From Bruhat intervals to intersection lattices and a conjecture of Postnikov

Authors:
Axel Hultman;Svante Linusson;John Shareshian;Jonas Sjöstrand
Affiliations:
Department of Mathematics, KTH-Royal Institute of Technology, SE-100 44, Stockholm, Sweden;Department of Mathematics, KTH-Royal Institute of Technology, SE-100 44, Stockholm, Sweden;Department of Mathematics, Washington University, St. Louis, MO 63130, USA;Department of Mathematics and Physics, Mälardalen University, Box 883, SE-721 23, Västerås, Sweden
Venue:
Journal of Combinatorial Theory Series A
Year:
2009

Citing 2
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Bruhat order, smooth Schubert varieties, and hyperplane arrangements

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Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams

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Abstract

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w@?S"n 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.