  The study of the action of the Steenrod algebra on the mod $p$ cohomology of
spaces has many applications to the topological structure of those spaces. In
this paper we present combinatorial formulas for the action of Steenrod
operations on the cohomology of Grassmannians, both in the Borel and the
Schubert picture. We consider integral lifts of Steenrod operations, which lie
in a certain Hopf algebra of differential operators. The latter has been
considered recently as a realization of the Landweber-Novikov algebra in
complex cobordism theory; it also has connections with the action of the
Virasoro algebra on the boson Fock space. Our formulas for Steenrod operations
are based on combinatorial methods which have not been used before in this
area, namely Hammond operators and the combinatorics of Schur functions. We
also discuss several applications of our formulas to the geometry of
Grassmannians.
