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arXiv · 2303.16569

Shi arrangements and low elements in Coxeter groups

Matthew Dyer·Christophe Hohlweg·Susanna Fishel·Alice Mark
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Abstract

Given an arbitrary Coxeter system $(W,S)$ and a nonnegative integer $m$, the $m$-Shi arrangement of $(W,S)$ is a subarrangement of the Coxeter hyperplane arrangement of $(W,S)$. The classical Shi arrangement ($m=0$) was introduced in the case of affine Weyl groups by Shi to study Kazhdan-Lusztig cells for $W$. As two key results, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in $W$ and that the union of their inverses form a convex subset of the Coxeter complex. The set of $m$-low elements in $W$ were introduced to study the word problem of the corresponding Artin-Tits (braid) group and they turn out to produce automata to study the combinatorics of reduced words in $W$. In this article, we generalize and extend Shi's results to any Coxeter system for any $m$: (1) the set of minimal length elements of the regions in a $m$-Shi arrangement is precisely the set of $m$-low elements, settling a conjecture of the first and third authors in this case; (2) the union of the inverses of the ($0$-)low elements form a convex subset in the Coxeter complex, settling a conjecture by the third author, Nadeau and Williams.

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Matthew Dyer, Christophe Hohlweg, Susanna Fishel, Alice Mark. 2023-03-29. Shi arrangements and low elements in Coxeter groups. https://doi.org/10.1112/plms.12624

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