AI 中文总结
本文针对仿射型$\tilde{A}_n$的仿射对称群$\tilde{S}_N$，研究辫群在简约反射分解集合上的Hurwitz作用，确定该作用的所有轨道，解决了有限Coxeter群中Hurwitz作用传递性相关问题的仿射情形。
AI 中文摘要
设W是型为$\tilde{A}_n$的仿射Coxeter群，即N=n+1时的仿射对称群$\tilde{S}_N$，T是其反射集合，$\text{Red}_T(w)$是W中元素w的简约反射分解集合。辫群通过Hurwitz作用作用于$\text{Red}_T(w)$。对于有限Coxeter群，已知该作用何时传递，即恰对抛物类准Coxeter元素成立。本文针对仿射型$\tilde{A}_n$解决此问题，确定$\text{Red}_T(w)$的所有轨道。
英文摘要
Let $W$ be an affine Coxeter group of type $\widetilde{A}_n$, that is, the affine symmetric group $\widetilde{S}_N$ 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 $\widetilde{A}_n$ by determining all orbits of $\mathrm{Red}_T(w)$.
