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:

In[136]:=

Symmetric[{x1, x2, x3}]

Out[136]=

StrongGenSet[{x1, x2}, GenSet[Cycles[{x1, x2}], Cycles[{x2, x3}]]]

In[137]:=

Antisymmetric[{x1, x2, x3, x4}]

Out[137]=

StrongGenSet[{x1, x2, x3}, GenSet[-Cycles[{x1, x2}], -Cycles[{x2, x3}], -Cycles[{x3, x4}]]]

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

In[138]:=

RiemannSymmetric[{3, 5, 2, 9}]

Out[138]=

StrongGenSet[{2, 3, 5}, GenSet[Cycles[{3, 2}, {5, 9}], -Cycles[{3, 5}], -Cycles[{2, 9}]]]

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.

In[139]:=

PairSymmetric[{{1, 2}, {3, 4}, {5, 6}, {7, 8}}, -1, 0]

Out[139]=

StrongGenSet[{1, 3, 5, 7}, GenSet[-Cycles[{7, 1, 3, 5}, {8, 2, 4, 6}], Cycles[{7, 3, 5}, {8, 4, 6}], -Cycles[{5, 7}, {6, 8}]]]

In[140]:=

PairSymmetric[{{1, 2}, {3, 4}, {5, 6}, {7, 8}}, 0, 1]

Out[140]=

StrongGenSet[{1, 3, 5, 7}, GenSet[Cycles[{1, 2}], Cycles[{3, 4}], Cycles[{5, 6}], Cycles[{7, 8}]]]

In[141]:=

PairSymmetric[{{1, 2}, {3, 4}, {5, 6}, {7, 8}}, -1, 1]

Out[141]=

StrongGenSet[{1, 3, 5, 7}, GenSet[-Cycles[{7, 1, 3, 5}, {8, 2, 4, 6}], Cycles[{7, 3, 5}, {8, 4, 6}], -Cycles[{5, 7}, {6, 8}], Cycles[{1, 2}], Cycles[{3, 4}], Cycles[{5, 6}], Cycles[{7, 8}]]]

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

In[142]:=

PairSymmetric[{{1, 2}, {3, 4}}, 1, -1]

Out[142]=

StrongGenSet[{1, 3}, GenSet[Cycles[{1, 3}, {2, 4}], -Cycles[{1, 2}], -Cycles[{3, 4}]]]

In[143]:=

RiemannSymmetric[{1, 2, 3, 4}]

Out[143]=

StrongGenSet[{1, 2, 3}, GenSet[Cycles[{1, 3}, {2, 4}], -Cycles[{1, 2}], -Cycles[{3, 4}]]]

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.


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