I want to define rules to simplify symbolic totally antisymmetric tensors. For example the rule (in the Minkowski signature)
$$
\varepsilon^{\mu\nu\rho\lambda} \varepsilon_{\kappa\nu\rho\lambda} = - 6\,\delta^\mu_\kappa\,,
$$
and other similar things with 4, 2 or 1 contractions. I could simply define such a replacement rule but then there are a lot of cases involving all permutations of the contracted indices, which will differ only by signs. I would like to avoid having to type down all cases. Even if I define $\varepsilon$ as a totally antisymmetric tensor (such as g
in this thread Totally Antisymmetric Function) I'd still have all possibilities involving the position of the uncontracted index among the contracted ones.
My set up is the following: I have a symbolic totally symmetric tensor ε4[m,n,r,l]
and a symmetric tensor g[m,n]
(the metric) which will appear with symbolic arguments (their indices do not need to assume specific numbers). Then whatever expression I have I want to simplify it by repeatedly expanding and applying rules.