The Cohomology of the Grassmannian is a $gl_n$-module

Abstract

The integral singular cohomology ring of the Grassmann variety parametrizing $r$-dimensional subspaces in the $n$-dimensional complex vector space is naturally an irreducible representation of the Lie algebra of all the $n\times n$ matrices with integral entries. Using the notion of Schubert derivation, a distinguished Hasse-Schmidt derivation on an exterior algebra, we describe explicitly such a representation, indicating its relationship with the celebrated bosonic vertex representation of the Lie algebra of infinite matrices due to Date, Jimbo, Kashiwara and Miwa.