Artículos de revistas
The cohomology of the Grassmannian is a gln-module
Fecha
2019-01-01Registro en:
Communications in Algebra.
1532-4125
0092-7872
10.1080/00927872.2019.1640240
2-s2.0-85070961822
3355840219680031
0000-0001-5885-5034
Autor
Politecnico di Torino
Universidade Estadual Paulista (Unesp)
Institución
Resumen
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.