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Condition that All Irreducible Representations of a Compact Lie Group, if Restricted to a Subgroup, Contain no Representation More than Once

Published online by Cambridge University Press:  20 November 2018

Fredric E. Goldrich  and
Eugene P. Wigner
Fredric E. Goldrich
Affiliation:
Princeton University, Princeton, New Jersey
Eugene P. Wigner
Affiliation:
University of Massachusetts, Amherst, Massachusetts
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One of the results of the theory of the irreducible representations of the unitary group in n dimensions Un is that these representations, if restricted to the subgroup Un-1 leaving a vector (let us say the unit vector e1 along the first coordinate axis) invariant, do not contain any irreducible representation of this Un-1 more than once (see [1, Chapter X and Equation (10.21)]; the irreducible representations of the unitary group were first determined by I. Schur in his doctoral dissertation (Berlin, 1901)). Some time ago, a criterion for this situation was derived for finite groups [3] and the purpose of the present article is to prove the aforementioned result for compact Lie groups, and to apply it to the theory of the representations of Un.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 3 , June 1972 , pp. 432 - 438
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Hamermesh, M., Group theory and its application to physical problems (Addison Wesley, London, 1962).Google Scholar
2. Taussky, O. and Zassenhaus, H., On the similarity transformation between a matrix and its transpose, Pacific J. Math. 9 (1959), 893–896.Google Scholar
3. Wigner, E. P., Condition that the irreducible representations of a finite group, considered as representations of a subgroup, do not contain any representation more than once, Spectroscopic and Group Theoretical Methods in Physics, Loebl, F., Editor (North Holland, Amsterdam, 1968).Google Scholar