Schubert Class and cyclotomic nilHecke algebras

Abstract

Let $\ell ,n$ be positive integers such that $\ell \ge n$. 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 $H^{\ast }({\mathbb{G}}_{n,\ell },\mathbb{Z})$ 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 $H_{\ell ,n}^{(0)}=⟨{\psi }_1,\cdots ,{\psi }_{n-1},y_1,\cdots ,y_n⟩$. 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 $b_{\lambda }$ constructed by Jun Hu and Xinfeng Liang by purely algebraic method, where $0\le a_1\le a_2\le \cdots \le a_n\le \ell -n$ with $a_i\in \mathbb{Z}$ for each $i$, $\lambda$ 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.

arXiv commentcorrected a number of typos and added one corollary in the last section

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