Schubert Calculus on a Grassmann Algebra

Abstract

Abstract. The (classical, small quantum, equivariant) cohomology ring of the grassmannian G(k, n) is generated by certain derivations operating on an exterior algebra of a free module of rank n (Schubert calculus on a Grassmann algebra). Our main result gives, in a unified way, a presentation of all such cohomology rings in terms of generators and relations. Using results of Laksov and Thorup, it also provides a presentation of the universal factorization algebra of a monic polynomial of degree n into the product of two monic polynomials, one of degree k.

Equivariant Schubert calculus

“…Thus A= i≥0 A i . We apply the theory exposed in [7] to the T -equivariant cohomology of G(k, n), in the case that the T -action is induced by the diagonal linear action (t 1 , ..., t n )(z 1 , ..., z n ) = (t 1 z 1 , ..., t n z n ) of T on C n . If F is an equivariant vector bundle over G(k, n), denote by c T (F ) its equivariant Chern polynomial i≥0 c T i (F )t i , where, for each i≥0, c T i (F ) is the T -equivariant ith Chern class of F .…”

“…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.…”

“…In our formulation, Schubert calculus takes place in a subring A * ( M ) of the ring of endomorphisms of the exterior algebra of a free A-module M of rank n, as in [7], generated by derivations of M . The (equivariant) Schubert calculus on G(k, n) is obtained by considering the A * ( M )-module structure of k M , the degree-k direct summand of M .…”

Newton Binomial Formulas in Schubert Calculus

“…The enumerative geometry of pencils on P 1 with prescribed ramifications is ruled by the intersection theory on G(2, n + 2). By [8],…”

“…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.…”