From Bruhat intervals to intersection lattices and a conjecture of Postnikov

Details From Bruhat intervals to intersection lattices and a conjecture of Postnikov From Bruhat intervals to intersection lattices and a conjecture of Postnikov

2007-10-05

arxiv.org/abs/0710.1220v1

Mathematics

Combinatorics

24 pages

Scientific paper

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation $w\in \Sn$ is at most the number of elements below $w$ in the Bruhat order, and (B) that equality holds if and only if $w$ avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinatorial interpretation of the chromatic polynomial of the inversion graph of a permutation which avoids the above patterns.

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