From Bruhat intervals to intersection lattices and a conjecture of Postnikov

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:116

Article

Hultman, Axel¹  Linusson, Svante¹  Shareshian, John²  Sjostrand, Jonas³ 

[1] KTH Royal Inst Technol, Dept Math, SE-10044 Stockholm, Sweden

[2] Washington Univ, Dept Math, St Louis, MO 63130 USA

[3] Malardalen Univ, Dept Math & Phys, SE-72123 Vasteras, Sweden

关键词: Bruhat order;    Inversion arrangements;    Pattern avoidance;   

DOI  :  10.1016/j.jcta.2008.09.001

来源: Elsevier

【 摘 要 】

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w is an element of (sic)(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. (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

Free