In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged.
[1][2] The index subset must generally either be all covariant or all contravariant.
holds when the tensor is antisymmetric with respect to its first three indices.
If a tensor changes sign under exchange of each pair of its indices, then the tensor is completely (or totally) antisymmetric.
A completely antisymmetric covariant tensor field of order
may be referred to as a differential
-form, and a completely antisymmetric contravariant tensor field may be referred to as a
A tensor A that is antisymmetric on indices
has the property that the contraction with a tensor B that is symmetric on indices
is identically 0.
For a general tensor U with components
and a pair of indices
U has symmetric and antisymmetric parts defined as: Similar definitions can be given for other pairs of indices.
As the term "part" suggests, a tensor is the sum of its symmetric part and antisymmetric part for a given pair of indices, as in
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,
}}(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{aligned}M_{[ab]}&={\frac {1}{2!
}}\,\delta _{ab}^{cd}M_{cd},\\[2pt]T_{[abc]}&={\frac {1}{3!
}}\,\delta _{abc}^{def}T_{def}.\end{aligned}}}
is the generalized Kronecker delta, and the Einstein summation convention is in use.
More generally, irrespective of the number of dimensions, antisymmetrization over
indices may be expressed as
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.
Totally antisymmetric tensors include: