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MiS Preprint

Hopf Algebra Structure of the Character Rings of Orthogonal and Symplectic Groups

Bertfried Fauser, Peter D. Jarvis and Ronald C. King


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.

MSC Codes:
16W30, 11E57
Orthogonal group, symplectic group, irreducible characters, symmetric functions, representation rings, hopf algebra, group characters

Related publications

2007 Repository Open Access
Bertfried Fauser, Peter D. Jarvis and Ronald C. King

Hopf algebra structure of the character rings of orthogonal and symplectic groups