  Let $\ell, n$ be positive integers such that $\ell\geq n$. Let
$\mathbb{G}_{n,\ell}$ be the Grassmannian which consists of the set of
$n$-dimensional subspaces of $\mathbb{C}^{\ell}$. There is a
$\mathbb{Z}$-graded algebra isomorphism between the cohomology
$H^*(\mathbb{G}_{n,\ell},\mathbb{Z})$ of $\mathbb{G}_{n,\ell}$ and a natural
$\mathbb{Z}$-form $B$ of the $\mathbb{Z}$-graded basic algebra of the type $A$
cyclotomic nilHecke algebra
$H_{\ell,n}^{(0)}=\langle\psi_1,\cdots,\psi_{n-1},y_1,\cdots,y_n\rangle$. 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 $a_i\in\mathbb{Z}$ for each $i$, $\mathbf{\lambda}$ is
the $\ell$-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
$\mathbb{G}_{\ell-n,\ell}$ and the $b_{\lambda}$'s basis of the natural
$\mathbb{Z}$-form $B$ of the $\mathbb{Z}$-graded basic algebra of
$H_{\ell,n}^{(0)}$ is also obtained. As an application, we obtain a second
version of Giambelli formula for Schubert classes.
