The subject of this paper arises from the familiar process whereby an automorphism of a field generates new representations from old. One may think of that process spatially, as a change of vector space structure in the representation space by means of the automorphism. The operators of the representation acting in the “new“ space then constitute the new representation. This point of view makes visible an algebraic structure we call a scalar action. A scalar action f of a ring R (with unity) in an abelian group Kis a ring homomorphism f:R → End(V) taking the unity element of R to the identity operator in End(V). If f is a scalar action of a field F and ϕ is an automorphism of F then f ∘ ϕ is another scalar action of F, and it is this construction which is used to define the “new” representation space mentioned above. But the variety of scalar actions goes rather beyond that construction.

Below we work out the algebra structure of some Hopf algebras which arise concretely in restricting representations of the symmetric group to certain subgroups. The basic idea generalizes that used by Adams [1] for H*(BSU). The question arose in discussions with H. K. Farahat. I would like to thank him for his interest in the work and to acknowledge the usefulness of several stimulating conversations with him.

1. Review and statement of results. A homogeneous element of a graded abelian group will have its gradation referred to as its dimension. In all such groups below there will be no non-zero elements with negative or odd dimension. A graded algebra (resp. coalgebra) will be associative (resp. coassociative), strictly commutative (resp. co-commutative) and in dimension zero will be isomorphic to the ground ring F, providing the unit (resp. counit). We shall deal amost entirely with F = Z or F = Z/p for a prime p; the cases F = 0 or a localization of Z will occur briefly. In every case, the component in each dimension will be a finitely generated free F-module, so dualization works simply.