Equivariant Giambelli and determinantal restriction formulas for the Grassmannian

Abstract

Let V be an n-dimensional complex vector space and Gd,n the Grassmannian of d-dimensional linear subspaces of V. The authors consider the T-equivariant integral cohomology ring H of Gd,n. It is known that the equivariant Schubert classes form a basis for H over the T-equivariant cohomology ring of a point. The main result of the article is a determinantal formula for the restriction to a torus fixed point of the equivariant class of a Schubert subvariety in H. To prove this result, the authors use Gröbner degeneration technique. As a corollary, an equivariant version of Giambelli formula is obtained.