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Lie Groups and Lie AlgebrasA Physicist's Perspective$
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Adam M. Bincer

Print publication date: 2012

Print ISBN-13: 9780199662920

Published to Oxford Scholarship Online: January 2013

DOI: 10.1093/acprof:oso/9780199662920.001.0001

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The symmetric group Sr and Young tableaux

The symmetric group Sr and Young tableaux

(p.107) 14 The symmetric group Sr and Young tableaux
Lie Groups and Lie Algebras

Adam M. Bincer

Oxford University Press

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

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