An effective decomposition theorem for Schubert varieties

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.