Inversion arrangements and Bruhat intervals

TL;DR

This paper provides a type-independent combinatorial criterion to determine when the principal Bruhat order ideal of an element in a finite reflection group matches the number of regions in its inversion hyperplane arrangement, linking poset properties and geometric arrangements.

Contribution

It introduces a universal combinatorial criterion characterizing elements satisfying a specific Bruhat order and hyperplane arrangement relation, extending previous pattern avoidance results and connecting to geometric smoothness.

Findings

The criterion is poset-theoretic and independent of group type.

For type A, the criterion aligns with known pattern avoidance characterizations.

Rational smoothness of Schubert varieties implies the criterion is satisfied.

Abstract

Let $W$ be a finite reflection group. For a given $w \in W$, 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 $w\in W$ 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 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 $w \in W$ is rationally smooth, then $w$ satisfies (*).