Equivariant Schubert calculus for ordinary and weighted projective space

Schubert calculus computes the linear subspaces of a complex n-dimensional vector space satisfying various intersection conditions. From a more algebraic perspective, it asks for the structure constants of the cohomology ring of a Grassmannian in terms of its particularly nice basis of Schubert classes. For the projective space of lines through the origin in complex n-space, the answer is one of the first things computed in an algebraic topology class: the cohomology ring has a unique generator in each degree, and as long as the degrees add correctly, the structure constant is 1.

The equivariant cohomology ring is more complicated than the ordinary cohomology: it is a module over a polynomial ring rather than over the integers, and fewer of its structure constants vanish. We give an extraordinarily simple formula for equivariant structure constants, in terms of a sequence of divided difference operators applied to a particular polynomial. We also generalize these results to weighted projective space, which is the collection of particular curves through the origin rather than lines through the origin.