The Hurwitz Action in the Affine Symmetric Group

Abstract

Let $W$ be an affine Coxeter group of type ${\tilde{A}}_n$, that is, the affine symmetric group ${\tilde{S}}_N$ with $N=n+1$, let $T$ be its set of reflections, and let ${\mathrm{Red}}_T(w)$ be the set of reduced reflection factorizations of an element $w\in W$. The braid group acts on ${\mathrm{Red}}_T(w)$ by the Hurwitz action. For finite Coxeter groups it is known exactly when this action is transitive, namely precisely for the parabolic quasi-Coxeter elements. We address this problem for the affine type ${\tilde{A}}_n$ by determining all orbits of ${\mathrm{Red}}_T(w)$.