Betti numbers of smooth Schubert varieties and the remarkable formula of Kostant,Macdonald,Shapiro and Steinberg

TL;DR

This paper refines a product formula for the Poincare polynomial of smooth Schubert varieties, leading to a new elementary criterion for their smoothness based on Bruhat intervals in Weyl groups.

Contribution

It provides a refined formula for the Poincare polynomial and a necessary condition for smoothness of Schubert varieties, connecting algebraic geometry and combinatorics.

Findings

Factorization of the number of elements in Bruhat intervals for smooth Schubert varieties

Elementary necessary condition for Schubert variety smoothness

Refinement of existing product formula for Poincare polynomials

Abstract

The purpose of this note is to give a refinement of the product formula proved in [1] for the Poincare polynomial of a smooth Schubert variety in the flag variety of an algebraic group G over C. This yields a factorization of the number of elements in a Bruhat interval [e,w] in the Weyl group W of G provided the Schubert variety associated to w is smooth. This gives an elementary necessary condition for a Schubert variety in the flag variety to be smooth.