Inversion arrangements and Bruhat intervals

Authors:
Axel Hultman
Affiliations:
Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden
Venue:
Journal of Combinatorial Theory Series A
Year:
2011

Citing 3
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Bruhat intervals as rooks on skew Ferrers boards

Journal of Combinatorial Theory Series A
Bruhat order, smooth Schubert varieties, and hyperplane arrangements

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From Bruhat intervals to intersection lattices and a conjecture of Postnikov

Journal of Combinatorial Theory Series A

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Abstract

Let W be a finite Coxeter 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:(1)The 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 Sjostrand. 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 (@?).