2019/08/23 by Letterio Gatto, Parham Salehyan
Mathematics · #Commutative Algebra and Its Applications #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models
paper · doi:10.1080/00927872.2019.1640240
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 gln(Z) 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, r=∞, corresponds to the bosonic vertex representation of the Lie algebra gl∞(Z) 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