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The character ring \CGL of covariant irreducible tensor representations ofthe general linear group admits a Hopf algebra structure isomorphic to the Hopfalgebra \Sym$ of symmetric functions. Here we study the character rings \CO and\CSp of the orthogonal and symplectic subgroups of the general linear groupwithin the same framework of symmetric functions. We show that \CO and \CSpalso admit natural Hopf algebra structures that are isomorphic to that of \CGL,and hence to \Sym. The isomorphisms are determined explicitly, along with thespecification of standard bases for \CO and \CSp analogous to those used for\Sym. A major structural change arising from the adoption of these bases is theintroduction of new orthogonal and symplectic Schur-Hall scalar products.Significantly, the adjoint with respect to multiplication no longer coincides,as it does in the \CGL case, with a Foulkes derivative or skew operation. Theadjoint and Foulkes derivative now require separate definitions, and theirproperties are explored here in the orthogonal and symplectic cases. Moreover,the Hopf algebras \CO and \CSp are not self-dual. The dual Hopf algebras \CO^*and \CSp^* are identified. Finally, the Hopf algebra of the universal rationalcharacter ring \CGLrat of mixed irreducible tensor representations of thegeneral linear group is introduced and its structure maps identified.

The Hopf Algebra Structure of the Character Rings of Classical Groups