Abstract:

With a permutation w of the set {1, ..., n} we may associate an arrangement of hyperplanes in R^n by including the hyperplanes given by x_i = x_j whenever (i,j) is an inversion of w. A conjecture of A. Postnikov asserts that the number of regions in the complement of this inversion arrangement is at most the number of elements weakly below w in the Bruhat order. He also conjectured a characterization of the permutations for which the two numbers are equal; intriguingly these permutations already have appeared in seemingly non-related work of Sjöstrand and Reiner-Gasharov.

In the talk, I will sketch a proof of Postnikov's conjecture. Generalizations beyond symmetric groups, and byproducts such as inequalities relating Betti numbers of complexified inversion arrangements and Schubert varieties will be touched upon.

This is joint work with S. Linusson, J. Shareshian and J. Sjöstrand.