Inversion arrangements and the weak Bruhat order

TL;DR

This paper explores the relationship between inversion hyperplane arrangements and the weak Bruhat order, establishing inequalities and pattern-avoidance characterizations that connect geometric and combinatorial properties of permutations.

Contribution

It proves a new inequality relating the number of regions of inversion arrangements to the weak Bruhat order and characterizes equality cases via pattern avoidance.

Findings

Number of regions of inversion arrangements is at least the number of permutations below w in the weak Bruhat order.

Equality holds if and only if w avoids patterns 231 and 312.

Connects geometric hyperplane arrangements with permutation pattern avoidance.

Abstract

For each permutation $w$, we can construct a collection of hyperplanes $\mathcal{A}_w$ according to the inversions of $w$, which is called the inversion hyperplane arrangement associated to $w$. It was conjectured by Postnikov and confirmed by Hultman, Linusson, Shareshian and Sj\"{o}strand that the number of regions of $\mathcal{A}_w$ is less than or equal to the number of permutations below $w$ in the Bruhat order, with the equality holds if and only if $w$ avoids the four patterns 4231, 35142, 42513 and 351624. In this paper, we show that the number of regions of $\mathcal{A}_w$ is greater than or equal to the number of permutations below $w$ in the weak Bruhat order, with the equality holds if and only if $w$ avoids the patterns 231 and 312.