Cadabra manual: AntiSymmetric

AntiSymmetric

Make an object anti-symmetric in all indices.

Make an object anti-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 anti-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={1,1}, 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 two rows, each containing 1 box, which is the fully anti-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.