Condition That All Irreducible Representations of a Compact Lie Group, If Restricted to a Subgroup, Contain No Representation More Than Once
One of the results of the theory of the irreducible representations of the unitary group in n dimensions U n is that these representations, if restricted to the subgroup U n − 1 leaving a vector (let us say the unit vector e l along the first coordinate axis) invariant, do not contain any irreducible representation of this U n − 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  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 U n .
KeywordsCharacteristic Vector Irreducible Representation Finite Group Unitary Group Group Volume
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- 3.E. P. Wigner, 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, F. Loebl, Editor ( North Holland, Amsterdam, 1968 ).Google Scholar