9.6. Symmetrization of an expression

ImposeSymmetry            Symmetrize an expression as given by a permutations group
Symmetrize                Symmetrize a set of indices
Antisymmetrize            Antisymmetrize a set of indices
PairSymmetrize            Symmetrize a set of pairs of indices
PairAntisymmetrize        Antisymmetrize a set of pairs of indices

Symmetrization

Symmetrization of an expression under a given group of permutations. Numbers on the third argument refer to positions in the second argument of ImposeSymmetry. Note that ToCanonical is always applied after the symmetrization functions in order to take into account the antisymmetry of the tensor U. It is not automatically implemented into ImposeSymmetry because for large numbers of indices it would take ages to canonicalize all terms!

In[952]:=

ImposeSymmetry[U[-a, -b, -c], {-a, -b}, Symmetric[{1, 2}]]//ToCanonical

Out[952]=

0

In[953]:=

ImposeSymmetry[U[-a, -b, -c], {-a, -b}, Antisymmetric[{1, 2}]]//ToCanonical

Out[953]=

U_abc^   

In[954]:=

ImposeSymmetry[U[-a, -b, -c] U[-d, -e, -f], {-a, -d}, Antisymmetric[{1, 2}]]//ToCanonical

Out[954]=

-1/2 U_aef^    U_bcd^    + 1/2 U_abc^    U_def^   

In[955]:=

Symmetrize[U[-a, -b, -c] U[-d, -e, -f], {-a, -d}]//ToCanonical

Out[955]=

1/2 U_aef^    U_bcd^    + 1/2 U_abc^    U_def^   

In[956]:=

Symmetrize[U[-a, -b, -c] U[-d, -e, -f], {-e, -d}]//ToCanonical

Out[956]=

0

In[957]:=

Antisymmetrize[U[-a, -b, -c] U[-d, -e, -f], {-a, -d}]//ToCanonical

Out[957]=

-1/2 U_aef^    U_bcd^    + 1/2 U_abc^    U_def^   

In[958]:=

Antisymmetrize[U[-a, -b, -c] U[-d, -e, -f], {-e, -d}]//ToCanonical

Out[958]=

U_abc^    U_def^   

In[959]:=

ps = PairSymmetrize[U[-a, -b, -c] U[-d, -e, -f], {{-a, -b}, {-c, -d}, {-e, -f}}]//ToCanonical

Out[959]=

In[960]:=

pa = PairAntisymmetrize[U[-a, -b, -c] U[-d, -e, -f], {{-a, -b}, {-c, -d}, {-e, -f}}]//ToCanonical

Out[960]=

In[961]:=

ps - ReplaceIndex[Evaluate[ps], {-a→ -c, -b→ -d, -c→ -a, -d→ -b}]//Simplification

Out[961]=

0

In[962]:=

pa + ReplaceIndex[Evaluate[pa], {-a→ -c, -b→ -d, -c→ -a, -d→ -b}]//Simplification

Out[962]=

0

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