| Abstract: Given a permutation (or more generally an element in a finite reflection group) $w$, one can define a hyperplane arrangement $\mathcal{A}_w\subseteq\mathbb{R}^n$ called the inversion arrangement. On the symmetric group (or any finite reflection group), one can define a partial order known as Bruhat order. Hultman showed that the number of regions $\mathcal{A}_w$ cuts $\mathbb{R}^n$ into is always at most the number of elements less than or equal to $w$ in Bruhat order, and gave a condition on the Bruhat graph (a graph related to Bruhat order) for when equality occurs. We call elements satisfying this condition Hultman elements. This result of Hultman generalizes work of Hultman, Linusson, Shareshian, and Sjöstrand in the case of permutations. In this case, they show equality occurs precisely when $w$ pattern avoids the 4 permutations 4231, 35142, 42513, and 351624. This
set of permutations was earlier studied in a different contexts by
Sjöstrand and by Gasharov and Reiner. I will describe analogues of the Sjöstrand and Gasharov-Reiner conditions that characterize the
Hultman elements in type B, as well as a pattern avoidance criterion. |