Preprint 36/2007

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

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

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Submission date: 10. Apr. 2007
Pages: 36
MSC-Numbers: 16W30, 11E57
Keywords and phrases: Orthogonal group, symplectic group, irreducible characters, symmetric functions, representation rings, hopf algebra, group characters
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We study the character rings formula15 and formula17 of the orthogonal and symplectic subgroups of the general linear group, within the framework of symmetric functions. We show that formula15 and formula17 admit natural Hopf algebra structures, and Hopf algebra isomorphisms from the general linear group character ring formula23 (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 formula15 and formula15 (symmetric functions of orthogonal and symplectic type) are defined, together with additional bases which generalise different attributes of the standard bases of the formula15 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 formula15 and formula17 are not self-dual, and the dual Hopf algebras formula35 and formula37 are identified.

21.02.2013, 01:42