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arXiv · 1902.03824

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

Letterio Gatto·Parham Salehyan
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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.

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Letterio Gatto, Parham Salehyan. 2019-02-11. The Cohomology of the Grassmannian is a $gl_n$-module. https://arxiv.org/abs/1902.03824

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