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New branching rules induced by plethysm


Fauser, B and Jarvis, PD and King, RC and Wybourne, BG, New branching rules induced by plethysm, Journal of Physics A: Mathematical and General, 39, (11) pp. 2611-2655. ISSN 0305-4470 (2006) [Refereed Article]

DOI: doi:10.1088/0305-4470/39/11/006


We derive group branching laws for formal characters of subgroups of leaving invariant an arbitrary tensor Tπ of Young symmetry type π where π is an integer partition. The branchings and fixing a vector vi, a symmetric tensor gij = gji and an antisymmetric tensor fij = -fji, respectively, are obtained as special cases. All new branchings are governed by Schur function series obtained from plethysms of the Schur function sπ ≡ {π} by the basic M series of complete symmetric functions and the L = M -1 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 π-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. © 2006 IOP Publishing Ltd.

Item Details

Item Type:Refereed Article
Research Division:Physical Sciences
Research Group:Other physical sciences
Research Field:Other physical sciences not elsewhere classified
Objective Division:Expanding Knowledge
Objective Group:Expanding knowledge
Objective Field:Expanding knowledge in the physical sciences
UTAS Author:Jarvis, PD (Dr Peter Jarvis)
ID Code:39301
Year Published:2006
Funding Support:Australian Research Council (DP0208808)
Web of Science® Times Cited:20
Deposited By:Physics
Deposited On:2006-08-01
Last Modified:2007-03-14

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