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MiS Preprint
36/2007
Hopf Algebra Structure of the Character Rings of Orthogonal and Symplectic Groups
Bertfried Fauser, Peter D. Jarvis and Ronald C. King
Abstract
We study the character rings ${\rm Char-}O$ and ${\rm Char-}Sp$ of the orthogonal and symplectic subgroups of the general linear group, within the framework of symmetric functions. We show that ${\rm Char-}O$ and ${\rm Char-}Sp$ admit natural Hopf algebra structures, and Hopf algebra isomorphisms from the general linear group character ring ${\rm Char-}GL$ (that is, the Hopf algebra of symmetric functions with respect to outer product) are determined. A major structural change is the introduction of new orthogonal and symplectic Schur-Hall scalar products. Standard bases for ${\rm Char-}O$ and ${\rm Char-}O$ (symmetric functions of orthogonal and symplectic type) are defined, together with additional bases which generalise different attributes of the standard bases of the ${\rm Char-}O$ case. Significantly, the adjoint with respect to outer multiplication no longer coincides with the Foulkes derivative (symmetric function 'skew'), which now acquires a separate definition. The properties of the orthogonal and symplectic Foulkes derivatives are explored. Finally, the Hopf algebras ${\rm Char-}O$ and ${\rm Char-}Sp$ are not self-dual, and the dual Hopf algebras ${\rm Char-}O^*$ and ${\rm Char-}Sp^*$ are identified.
