From Bruhat intervals to intersection lattices and a conjecture of Postnikov

TL;DR

This paper proves Postnikov's conjecture relating the number of regions in inversion hyperplane arrangements to Bruhat order, extending it to all finite reflection groups, and explores related combinatorial and topological implications.

Contribution

It confirms Postnikov's conjecture, extends it to finite reflection groups, and provides new combinatorial and topological insights into inversion arrangements and related structures.

Findings

Number of regions in inversion arrangements is at most the Bruhat order elements below w.

Equality holds iff w avoids specific patterns.

Derived inequalities for Betti numbers and a combinatorial interpretation of chromatic polynomials.

Abstract

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation $w\in \Sn$ is at most the number of elements below $w$ in the Bruhat order, and (B) that equality holds if and only if $w$ avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinatorial interpretation of the chromatic polynomial of the inversion graph of a permutation which avoids the above patterns.