The Symmetric Group

Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 78)


In this chapter, fundamentals of the symmetric group, namely, classes of permutations, Young diagrams, irreducible characters, and the construction of irreducible representations and their bases, are reviewed and summarized.

The relation between the irreducible representations of the symmetric group and tensor representations of the unitary group is essential in understanding the wavefunctions of a many-electron atom with a definite magnitude of spin S. We have also seen that the concepts of symmetric and antisymmetric product representations play important roles in other fields covered in this book.


Irreducible Representation Symmetric Group Unitary Group Young Diagram Irreducible Character 
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  1. 15.1
    D. E. Rutherford:Substitutional Analysis (Edinburgh University Press, Edinburgh 1948) pp. 23–44zbMATHGoogle Scholar
  2. 15.2
    T. Yamanouchi: Proc. Phys.-Math. Soc. Jpn, 19, 436 (1937)zbMATHGoogle Scholar
  3. 15.3
    H. Weyl: The Classical Groups (Princeton University Press, Princeton, 1939) pp. 115–136Google Scholar
  4. 15.4
    G. Racah: Phys. Rev. 76, 1352 (1949)ADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  1. 1.Japan Women’s UniversityBunkyo-ku, Tokyo 112Japan
  2. 2.Department of Physics, School of Science and TechnologyMeiji UniversityTama-ku, Kawasaki 214Japan

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