Inversion hyperplane arrangements and intervals in Bruhat order for the hyperoctahedral group

par Alexander Woo (University of Idaho)

mardi 4 juil. 2017, 10:45 → 11:45 Europe/Paris

Bât. Braconnier, salle séminaire 2 (ICJ, Université Lyon 1)

To each permutation in S_n, one can associate an arrangement of hyperplanes in R^n called the inversion arrangement. A hyperplane arrangement cuts R^n into connected components called chambers. One would like to know the number of chambers in the inversion arrangement of a given permutation. On the set of all permutations in S_n, one can define a graph called the Bruhat graph and a partial order called Bruhat order. Given one permutation w, one can count the number of permutations less than or equal to w in Bruhat order. Hultman, Linusson, Shareshian, and Sjostrand (HLSS) show that the number of permutations less than or equal to w is an upper bound for the number of chambers for the inversion arrangement of w, with equality holding if and only if w avoids 4231, 35142, 42513, and 351624. The set of permutations avoiding 4231, 35142, 42513, and 351624 has an alternate characterization due to Gasharov and Reiner in terms of the minimal permutations not smaller than w (in Bruhat order). On the other hand, Hultman extended the HLSS theorem to arbitrary finite reflection groups using a Bruhat graph criterion to describe when equality holds. After explaining all of the above with several examples, I will state at the very end a new theorem giving a Gasharov--Reiner style criterion for equality (of chamber counts and Bruhat interval sizes) for elements of the symmetry group of the n-dimensional cube.