Schubert Class and Cyclotomic NilHecke Algebras

Let ℓ and n be positive integers such that ℓ≥n, and let Gn,ℓ be the Grassmannian which consists of the set of n-dimensional subspaces of Cℓ. There is a Z-graded algebra isomorphism between the cohomology H(Gn,ℓ,M) of G n,ℓ and a natural Z-form B of the Z-graded basic algebra of the type A cyclotomic nilHecke algebra Hℓ,n(0)= ℓ ψ1,.., ψ n-1, y1,..,yn. We show that the isomorphism can be chosen such that the image of each (geometrically defined) Schubert class (a1,.., an) coincides with the basis element bλ constructed by Hu and Liang by purely algebraic method, where 0≤a1≤ a2≤⋯≤an≤ℓ-n with ai Z for each i, and λ is the ℓ-multipartition of n associated to (ℓ+1-(an+n),ℓ+1-(an-1+n-1),..,ℓ+1-(a1+1)). A similar correspondence between the Schubert class basis of the cohomology of the Grassmannian Gℓ-n,ℓ and the bλ's basis (λ is an ℓ-multipartition of n with each component being either (1) or empty) of the natural Z-form B of the Z-graded basic algebra of Hℓ,n (0) is also obtained. As an application, we obtain a second version of the Giambelli formula for Schubert classes.