From Bruhat intervals to intersection lattices and a conjecture of Postnikov
2009 (English)In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 116, no 3, 564-580 p.Article in journal (Refereed) Published
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.
Place, publisher, year, edition, pages
2009. Vol. 116, no 3, 564-580 p.
Bruhat order, Inversion arrangements, Pattern avoidance, smooth schubert varieties, arrangements
IdentifiersURN: urn:nbn:se:kth:diva-18267DOI: 10.1016/j.jcta.2008.09.001ISI: 000264406900004ScopusID: 2-s2.0-60649114279OAI: oai:DiVA.org:kth-18267DiVA: diva2:336313
QC 201005252010-08-052010-08-052011-01-10Bibliographically approved