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From Bruhat intervals to intersection lattices and a conjecture of Postnikov
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0001-6339-2230
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
Abstract [en]

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.
Keyword [en]
Bruhat order, Inversion arrangements, Pattern avoidance, smooth schubert varieties, arrangements
Identifiers
URN: urn:nbn:se:kth:diva-18267DOI: 10.1016/j.jcta.2008.09.001ISI: 000264406900004Scopus ID: 2-s2.0-60649114279OAI: oai:DiVA.org:kth-18267DiVA: diva2:336313
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved

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  • nn-NB
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