Cadabra
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Symmetric

Make an object symmetric in all indices.
Make an object symmetric in all indices. This information is then subsequently used by various algorithms, for instance canonicalise. An example:
A_{m n}::AntiSymmetric. B_{m n}::Symmetric. ex:=A_{m n} B_{m n};
\(\displaystyle{}A_{m n} B_{m n}\)
canonicalise(_);
\(\displaystyle{}0\)
If you need symmetry in only a subset of all indices of a tensor, you need to use the TableauSymmetry property. A quick example:
C_{a n p}::TableauSymmetry(shape={2}, indices={1,2});
\(\displaystyle{}\text{Attached property TableauSymmetry to }C_{a n p}.\)
This gives indices 1 and 2 (counting starts from 0) the symmetry of the Young Tableau formed by one row of 2 boxes, which is the fully symmetric representation of the permutation group. Now you get, as expected,
ex:=C_{a n p} - C_{a p n};
\(\displaystyle{}C_{a n p}-C_{a p n}\)
canonicalise(_);
\(\displaystyle{}0\)
For more information see the TableauSymmetry documentation.
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