Hultman elements for the hyperoctahedral groups

TL;DR

This paper explores characterizations of elements in hyperoctahedral groups that satisfy certain combinatorial criteria related to pattern avoidance, Bruhat order, and inversion arrangements, extending previous work on symmetric groups.

Contribution

It provides new pattern avoidance characterizations for hyperoctahedral groups, generalizing existing criteria from symmetric groups to a broader class of finite reflection groups.

Findings

Characterizations of hyperoctahedral group elements satisfying Hultman's criterion

Pattern avoidance criteria using Billey and Postnikov's notion

Connections between inversion arrangements and Bruhat order in hyperoctahedral groups

Abstract

Hultman, Linusson, Shareshian, and Sj\"ostrand gave a pattern avoidance characterization of the permutations for which the number of chambers of its associated inversion arrangement is the same as the size of its lower interval in Bruhat order. Hultman later gave a characterization, valid for an arbitrary finite reflection group, in terms of distances in the Bruhat graph. On the other hand, the pattern avoidance criterion for permutations had earlier appeared in independent work of Sj\"ostrand and of Gasharov and Reiner. We give characterizations of the elements of the hyperoctahedral groups satisfying Hultman's criterion that is in the spirit of those of Sj\"ostrand and of Gasharov and Reiner. We also give a pattern avoidance criterion using the notion of pattern avoidance defined by Billey and Postnikov.