Skew-symmetric tensor

A tensor over an $ n $- dimensional vector space $ E $ that is invariant under the operation of alternation with respect to some group of its indices. The components of a skew-symmetric tensor are skew-symmetric with respect to the corresponding group of indices, i.e. if two indices are exchanged the components change sign (in the sense of the additive law of the field $ K $ over which $ E $ is defined), and if two indices are equal the components vanish.

The most important skew-symmetric tensors are those that remain invariant under alternation with respect to the entire group of covariant or contravariant indices. A contravariant (covariant) skew-symmetric tensor of valency $ r $ is an $ r $- vector or multi-vector over $ E $( respectively, over $ E ^ {*} $, the space dual to $ E $); they are elements of the exterior algebra of the vector space $ E $. The exterior algebra over $ E ^ {*} $ is usually called the algebra of exterior forms, identifying covariant skew-symmetric tensors of valency $ r $ with $ r $- forms.

For references see Exterior algebra.