Palindromic intervals in Bruhat order and hyperplane arrangements

Abstract

An element $w$ of the Weyl group is called rationally smooth if the corresponding Schubert variety is rationally smooth. This happens exactly when the lower interval $[id,w]$ in the Bruhat order is palindromic. For each element $w$ of the Weyl group, we construct a certain hyperplane arrangement. After analyzing the palindromic intervals inside the maximal quotients, we use this result to show that the generating function for regions of the arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth.

12 pages, 10 figures

References in corpus (2)

• Rationally smooth Schubert varieties and inversion hyperplane arrangements
• Pattern avoidance and fiber bundle structures on Schubert varieties