Bruhat order, smooth Schubert varieties, and hyperplane arrangements

Authors:
Suho Oh;Alexander Postnikov;Hwanchul Yoo
Affiliations:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA;Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA;Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA
Venue:
Journal of Combinatorial Theory Series A
Year:
2008

Citing 3
Cited 3

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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

The aim of this article is to link Schubert varieties in the flag manifold with hyperplane arrangements. For a permutation, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincare polynomial of the corresponding Schubert variety if and only if the Schubert variety is smooth. We give an explicit combinatorial formula for the Poincare polynomial. Our main technical tools are chordal graphs and perfect elimination orderings.