Equivariant Schubert calculus (Q2380811)

Let \(G(k,n)\) be the complex Grassmannian space of all \(k\)-dimensional vector subspaces of \(\mathbb C^n\). Suppose that \(n\)-dimensional torus \(T\) acts on \(G(k,n)\) such that the fixed point locus is isolated. Then the integral \(T\)-equivariant cohomogy \(H^*_T (G(k,n))\) of complex Grassmannian space \(G(k,n)\) is isomorphic to a free module over the ring \(A:= H^*_T (pt)\) which is the integral \(T\)-equivariant cohomogy of a point. In this work the main result says that the ring structure of the \(A\)-algebra \(H^*_T (G(k,n))\) can be described by derivations on the exterior algebra of a free \(A\)-module of rank \(n\). Actually these derivations on the exterior algebra compute the coefficients in the \(T\)-equivariant Schubert calculus. Also the authors are giving the answer for determining of \(T\)-equivariant Pieri formula.