A Hopf algebraic approach to the theory of group branchings

Bertfried Fauser, Peter D. Jarvis, and Ronald C. King
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Submission date: 16. Aug. 2005
Pages: 15
MSC-Numbers: 05E05, 16W30, 20G10, 11E57
Keywords and phrases: group branchings, symmetric functions, plethysm,, hopf algebra, representation rings
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Abstract:
H GL We describe a Hopf algebraic approach to the Grothendieck ring of representations of subgroups of the general linear group which stabilize a tensor of Young symmetry . It turns out that the representation ring of the subgroup can be described as a Hopf algebra twist, with a 2-cocycle derived from the Cauchy kernel 2-cocycle using plethysms. Due to Schur-Weyl duality we also need to employ the coproduct of the inner multiplication. In this paper we focus on the Hopf algebraic treatment, and a more formal approach to representation rings and symmetric functions.