Séminaire Lotharingien de Combinatoire, 91B.19 (2024), 12 pp.
Matthew Dyer, Susanna Fishel, Christophe Hohlweg and Alice Mark
Shi Arrangements and Low Elements in Coxeter Groups
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.
We generalize and extend Shi's results to any Coxeter system. First,
for m ∈ N 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. Second, for m=0 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.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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