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Beyond the Enveloping Algebra of sl 3

Published online by Cambridge University Press:  20 November 2018

Daniel E. Flath  and
L. C. Biedenharn
Daniel E. Flath
Affiliation:
Duke University, Durham, North Carolina
L. C. Biedenharn
Affiliation:
Duke University, Durham, North Carolina
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The problem which motivated the writing of this paper is that of finding structure behind the decomposition of the sl 3 representation spaces V* ⊗ W = Hom(V, W) for finite dimensional irreducible sl 3-modules V and W. For sl 2 this extends the classical Clebsch-Gordon problem. The question has been considered for sl 3 in a computational way in [5]. In this paper we build a conceptual algebraic framework going beyond the enveloping algebra of sl 3.

For each dominant integral weight α let V α be an irreducible representation of sl 3 of highest weight α. It is well known that, for weights α, μ, λ, the multiplicity of Vλ in Hom(Vα, Vα+μ ) is bounded by the multiplicity of μ in Vλ ,with equality for generic α. This suggests the possibility of a single construction of highest weight vectors of weight X in Hom(Vα, Vα+μ ) which is valid for all a.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 4 , 01 August 1985 , pp. 710 - 729
Copyright
Copyright © Canadian Mathematical Society 1985

References

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