The Hopf algebra structure of the character rings of classical groups

Abstract

The character ring of covariant irreducible tensor representations of the general linear group admits a Hopf algebra structure isomorphic to the Hopf algebra of symmetric functions. Here we study the character rings and of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions. We show that and also admit natural Hopf algebra structures that are isomorphic to that of , and hence to . The isomorphisms are determined explicitly, along with the specification of standard bases for and analogous to those used for . A major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic Schur–Hall scalar products. Significantly, the adjoint with respect to multiplication no longer coincides, as it does in the case, with a Foulkes derivative or skew operation. The adjoint and Foulkes derivative now require separate definitions, and their properties are explored here in the orthogonal and symplectic cases. Moreover, the Hopf algebras and are not self-dual. The dual Hopf algebras and are identified. Finally, the Hopf algebra of the universal rational character ring of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified.