Abstract
A formalism of decomposing tensor product of irreducible representations of compact semisimple Lie groups is presented, using holomorphic induction techniques. The case of SU(n) is examined in detail. Irreducible representation spaces are realized as polynomial functions over GL(n,ℂ) group variables and it is shown how to generate invariant spaces labelled by double cosets, with one double coset subspace isomorphic to the original tensor product space. Global and noninductive procedure for constructing orthogonal polynomial bases is presented and is compared with the Gelfand-Žetlin procedure. Clebsch-Gordan (Wigner) coefficients are computed and are used to provide a resolution of the multiplicity problem occurring in the tensor product decompositions of U(n) groups.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. C. Biedenharn, A. Giovannini, and J. D. Louck, J. Math. Phys. t.8:691 (1967), and references cited therein.
D. P. Zelobenko, Compact Lie groups and their representations, “Nauka”, Moscow, 1970; English transl., Transl. Math. Monographs, vol. 40, A.M.S., Providence, R. I., 1973, and references cited therein.
Harish-Chandra, Differential operators on a semisimple Lie algebra, Amer. J. Math. 79: 87–120 (1957).
B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85: 327–404 (1963).
S. Helgason, Invariants and fundamental functions, Acta Math. 109: 241–258 (1963).
Tuong Ton-That, Lie group representations and harmonic polynomials of a matrix variable, Trans. Amer. Math. Soc. 216: 1–46 (1976).
G. Warner, Harmonic Analysis on Semisimple Lie Groups, Vol. I, Springer-Verlag, Berlin (1972), and references cited therein.
J. Humphreys, Introduction to Lie Algebras and Representation Theory, 2nd ed., Springer-Verlag, Berlin (1972), and references cited therein.
W. H. Klink and T. Ton-That, Holomorphic induction and tensor product decomposition of irreducible representations of compact groups I. SU(n) groups, Ann. Inst. Henri Poincaré 31: 77–97 (1979).
D. King, The geometric structure of the tensor product of irreducible representations of a complex semisimple Lie algebra (preprint).
W. H. Klink and T. Ton-That, On the resolution of the multiplicity problem for U(n) (submitted for publication).
I. M. Gelfand and M. I. Graev, Finite dimensional irreducible representations of the unitary group and the full linear groups and related special functions, Izv. Akad. Nauk. SSSR Ser. Mat. t.29:1329–1356 (1965); English transl., Amer. Math. Soc. Transl., t. 64: 116–146 (1967).
W. H. Klink and T. Ton-That, Construction explicite non itérative des bases de GL(n,ℂ)-modules, C.R. Acad. Sci. Paris, Ser. B 289: 115–118 (1979).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1981 Plenum Press, New York
About this chapter
Cite this chapter
Ton-That, T., Klink, W.H. (1981). Tensor Product Decomposition of Holomorphically Induced Representations and Clebsch-Gordan Coefficients. In: Gustafson, K.E., Reinhardt, W.P. (eds) Quantum Mechanics in Mathematics, Chemistry, and Physics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3258-9_32
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3258-9_32
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3260-2
Online ISBN: 978-1-4613-3258-9
eBook Packages: Springer Book Archive