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Abstract: Let $W$ be a finite reflection group. For a given $w \in W$, the followingassertion may or may not be satisfied:* The principal Bruhat order ideal of $w$ contains as many elements asthere are regions in the inversion hyperplane arrangement of $w$.We present a type independent combinatorial criterion which characterises theelements $w\in W$ that satisfy *. A couple of immediate consequences arederived: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 patternavoidance, was previously given in collaboration with Linusson, Shareshian andSj\-ostrand. We obtain a short and simple proof of that result.3 If $W$ is a Weyl group and the Schubert variety indexed by $w \in W$ isrationally smooth, then $w$ satisfies *.

Autor: Axel Hultman


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