complex reflection arrangements are \({K(π,1)}\), Ann of Math. (2) 181 (2015), no. 3, 809-904. · Zbl 1372.20036
[5] Bessis D., Digne F. and Michel J., Springer theory in braid groups and the Birman-Ko-Lee monoid, Pacific J. Math. 205 (2002), 287-309. · Zbl 1056.20023
[6] Björner A. and Brenti F., Combinatorics of Coxeter Groups, Grad. Texts in Math. 231, Springer, New York, 2005.
[7] Bourbaki N., Groupes et Algèbres de Lie, Chapitres 4-6, Hermann, Paris, 1968.
[8] Brady T., A partial order on the symmetric group and new \({K(π,1)}\)’s for the braid groups, Adv. Math. 161 (2001), no. 1, 20-40. · Zbl 1011.20040
[9] Brady N., McCammond J. P., Mühlherr B. and Neumann W. D., Rigidity of Coxeter groups and Artin groups, Geom. Dedicata 94 (2002), no. 1, 91-109. · Zbl 1031.20035
[10] Brady T. and Watt C., \({K(π,1)}\)’s for Artin groups of finite type, Geom. Dedicata 94 (2002), 225-250. · Zbl 1053.20034
[11] Brieskorn E., Autmomorphic sets and braids and singularities, Braids (Santa Cruz 1986), Contemp. Math. 78, American Mathematical Society, Providence (1988), 45-115.
[12] Carter R. W., Conjugacy classes in the Weyl group, Compos. Math. 25 (1972), 1-59. · Zbl 0254.17005
[13] Carter R. W. and Elkington G. B., A note on the parametrization of conjugacy classes, J. Algebra 20 (1972), 350-354. · Zbl 0239.20053
[14] Digne F. and Gobet T., Dual braid monoids, Mikado braids and positivity in Hecke algebras, Math. Z., to appear. · Zbl 1400.20029
[15] Douglass J. M., Pfeiffer G. and Röhrle G., On reflection subgroups of finite Coxeter groups, Comm. Algebra 41 (2013), no. 7, 2574-2592. · Zbl 1282.20040
[16] Dyer M. J., Reflection subgroups of Coxeter systems, J. Algebra 135 (1990), no. 1, 57-73. · Zbl 0712.20026
[17] Ebeling W., Lattices and Codes, Vieweg, Wiesbaden, 2002. · Zbl 1051.62093
[18] Franzsen W. N., Howlett R. B. and Mühlherr B., Reflections in abstract Coxeter groups, Comment. Math. Helv. 81 (2006), no. 3, 665-697. · Zbl 1101.20024
[19] Hou X., Hurwitz equivalence in tuples of generalized quaternion groups and dihedral groups, Electron. J. Combin. 15 (2008), no. 1, Paper No. R80. · Zbl 1188.20032
[20] Hubery A. and Krause H., A categorification of non-crossing partitions, preprint 2013, ; to appear in J. Eur. Math. Soc.
[21] Humphreys J. E., Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math. 29, Cambridge University Press, Cambridge, 1990.
[22] Humphries S. P., Finite Hurwitz braid group actions on sequences of Euclidean reflections, J. Algebra 269 (2003), no. 2, 556-588. · Zbl 1040.20031
[23] Hurwitz A., Über Riemannsche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), 1-61. · JFM 23.0429.01
[24] Igusa K. and Schiffler R., Exceptional sequences and clusters, J. Algebra 323 (2010), 2183-2202. · Zbl 1239.16019
[25] James A., Magaard K. and Shpectorov S., The lift invariant distinguishes components of Hurwitz spaces for \({A_{5}}\), Proc. Amer. Math. Soc. 143 (2015), no. 4, 1377-1390. · Zbl 1373.14028
[26] Kane R., Reflection Groups and Invariant Theory, CMS Books Math./Ouvrages Math. SMC 5, Springer, New York, 2001. · Zbl 0986.20038
[27] Kluitmann P., Geometrische Basen des Milnorgitters einer einfach elliptischen Singularität, Diploma thesis, Universität Bonn, Bonn, 1983.
[28] Kulikov S. V. and Teicher M., Braid monodromy factorizations and diffeomorphism types, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), no. 2, 89-120. · Zbl 1004.14005
[29] Liberman E. and Teicher M., The Hurwitz equivalence problem is undecidable, preprint 2004, .
[30] Matsumoto H., Générateurs et relations des groupes de Weyl généralisés, C. R. Math. Acad. Sci. Paris 258 (1964), 3419-3422. · Zbl 0128.25202
[31] Michel J., Hurwitz action on tuples of Euclidean reflections, J. Algebra 295 (2006), no. 1, 289-292. · Zbl 1172.20306
[32] Nuida K., On the direct indecomposability of infinite irreducible Coxeter groups and the isomorphism problem of Coxeter groups, Comm. Algebra 34 (2006), no. 7, 2559-2595. · Zbl 1104.20038
[33] Qi D., A note on parabolic subgroups of a Coxeter group, Expo. Math. 25 (2007), no. 1, 77-81. · Zbl 1121.20031
[34] Reiner V., Ripoll V. and Stump C., On non-conjugate Coxeter elements in well-generated reflection groups, preprint 2014, .
[35] Sia C., Hurwitz equivalence in tuples of dihedral groups, dicyclic groups, and semidihedral groups, Electron. J. Combin. 16 (2009), no. 1, Paper No. R95. · Zbl 1191.20035
[36] Tits J., Buildings of Spherical Type and Finite Bn-Pairs, Lecture Notes in Math. 386, Springer, Berlin, 1974. · Zbl 0295.20047
[37] Voigt E., Ausgezeichnete Basen von Milnorgittern einfacher Singularitäten, Bonner Math. Schriften 160 (1985). · Zbl 0576.14004
[38] The GAP Group, GAP-Groups, Algorithms, and Programming, Version 4.7.7, 2015, .
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