Schubert Class and cyclotomic nilHecke algebras

Let $\mathrm{\ell },n$ be positive integers such that $\mathrm{\ell }\ge n$. Let ${\mathbb{G}}_{n,\mathrm{\ell }}$ be the Grassmannian which consists of the set of $n$-dimensional subspaces of ${\mathbb{C}}^{\mathrm{\ell }}$. There is a $\mathbb{Z}$-graded algebra isomorphism between the cohomology $H^{\ast }({\mathbb{G}}_{n,\mathrm{\ell }},\mathbb{Z})$ of ${\mathbb{G}}_{n,\mathrm{\ell }}$ and a natural $\mathbb{Z}$-form $B$ of the $\mathbb{Z}$-graded basic algebra of the type $A$ cyclotomic nilHecke algebra $H_{\mathrm{\ell },n}^{(0)}=⟨{\psi }_1,\cdots ,{\psi }_{n-1},y_1,\cdots ,y_n⟩$. In this paper, we show that the isomorphism can be chosen such that the image of each (geometrically defined) Schubert class $(a_1,\cdots ,a_n)$ coincides with the basis element $b_{\mathbf{\lambda }}$ constructed by Jun Hu and Xinfeng Liang by purely algebraic method, where $0\leq a_1\leq a_2\leq\cdots\leq a_{n}\leq \ell-n$with$a_i\in$\mathbb{Z}$$foreach$i$,$$\mathbf{\lambda }$$ is the $\mathrm{\ell }$-multipartition of $n$ associated to $(\ell+1-(a_{n}+n), \ell+1-(a_{n-1}+n-1),\cdots,\ell+1-(a_1+1))$. A similar correspondence between the Schubert class basis of the cohomology of the Grassmannian ${\mathbb{G}}_{\mathrm{\ell }-n,\mathrm{\ell }}$ and the $b_{\lambda }$'s basis of the natural $\mathbb{Z}$-form $B$ of the $\mathbb{Z}$-graded basic algebra of $H_{\mathrm{\ell },n}^{(0)}$ is also obtained. As an application, we obtain a second version of Giambelli formula for Schubert classes.