Mathematics – Representation Theory
Scientific paper
2007-04-16
Mathematics
Representation Theory
33 pages, uses pstricks
Scientific paper
We study the character rings Char-O and Char-Sp of the orthogonal and symplectic subgroups of the general linear group, within the framework of symmetric functions. We show that Char-O and Char-Sp admit natural Hopf algebra structures, and Hopf algebra isomorphisms from the general linear group character ring 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 Char-O and Char-Sp (symmetric functions of orthogonal and symplectic type) are defined, together with additional bases which generalise different attributes of the standard bases of the Char-Gl 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 Char-O and Char-Sp are not self-dual, and the dual Hopf algebras Char-O^* and Char-Sp^* are identified.
Fauser Bertfried
Jarvis Peter D.
King Ronald C.
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