

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
Bibtex
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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Abstract:
We study the character rings and
of the
orthogonal and symplectic subgroups of the general linear group, within the
framework of symmetric functions. We show that
and
admit natural Hopf algebra structures, and Hopf algebra
isomorphisms from the general linear group character ring
(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
and
(symmetric functions of orthogonal
and symplectic type) are defined, together with additional bases which
generalise different attributes of the standard bases of the
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
and
are not self-dual, and the dual Hopf
algebras
and
are identified.