The restricted Hurwitz action on cameral decompositions of Coxeter elements

Source abstract

Given a dual Coxeter system (W, T) of rank n, the Hurwitz action of the braid group on the reduced T-decompositions of a Coxeter element is transitive. When we choose a geometric representation of W, the reflections with respect to the n walls of one given chamber form a generating set S that we call cameral. An ordering of S is called a cameral decomposition and the product a cameral Coxeter element, which depends only on the quiver obtained by orienting the Coxeter diagram with vertices S according to the ordering. We show that the reduced decompositions of a cameral Coxeter elements for which the underlying set S satisfies that (W, S ) is a Coxeter system, are necessarily cameral, and we count the number of chambers from which a cameral Coxeter element can be obtained. We show that one can go through all these chambers using the restricted Hurwitz action on the set of cameral decompositions, which is generated by transpositions of adjacent commuting reflections and by the Garside element δ of the dual Garside structure, which acts by a quiver mutation of the orientation of the Coxeter diagram.