The Lehmer complex of a Bruhat interval

TL;DR

This paper introduces Lehmer codes for finite Coxeter groups, linking Bruhat intervals to multicomplexes and simplicial complexes, and classifies Poincaré polynomials of smooth Schubert varieties using unimodal permutations.

Contribution

It develops Lehmer codes for Coxeter groups, connects Bruhat intervals to multicomplexes and simplicial complexes, and classifies Poincaré polynomials of smooth Schubert varieties.

Findings

Poincaré polynomials are h-polynomials of vertex-decomposable complexes

Lehmer codes provide immersions in Bruhat order for classical Weyl groups

Classification of Poincaré polynomials via unimodal permutations

Abstract

We introduce Lehmer codes, with immersions in the Bruhat order, for several finite Coxeter groups, including all the classical Weyl groups. This allows to associate to each lower Bruhat interval of these groups a multicomplex whose f-polynomial is the Poincar\'e polynomial of the interval. Via a general construction, we prove that these polynomials are h-polynomials of vertex-decomposable simplicial complexes. Moreover we provide a classification, in terms of unimodal permutations, of Poincar\'e polynomials of smooth Schubert varieties in flag manifolds.