Polynomial identities related to Special Schubert varieties

TL;DR

This paper derives explicit polynomial identities relating the intersection cohomology Poincaré polynomials of special Schubert varieties and their strata, revealing new algebraic relations among Grassmannian invariants.

Contribution

It provides explicit formulas for Poincaré polynomials of intersection cohomology of single condition Schubert varieties, building on recent decomposition theorem results.

Findings

Derived polynomial identities linking Poincaré polynomials of Grassmannians.

Explicit formulas for intersection cohomology Poincaré polynomials of Schubert varieties.

Symbolic analysis of specific polynomial identities.

Abstract

Let $\mathcal S$ be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincar\'{e} polynomials of the intersection cohomology of $\mathcal S$ by means of the Poincar\'{e} polynomials of its strata, obtaining interesting polynomial identities relating Poincar\'{e} polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities.