Shi arrangements and low elements in Coxeter groups

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.