A determinantal formula for the exterior powers of the polynomial ring

Equivariant Schubert calculus

“…This can be seen either by integration by parts (as in [7]) or by noticing, as in [13], that Giambelli's formula…”

“…As most expositions on (equivariant) Schubert calculus do, including [9] and [16], we give a quick look, at the end of the paper, to the example of G(2, 4), the first Grassmannian which is not a projective space. We revisit the example at p. 231 of [9], depicting the basis of the equivariant cohomology of G(2, 4), using solely Leibniz's rule and a bit of integration by parts (if one wants to avoid the general Giambelli's formula due to Laksov and Thorup [13], Theorem 0.1 (2)).…”

“…"Integration by parts" (as in [7], Remark 3.12) or the general determinantal formula due to Laksov and Thorup [13], yields…”

Schubert Calculus via Hasse-Schmidt Derivations

“…The results of this work have been recently improved and generalized by Laksov and Thorup ( [6]) to grassmannian bundles, using the theory of symmetric functions and of splitting algebras, allowing them to study, in general, the cohomology of (partial) flag varieties of a finite dimensional vector space over an algebraically closed field ( [7]). …”

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.…”