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.
For example,
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 -vector field.
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,
and for an order 3 covariant tensor T,
In any 2 and 3 dimensions, these can be written as
where is the generalized Kronecker delta, and we use the Einstein notation to summation over like indices.
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:
This decomposition is not in general true for tensors of rank 3 or more, which have more complex symmetries.
Totally antisymmetric tensors include:
- Antisymmetric matrix – Form of a matrixPages displaying short descriptions of redirect targets
- Exterior algebra – Algebra of exterior/ wedge products
- Levi-Civita symbol – Antisymmetric permutation object acting on tensors
- Ricci calculus – Tensor index notation for tensor-based calculations
- Symmetric tensor – Tensor invariant under permutations of vectors it acts on
- Symmetrization – process that converts any function in n variables to a symmetric function in n variablesPages displaying wikidata descriptions as a fallback