The Hopf algebra structure of the character rings of classical groups

The character ring Char-GL of covariant irreducible tensor representations of the general linear group admits a Hopf algebra structure isomorphic to the Hopf algebra Symm-^ of symmetric functions. Here we study the character rings Char-O and Char-Sp of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions. We show that Char-O and Char-Sp also admit natural Hopf algebra structures that are isomorphic to that of Char-GL, and hence to Symm-^. The isomorphisms are determined explicitly, along with the specification of standard bases for Char-O and Char-Sp analogous to those used for Symm-^. 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 Char-GL 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 Char-O and Char-Sp are not self-dual. The dual Hopf algebras Char-O∗ and Char-Sp∗ are identified. Finally, the Hopf algebra of the universal rational character ring Char-GLrat of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified.