Combinatorics Seminar: "Hultman elements for type B"

Abstract: Given a permutation (or more generally an element in a finite reflection group) w, one can define a hyperplane arrangement 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 defined by the inversion arrangement for w 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 these elements, where equality occurs, the Hultman elements.

 This result of Hultman generalizes work of Hultman, Linusson, Shareshian, and Sjostrand 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 context by Gasharov and Reiner. I will talk about the generalization of the Gasharov--Reiner-like as well as a pattern avoidance criterion for the Hultman elements for type B (the symmetries of the cube).

Host: Martha Precup