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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

  • F. E. Goldrich
  • E. P. Wigner
Part of the The Collected Works of Eugene Paul Wigner book series (WIGNER, volume A / 1)

Abstract

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 [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 U n .

Keywords

Characteristic Vector Irreducible Representation Finite Group Unitary Group Group Volume 
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References

  1. 1.
    M. Hamermesh, Group theory and its application to physical problems ( Addison Wesley, London, 1962 ).MATHGoogle Scholar
  2. 2.
    O. Taussky and H. Zassenhaus, On the similarity transformation between a matrix and its transpose, Pacific J. Math. 9 (1959), 893–896.MathSciNetMATHGoogle Scholar
  3. 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

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • F. E. Goldrich
    • 1
  • E. P. Wigner
    • 2
  1. 1.Princeton UniversityPrincetonUSA
  2. 2.University of MassachusettsAmherstUSA

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