Bruhat order, smooth Schubert varieties, and hyperplane arrangements

TL;DR

This paper establishes a connection between the smoothness of Schubert varieties and hyperplane arrangements, providing a combinatorial formula for their Poincare polynomial using chordal graphs.

Contribution

It introduces a novel graphical hyperplane arrangement construction linked to permutations and characterizes smooth Schubert varieties through arrangement regions.

Findings

Generating function for arrangement regions matches Poincare polynomial for smooth varieties

Provides explicit combinatorial formula for Poincare polynomial

Uses chordal graphs and perfect elimination orderings as key tools

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.