Equivariant cohomology of juggling varieties in rank one

TL;DR

This paper computes the equivariant cohomology ring of rank-one juggling varieties, providing explicit generators, relations, and structure constants, and shows their integrality.

Contribution

It introduces a basis for the cohomology of these varieties and explicitly describes their ring structure in terms of generators and relations.

Findings

Explicit ring structure with generators and relations

Construction of a Knutson--Tao type basis

Proved integrality of structure constants

Abstract

We determine the ring structure of the torus-equivariant cohomology of rank-one juggling varieties with rational coefficients. By realizing these varieties as cyclic quiver Grassmannians, we construct a Knutson--Tao type basis for their equivariant cohomology. Using this basis, we give an explicit description of the ring structure in terms of generators and relations, and compute the corresponding structure constants. Finally, we show that these structure constants are integral.