View source for Skew-symmetric tensor

A tensor over an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085730/s0857301.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085730/s0857302.png" /> that is invariant under the operation of [[Alternation|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085730/s0857303.png" /> over which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085730/s0857304.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085730/s0857305.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085730/s0857307.png" />-vector or multi-vector over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085730/s0857308.png" /> (respectively, over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085730/s0857309.png" />, the space dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085730/s08573010.png" />); they are elements of the exterior algebra of the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085730/s08573011.png" />. The exterior algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085730/s08573012.png" /> is usually called the algebra of exterior forms, identifying covariant skew-symmetric tensors of valency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085730/s08573013.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085730/s08573014.png" />-forms.

For references see [[Exterior algebra|Exterior algebra]].