6. Some important groups of symmetry

There are some groups which are used very often. xPerm` defines special ways to generate their associated SGSs. By default the permutations are given in Cycles notation:

Strong Generating Set for the symmetric or antisymmetric groups of several points:

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Strong Generating Set for the Riemann symmetry group of four points. Note the position of the two antisymmetric pairs:

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The function PairSymmetric is far more general. It has two switches which control symmetry and antisymmetry under exchange of pairs and of members of a given pair.

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The symmetry of a Riemann tensor would be given as follows, and differs from RiemannSymmetric in the base:

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Symmetric                Give a SGS for a symmetric group
Antisymmetric            Give a SGS for an alternating group
PairSymmetric            Give a SGS for an group of permutations of pairs and/or their elements
RiemannSymmetric        Give a SGS for the group of symmetries of the Riemann tensor

Some important SGSs.

 Created by Mathematica  (May 16, 2008)