Chapter

The symmetric group S<sub>r</sub> and Young tableaux

Adam M. Bincer

in Lie Groups and Lie Algebras

Published in print October 2012 | ISBN: 9780199662920
Published online January 2013 | e-ISBN: 9780191745492 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199662920.003.0014
The symmetric group Sr and Young tableaux

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The symmetric group Sr and Young tableaux are defined. It is shown that the actions of the symmetric group and unitary group commute, therefore SU(n) tensors can be classified by the symmetric group. The cycle notation is described. Symmetrizer and antisymmetrizer are defined and expressed in terms of Young tableaux. Rules are given for forming Young tableaux for a given Young pattern. The general Young pattern provides an irreducible representation of the symmetric group and can be specified by the partition (f 1, f 2,…, fn). The number of different tableaux for a given pattern gives the dimension of the corresponding representation of the symmetric group Sr. Biographical notes on Young are given.

Keywords: symmetric group; cycle; symmetrizer; antisymmetrizer; Young tableaux; Young pattern; partition

Chapter.  2906 words. 

Subjects: Mathematical and Statistical Physics

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