Inversion arrangements and Bruhat intervals

9 years 4 months ago
Inversion arrangements and Bruhat intervals
Let W be a finite reflection group. For a given w ∈ W, the following assertion may or may not be satisfied: (∗) The principal Bruhat order ideal of w contains as many elements as there are regions in the inversion hyperplane arrangement of w. We present a type independent combinatorial criterion which characterises the elements w ∈ W that satisfy (∗). A couple of immediate consequences are derived: criterion only involves the order ideal of w as an abstract poset. In this sense, (∗) is a poset-theoretic property. (2) For W of type A, another characterisation of (∗), in terms of pattern avoidance, was previously given in collaboration with Linusson, Shareshian and Sj¨ostrand. We obtain a short and simple proof of that result. (3) If W is a Weyl group and the Schubert variety indexed by w ∈ W is rationally smooth, then w satisfies (∗).
Axel Hultman
Added 14 May 2011
Updated 14 May 2011
Type Journal
Year 2011
Where JCT
Authors Axel Hultman
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