The low-element characterization of Shi regions in Coxeter systems

Let $(W,S)$ be a Coxeter system with positive-root system $4\Phi^+4$, inversion sets $N(w)$, and finite set of small roots $4\Sigma(W,S)4$. Define the Shi arrangement by

$$\operatorname{Shi}(W,S)=\{H_\alpha\mid \alpha\in\Sigma(W,S)\}.$$

Let $L_{\operatorname{Shi}}(W,S)$ consist of the elements of minimal length in their Shi regions, and let $L(W,S)$ be the set of low elements, namely those $w\in W$ satisfying

$$N(w)=\operatorname{cone}(\Sigma(W,S)\cap N(w))\cap\Phi.$$

Shi-region low-element conjecture. One has $L_{\operatorname{Shi}}(W,S)=L(W,S)$. Moreover, every Shi region contains a unique low element, which is its unique element of minimal length.

The two assertions are equivalent and imply that the Shi-region minimal elements form a Garside shadow. The conjecture is known for complete-graph and right-angled Coxeter systems, for rank-three systems, and, as established in this paper, for affine Coxeter systems; it remains open in general.