An effective decomposition theorem for Schubert varieties

TL;DR

This paper introduces a new decomposition theorem for Schubert varieties, providing detailed insights into their derived pushforwards, Poincaré polynomial expressions, and an algorithm to compute these terms, revealing fewer summands than supports.

Contribution

It presents a novel decomposition theorem for Schubert varieties, enhancing understanding of their derived pushforwards and offering an algorithm for computing Poincaré polynomial terms.

Findings

Derived pushforward decompositions for Schubert varieties clarified

Algorithm for computing Poincaré polynomial terms developed

Number of direct summands is less than the number of supports

Abstract

Given a Schubert variety $\mathcal{S}$ contained in a Grassmannian $\mathbb{G}_{k}(\mathbb{C}^{l})$, we show how to obtain further information on the direct summands of the derived pushforward $R \pi_{*} \mathbb{Q}_{\tilde{\mathcal{S}}}$ given by the application of the decomposition theorem to a suitable resolution of singularities $\pi: \tilde{\mathcal{S}} \rightarrow \mathcal{S}$. As a by-product, Poincar\'e polynomial expressions are obtained along with an algorithm which computes the unknown terms in such expressions and which shows that the actual number of direct summands happens to be less than the number of supports of the decomposition.