Bruhat order, smooth Schubert varieties, and hyperplane arrangements

Details Bruhat order, smooth Schubert varieties, and hyperplane arrangements Bruhat order, smooth Schubert varieties, and hyperplane arrangements

2007-09-20

arxiv.org/abs/0709.3259v1

Mathematics

Combinatorics

14 pages, 2 figures

Scientific paper

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.

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