Shi arrangements and low elements in affine Coxeter groups

TL;DR

This paper proves that in affine Coxeter groups, the minimal length elements in Shi arrangement regions are exactly the low elements, confirming a conjecture and linking geometric and combinatorial structures.

Contribution

It establishes that minimal length region elements in affine Shi arrangements are precisely the low elements, confirming a conjecture by Dyer and the second author.

Findings

The set of minimal length elements in Shi arrangement regions equals the set of low elements.

Descent walls of regions correspond to descent walls of their associated low elements.

The result confirms a conjecture in the case of affine Coxeter groups.

Abstract

Given an affine Coxeter group $W$, the corresponding Shi arrangement is a refinement of the corresponding Coxeter hyperplane arrangements that was introduced by Shi to study Kazhdan-Lusztig cells for $W$. In particular, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in $W$. Low elements in $W$ were introduced to study the word problem of the corresponding Artin-Tits (braid) group and turns out to produce automata to study the combinatorics of reduced words in $W$. In this article, we show in the case of an affine Coxeter group that the set of minimal length elements of the regions in the Shi arrangement is precisely the set of low elements, settling a conjecture of Dyer and the second author in this case. As a byproduct of our proof, we show that the descent-walls -- the walls that separate a region from the fundamental alcove -- of any region in the Shi arrangement are precisely the descent walls of the alcove of its corresponding low element.