Schubert Class and cyclotomic nilHecke algebras

TL;DR

This paper establishes an algebraic correspondence between Schubert classes in Grassmannian cohomology and basis elements of cyclotomic nilHecke algebras, providing new algebraic tools and a second Giambelli formula.

Contribution

It constructs an explicit algebraic isomorphism aligning Schubert classes with basis elements of cyclotomic nilHecke algebras, enhancing understanding of their structure.

Findings

Explicit algebraic correspondence between Schubert classes and algebra basis elements.

Construction of a second Giambelli formula for Schubert classes.

Identification of basis elements with geometrically defined Schubert classes.

Abstract

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.