  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.
