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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
GAP
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