$\mathrm{GL}_n$-structure and principal $\mathfrak{sl}_2$-triple on the cohomology ring of complex Grassmannian

TL;DR

This paper describes the $ ext{GL}_n$-module structure of the cohomology ring of complex Grassmannians, providing explicit formulas for the principal $ ext{sl}_2$-triple operators and connecting them to classical geometric operators.

Contribution

It explicitly characterizes the $ ext{GL}_n$-module structure of the Grassmannian cohomology ring and relates the $ ext{sl}_2$-triple operators to known geometric operators.

Findings

Cohomology ring is isomorphic to the $k$-th exterior power of the standard representation.

Explicit formulas for principal $ ext{sl}_2$-triple operators.

Operators correspond to degree shift and Lefschetz map.

Abstract

In this note we describe the cohomology ring of the Grassmannian of $k$-planes in $n$-dimensional complex vector space as an $\mathrm{GL}_n$-module. We give explicit formulas for the operators of its principal $\mathfrak{sl}_2$-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 $\mathrm{GL}_n$-representation is isomorphic to the $k$-th exterior power of the standard $n$-dimensional representation.