Schubert Calculus via Hasse-Schmidt Derivations

Abstract

Abstract. A natural Hasse-Schmidt derivation on the exterior algebra of a free module realizes the (small quantum) cohomology ring of the grassmannian G k (C n ) as a ring of operators on the exterior algebra of a free module of rank n. Classical Pieri's formula can be interpreted as Leibniz's rule enjoyed by special Schubert cycles with respect to the wedge product.

Equivariant Schubert calculus

“…equation (1) can be more compactly phrased by saying that D t is an A-algebra homomorphism ( [5], [6], [7], [18]), i.e., more explicitly, that…”

“…It turns out that H * T (G(k, n)), the integral T -equivariant cohomology ring of G(k, n), is a finite free module over the ring A:=H * T (pt), the T -equivariant cohomology of a point. Our main result (Theorem 2.2) shows that the multiplicative structure of the A-algebra H * T (G(k, n)), for all 1≤k≤n, can be described through derivations on the Grassmann algebra of a free A-module of rank n, in the same spirit as [2], [5], [6] and [7]. We stress that the description of the product structure of H * T (G(k, n)) heavily depends on the particular choice of an Abasis for it.…”

A formalism for equivariant Schubert calculus

“…The observation that equivariant cohomology could be interpreted within the framework of exterior powers was made in [Gatto and Santiago 2006] and [Santiago 2006]. In the latter reference it was proved that there exists an isomorphism between Schubert calculus on exterior powers, that is, Schubert calculus in a setting similar to the Grass l (n) case mentioned above, and equivariant cohomology for Grassmann manifolds, and for simple examples (projective space k = 1, and the Knutson-Tao [2003] example with k = 2, n = 4) it was indicated what the isomorphism should look like; see [Gatto and Santiago 2006]. It was this work that inspired us to consider the equivariant cohomology of Grassmann schemes and to describe the explicit isomorphism in the general case.…”

“…. , T l ] sym in l variables over an arbitrary ring A acting on the exterior product l A A[T ] of the polynomial ring A[T ] in one variable (see also [Gatto 2005] and [Gatto and Santiago 2009]).…”

Newton Binomial Formulas in Schubert Calculus

“…In [6] (see also [7,17]), the intersection theory on G(k, n) (Schubert calculus) is rephrased via a natural derivation on the exterior algebra of a free Z-module of rank n. Classical Pieri's and Giambelli's formulas are recovered, respectively, from Leibniz's rule and integration by parts inherited from such a derivation. The generalization of [6] to the intersection theory on Grassmann bundles is achieved in [8], by suitably translating previous important work by Laksov and Thorup [13,14] regarding the existence of a canonical symmetric structure on the exterior algebra of a polynomial ring.…”