Equivariant Cohomology of Weighted Grassmannians and Weighted Schubert Classes

TL;DR

This paper explores the T_w-equivariant cohomology of weighted Grassmannians, introduces weighted Schubert classes, and provides explicit formulas for their structure constants, revealing positivity properties.

Contribution

It introduces equivariant weighted Schubert classes and derives explicit structure constants formulas, advancing the understanding of weighted Grassmannian cohomology.

Findings

Schubert classes form a basis of the equivariant cohomology

Explicit formulas for structure constants are provided

Structure constants are polynomials with non-negative coefficients

Abstract

In this paper, we study the T_w-equivariant cohomology of the weighted Grassmannians wGr(d,n) introduced by Corti-Reid where T_w is the n-dimensional torus that naturally acts on wGr(d,n). We introduce the equivariant weighted Schubert classes and, after we show that they form a basis of the equivariant cohomology, we give an explicit formula for the structure constants with respect to this Schubert basis. We also find a linearly independent subset {wu_1,...,wu_n} of Lie(T_w)^* such that those structure constants are polynomials in wu_i's with non-negative coefficients, up to a permutation on the weights.