The and Antisymmetrize and Symmetrize commands receive an algebraic expression T involving tensors and returns another algebraic expression that is respectively totally antisymmetric or totally symmetric with respect to any permutation of the free tensorial indices of T.

In the particular case where T is itself a product of totally antisymmetric covariant tensors, the expression returned by Antisymmetrize is also, by definition, the exterior product of these tensors.

When a second argument, idx, is indicated as a set or list of indices, the (anti)symmetrization is performed only with respect to these indices, otherwise, it is performed with regards to all the free indices of T. In either case, to perform the (anti)symmetrization it is necessary that the indices are all of the same kind, that is, all covariant or all contravariant.

By default, the (anti)symmetrization is performed without evaluation the tensors entering the result. However, when there are tensors in T that already have symmetry properties (see Define to define them), it may be convenient to evaluate these tensors (option evaluatetensor) or even simplify the result with regards to the summation convention for repeated indices and taking into account the (anti)symmetry properties of the tensors involved or their products. For these purposes, respectively pass the option evaluatetensor or simplifytensor (the latter supersedes the former).

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{Define}\left(A\left[\mathrm{\mu }\right]\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{A_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

$\mathrm{Define}\left(B\left[\mathrm{\mu },\mathrm{\nu }\right]\,\mathrm{symmetric}\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{A_{\mathrm{\mu }}\,B_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

$\mathrm{Define}\left(C\left[\mathrm{\mu },\mathrm{\nu },\mathrm{\rho }\right]\,\mathrm{antisymmetric}\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{A_{\mathrm{\mu }}\,B_{\mathrm{\mu },\mathrm{\nu }}\,C_{\mathrm{\mu },\mathrm{\nu },\mathrm{\rho }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

$A\left[\mathrm{\mu }\right]B\left[\mathrm{\alpha },\mathrm{\beta }\right]$

$A_{\mathrm{\mu }}{B}_{\mathrm{\alpha },\mathrm{\beta }}$

$\mathrm{Symmetrize}\left(\right)$

$\frac{1}{6}{A}_{\mathrm{\alpha }}{B}_{\mathrm{\beta },\mathrm{\mu }}+\frac{1}{6}{A}_{\mathrm{\alpha }}{B}_{\mathrm{\mu },\mathrm{\beta }}+\frac{1}{6}{A}_{\mathrm{\beta }}{B}_{\mathrm{\alpha },\mathrm{\mu }}+\frac{1}{6}{A}_{\mathrm{\beta }}{B}_{\mathrm{\mu },\mathrm{\alpha }}+\frac{1}{6}{A}_{\mathrm{\mu }}{B}_{\mathrm{\alpha },\mathrm{\beta }}+\frac{1}{6}{A}_{\mathrm{\mu }}{B}_{\mathrm{\beta },\mathrm{\alpha }}$

$\mathrm{Library}:-\mathrm{IsTensorialSymmetric}\left(\right)$

$\mathrm{true}$

$\mathrm{Symmetrize}\left(\,\mathrm{simplifytensor}\right)$

$\frac{A_{\mathrm{\alpha }}{B}_{\mathrm{\beta },\mathrm{\mu }}}{3}+\frac{A_{\mathrm{\beta }}{B}_{\mathrm{\alpha },\mathrm{\mu }}}{3}+\frac{A_{\mathrm{\mu }}{B}_{\mathrm{\alpha },\mathrm{\beta }}}{3}$

$\mathrm{Antisymmetrize}\left(\right)$

$0$

$A\left[\mathrm{\mu }\right]C\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }\right]$

$A_{\mathrm{\mu }}{C}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }}$

$\mathrm{Symmetrize}\left(\right)$

$0$

$\mathrm{Antisymmetrize}\left(\right)$

$-\frac{1}{24}{A}_{\mathrm{\alpha }}{C}_{\mathrm{\beta },\mathrm{\gamma },\mathrm{\mu }}+\frac{1}{24}{A}_{\mathrm{\alpha }}{C}_{\mathrm{\beta },\mathrm{\mu },\mathrm{\gamma }}+\frac{1}{24}{A}_{\mathrm{\alpha }}{C}_{\mathrm{\gamma },\mathrm{\beta },\mathrm{\mu }}-\frac{1}{24}{A}_{\mathrm{\alpha }}{C}_{\mathrm{\gamma },\mathrm{\mu },\mathrm{\beta }}-\frac{1}{24}{A}_{\mathrm{\alpha }}{C}_{\mathrm{\mu },\mathrm{\beta },\mathrm{\gamma }}+\frac{1}{24}{A}_{\mathrm{\alpha }}{C}_{\mathrm{\mu },\mathrm{\gamma },\mathrm{\beta }}+\frac{1}{24}{A}_{\mathrm{\beta }}{C}_{\mathrm{\alpha },\mathrm{\gamma },\mathrm{\mu }}-\frac{1}{24}{A}_{\mathrm{\beta }}{C}_{\mathrm{\alpha },\mathrm{\mu },\mathrm{\gamma }}-\frac{1}{24}{A}_{\mathrm{\beta }}{C}_{\mathrm{\gamma },\mathrm{\alpha },\mathrm{\mu }}+\frac{1}{24}{A}_{\mathrm{\beta }}{C}_{\mathrm{\gamma },\mathrm{\mu },\mathrm{\alpha }}+\frac{1}{24}{A}_{\mathrm{\beta }}{C}_{\mathrm{\mu },\mathrm{\alpha },\mathrm{\gamma }}-\frac{1}{24}{A}_{\mathrm{\beta }}{C}_{\mathrm{\mu },\mathrm{\gamma },\mathrm{\alpha }}-\frac{1}{24}{A}_{\mathrm{\gamma }}{C}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu }}+\frac{1}{24}{A}_{\mathrm{\gamma }}{C}_{\mathrm{\alpha },\mathrm{\mu },\mathrm{\beta }}+\frac{1}{24}{A}_{\mathrm{\gamma }}{C}_{\mathrm{\beta },\mathrm{\alpha },\mathrm{\mu }}-\frac{1}{24}{A}_{\mathrm{\gamma }}{C}_{\mathrm{\beta },\mathrm{\mu },\mathrm{\alpha }}-\frac{1}{24}{A}_{\mathrm{\gamma }}{C}_{\mathrm{\mu },\mathrm{\alpha },\mathrm{\beta }}+\frac{1}{24}{A}_{\mathrm{\gamma }}{C}_{\mathrm{\mu },\mathrm{\beta },\mathrm{\alpha }}+\frac{1}{24}{A}_{\mathrm{\mu }}{C}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }}-\frac{1}{24}{A}_{\mathrm{\mu }}{C}_{\mathrm{\alpha },\mathrm{\gamma },\mathrm{\beta }}-\frac{1}{24}{A}_{\mathrm{\mu }}{C}_{\mathrm{\beta },\mathrm{\alpha },\mathrm{\gamma }}+\frac{1}{24}{A}_{\mathrm{\mu }}{C}_{\mathrm{\beta },\mathrm{\gamma },\mathrm{\alpha }}+\frac{1}{24}{A}_{\mathrm{\mu }}{C}_{\mathrm{\gamma },\mathrm{\alpha },\mathrm{\beta }}-\frac{1}{24}{A}_{\mathrm{\mu }}{C}_{\mathrm{\gamma },\mathrm{\beta },\mathrm{\alpha }}$

$\mathrm{Simplify}\left(\right)$

$-\frac{A_{\mathrm{\alpha }}{C}_{\mathrm{\beta },\mathrm{\gamma },\mathrm{\mu }}}{4}+\frac{A_{\mathrm{\beta }}{C}_{\mathrm{\alpha },\mathrm{\gamma },\mathrm{\mu }}}{4}-\frac{A_{\mathrm{\gamma }}{C}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu }}}{4}+\frac{A_{\mathrm{\mu }}{C}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }}}{4}$

$B\left[\mathrm{\mu },\mathrm{\nu }\right]C\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }\right]$

$B_{\mathrm{\mu },\mathrm{\nu }}{C}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }}$

$\mathrm{Antisymmetrize}\left(\mathrm{Symmetrize}\left(\,\left[\mathrm{\mu }\,\mathrm{\alpha }\right]\right)\,\left[\mathrm{\nu }\,\mathrm{\beta }\right]\,\mathrm{simplifytensor}\right)$

$\frac{B_{\mathrm{\alpha },\mathrm{\nu }}{C}_{\mathrm{\beta },\mathrm{\gamma },\mathrm{\mu }}}{4}+\frac{B_{\mathrm{\mu },\mathrm{\nu }}{C}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }}}{4}-\frac{B_{\mathrm{\alpha },\mathrm{\beta }}{C}_{\mathrm{\gamma },\mathrm{\mu },\mathrm{\nu }}}{4}+\frac{B_{\mathrm{\beta },\mathrm{\mu }}{C}_{\mathrm{\alpha },\mathrm{\gamma },\mathrm{\nu }}}{4}$

$\mathrm{Library}:-\mathrm{IsTensorialSymmetric}\left(\,\left[\mathrm{\mu }\,\mathrm{\alpha }\right]\right)$

$\mathrm{true}$

$\mathrm{Library}:-\mathrm{IsTensorialAntisymmetric}\left(\,\left[\mathrm{\nu }\,\mathrm{\beta }\right]\right)$

$\mathrm{true}$