Geodesic
Condition that All Irreducible Representations of a Compact Lie Group, if Restricted to a Subgroup, Contain no Representation More than Once
Canadian journal of mathematics, Tome 24 (1972) no. 3, pp. 432-438

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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. 
Goldrich, Fredric E.; Wigner, Eugene P. Condition that All Irreducible Representations of a Compact Lie Group, if Restricted to a Subgroup, Contain no Representation More than Once. Canadian journal of mathematics, Tome 24 (1972) no. 3, pp. 432-438. doi: 10.4153/CJM-1972-036-3
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     title = {Condition that {All} {Irreducible} {Representations} of a {Compact} {Lie} {Group,} if {Restricted} to a {Subgroup,} {Contain} no {Representation} {More} than {Once}},
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[1] 1. Hamermesh, M., Group theory and its application to physical problems (Addison Wesley, London, 1962). Google Scholar

[2] 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] 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

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