Abstract
Given an arbitrary Coxeter system (Formula presented.) and a non-negative integer (Formula presented.), the (Formula presented.) -Shi arrangement of (Formula presented.) is a subarrangement of the Coxeter hyperplane arrangement of (Formula presented.). The classical Shi arrangement ((Formula presented.)) was introduced in the case of affine Weyl groups by Shi to study Kazhdan–Lusztig cells for (Formula presented.). As two key results, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in (Formula presented.) and that the union of their inverses form a convex subset of the Coxeter complex. The set of (Formula presented.) -low elements in (Formula presented.) 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 (Formula presented.). In this article, we generalize and extend Shi's results to any Coxeter system for any (Formula presented.) : (1) the set of minimal length elements of the regions in a (Formula presented.) -Shi arrangement is precisely the set of (Formula presented.) -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.
Original language | English (US) |
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Article number | e12624 |
Journal | Proceedings of the London Mathematical Society |
Volume | 129 |
Issue number | 2 |
DOIs | |
State | Published - Aug 2024 |
ASJC Scopus subject areas
- General Mathematics