The cohomology of the Grassmannian is a gln-module - Gatto, Letterio; Salehyan, Parham [UNESP] (Taylor & Francis, 2019)
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 (Formula presented.) 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, (Formula presented.) corresponds to the bosonic vertex representation of the Lie algebra (Formula presented.) 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.
Keywords: Bosonic vertex representation of Date-Jimbo-Kashiwara-Miwa; cohomology of the Grassmannian; Hasse-Schmidt derivations on exterior algebras; Schubert derivations; vertex operators