The cohomology of the Grassmannian is a gln-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 × n matrices with integral entries. The simplest case, r = 1, recovers the well known fact that any vector space is a module over the Lie algebra of its own endomorphisms. The other extremal case, corresponds to the bosonic vertex representation of the Lie algebra on the polynomial ring in infinitely many indeterminates, due to Date, Jimbo, Kashiwara and Miwa. In the present article we provide the structure of this irreducible representation explicitly, by means of a distinguished Hasse-Schmidt derivation on an exterior algebra, borrowed from Schubert Calculus