A footnote to a paper of Deodhar

TL;DR

This paper refines the decomposition theorem for Schubert varieties, providing a recursive method to extract Kazhdan-Lusztig polynomials and introducing a new basis in the Hecke algebra derived from equivariant resolutions.

Contribution

It offers an effective version of the decomposition theorem and a recursive procedure to compute Kazhdan-Lusztig polynomials without prior minimal set knowledge.

Findings

Recursive procedure for Kazhdan-Lusztig polynomial extraction

New basis in the Hecke algebra from equivariant resolutions

Method to compute transition matrix between bases

Abstract

Let $X\subseteq G\slash B$ be a Schubert variety in a flag manifold and let $\pi: \tilde X \rightarrow X$ be a Bott-Samelson resolution of $X$. In this paper we prove an effective version of the decomposition theorem for the derived pushforward $R \pi_{*} \mathbb{Q}_{\tilde{X}}$. As a by-product, we obtain recursive procedure to extract Kazhdan-Lusztig polynomials from the polynomials introduced by V. Deodhar in \cite{Deo}, which does not require prior knowledge of a minimal set. We also observe that any family of equivariant resolutions of Schubert varieties allows to define a new basis in the Hecke algebra and we show a way to compute the transition matrix, from the Kazhdan-Lusztig basis to the new one.