Abstract:
A couple of years ago we proved in joint work with Linusson, Shareshian and Sjöstrand [HLSS] the following results that had been conjectured by A. Postnikov:

(1) The number of regions in the inversion arrangement corresponding to an element w in a finite reflection group is no larger than the number of elements below w in Bruhat order.

(2) In symmetric groups, equality holds in the above statement iff w avoids the permutation patterns 4231, 35142, 42513 and 351624.

In this talk we revisit the key constructions from [HLSS]. A new characterization of equality, valid for any finite reflection group, is given. Among the consequences is a much simpler proof of the hard direction of statement (2). Another corollary is that equality holds whenever w corresponds to a rationally smooth Schubert variety. The latter assertion relates to recent work by Oh and Yoo, partly joint with Postnikov.