Equivariant Giambelli and Determinantal Restriction Formulas for the Grassmannian

Abstract

The main result of the paper is a determinantal formula for the restriction to a torus fixed point of the equivariant class of a Schubert subva- riety in the torus equivariant integral cohomology ring of the Grassmannian. As a corollary, we obtain an equivariant version of the Giambelli formula. The (torus) equivariant cohomology rings of flag varieties in general and of the Grassmannian in particular have recently attracted much interest. Here we con- sider the equivariant integral cohomology ring of the Grassmannian. Just as the ordinary Schubert classes form a module basis over the ordinary cohomology ring of a point (namely the ring of integers) for the ordinary integral cohomology ring of the Grassmannian, so do the equivariant Schubert classes form a basis over the equivariant cohomology of a point (namely the ordinary cohomology ring of the classifying space of the torus) for the equivariant cohomology ring (this is true for any generalized flag variety of any type, not just the Grassmannian). Again as in the ordinary case, computing the structure constants of the multiplication with respect to this basis is an interesting problem that goes by the name of Schubert calculus. There is a forgetful functor from equivariant cohomology to ordinary cohomology so that results about the former specialize to those about the latter. Knutson-Tao-Woodward (5) and Knutson-Tao (6) show that the structure con- stants, both ordinary and equivariant, count solutions to certain jigsaw puzzles, thereby showing that they are manifestly positive. In the present paper we take a very different route to computing the equivariant structure constants. Namely, we try to extend to the equivariant case the classical approach by means of the Pieri and Giambelli formulas. Recall, from (3, Eq.(10), p.146) for example, that the Gi- ambelli formula expresses an arbitrary Schubert class as a polynomial with integral coefficients in certain Schubert classes—the Chern classes of the tautolog- ical quotient bundle—and that the Pieri formula expresses as a linear combination of the Schubert classes the product of a special Schubert class with an arbitrary Schubert class. Together they can be used to compute the structure constants. We only partially succeed in our attempt: the first of the three theorems of this paper—see §2 below—is an equivariant Giambelli formula that specializes to the ordinary Giambelli formula as in (3, Eq.(10), p.146), but we still do not have a satisfactory equivariant Pieri formula—see, however, §7 below. The derivation in Fulton (2, §14.3) of the Giambelli formula can perhaps be extended to the equi- variant case, but this is not what we do. Instead, we deduce the Giambelli formula from our second theorem which gives a certain closed-form determinantal formula for the restriction to a torus fixed point of an equivariant Schubert class.