Antisymmetric tensor

Notation

A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. For example, in arbitrary dimensions, for an order 2 covariant tensor M, $${\displaystyle M_{[ab]}=\frac{1}{2!}(M_{ab}-M_{ba}),}$$ and for an order 3 covariant tensor T, $${\displaystyle T_{[abc]}=\frac{1}{3!}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}).}$$

In any 2 and 3 dimensions, these can be written as $${\displaystyle \begin{array}{cc}{M}_{[ab]}& =\frac{1}{2!}{\delta }_{ab}^{cd}{M}_{cd},\\ T_{[abc]}& =\frac{1}{3!}{\delta }_{abc}^{def}{T}_{def}.\end{array}}$$ where ${\displaystyle {\delta }_{ab\dots }^{cd\dots }}$ is the generalized Kronecker delta, and the Einstein summation convention is in use.

More generally, irrespective of the number of dimensions, antisymmetrization over ${\displaystyle p}$ indices may be expressed as $${\displaystyle T_{[a_1\dots a_p]}=\frac{1}{p!}{\delta }_{a_1\dots a_p}^{b_1\dots b_p}{T}_{b_1\dots b_p}.}$$

In general, every tensor of rank 2 can be decomposed into a symmetric and anti-symmetric pair as: $${\displaystyle T_{ij}=\frac{1}{2}(T_{ij}+T_{ji})+\frac{1}{2}(T_{ij}-T_{ji}).}$$

This decomposition is not in general true for tensors of rank 3 or more, which have more complex symmetries.

Examples

Totally antisymmetric tensors include:

• Trivially, all scalars and vectors (tensors of order 0 and 1) are totally antisymmetric (as well as being totally symmetric).
• The electromagnetic tensor, ${\displaystyle F_{\mu \nu }}$ in electromagnetism.
• The Riemannian volume form on a pseudo-Riemannian manifold.