New branching rules induced by plethysm
Bertfried Fauser, Peter D. Jarvis, Ronald C. King, and Brian G. Wybourne
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Submission date: 11. May. 2005
published in: Journal of physics / A, 39 (2006) 11, p. 2611-2655
DOI number (of the published article): 10.1088/0305-4470/39/11/006
MSC-Numbers: 05E05, 16W30, 20G10, 11E57
Keywords and phrases: group branchings, symmetric functions, plethysm, hopf algebra, schur function series, newell-littlewood theorem, representation theory
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We derive group branching laws for formal characters of subgroups of leaving invariant an arbitrary tensor of Young symmetry type where is an integer partition. The branchings , and fixing a vector , a symmetric tensor and an antisymmetric tensor , respectively, are obtained as special cases. All new branchings are governed by Schur function series obtained from plethysms of the Schur function by the basic M series of complete symmetric functions and the series of elementary symmetric functions. Our main technical tool is that of Hopf algebras, and our main result is the derivation of a coproduct for any Schur function series obtained by plethysm from another such series. Therefrom one easily obtains -generalized Newell-Littlewood formulae, and the algebra of the formal group characters of these subgroups is established. Concrete examples and extensive tabulations are displayed for , , and , showing their involved and nontrivial representation theory. The nature of the subgroups is shown to be in general affine, and in some instances non reductive. We discuss the complexity of the coproduct formula and give a graphical notation to cope with it. We also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2-cocycles. The algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for formal characters. Modification rules are beyond the scope of the present paper, but are briefly discussed.