GL_n-structure and principal \mathfraksl₂-triple on the cohomology ring of complex Grassmannian

2021/11/16 by Nhok Tkhai Shon Ngo, Ngo, Nhok Tkhai Shon
Mathematics · #Advanced Combinatorial Mathematics #Advanced Algebra and Geometry #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2111.08754

Abstract

In this note we describe the cohomology ring of the Grassmannian of k-planes in n-dimensional complex vector space as an GL_n-module. We give explicit formulas for the operators of its principal \mathfraksl₂-triple. It is proved that one of these operators corresponds to the shifted cohomology degree operator and the second operator coincides with the Lefschetz map on cohomology (as in the hard Lefschetz theorem). We check that the cohomology ring of the complex Grassmannian as a GL_n-representation is isomorphic to the k-th exterior power of the standard n-dimensional representation.

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