# Palindromic intervals in Bruhat order and hyperplane arrangements

## TL;DR

This paper explores the relationship between palindromic intervals in Bruhat order, hyperplane arrangements, and the rational smoothness of Schubert varieties, establishing a link between geometric properties and combinatorial structures.

## Contribution

It constructs hyperplane arrangements for Weyl group elements and proves their generating functions match Poincaré polynomials precisely when the Schubert variety is rationally smooth.

## Key findings

- Generating functions for arrangements match Poincaré polynomials in rationally smooth cases
- Palindromic intervals characterize rational smoothness in Bruhat order
- Hyperplane arrangements encode geometric properties of Schubert varieties

## 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 Poincar\'e polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth.