View source for Alternation

''skew symmetry, anti-symmetry, alternance''

One of the operations of tensor algebra, yielding a tensor that is skew-symmetric (over a group of indices) from a given tensor. Alternation is always effected over a few superscripts or over a few subscripts. A tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120601.png" /> with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120602.png" /> is the result of alternation of a tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120603.png" /> with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120604.png" />, for example, over superscripts, over a group of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120605.png" /> if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120606.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

The summation is conducted over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120607.png" /> rearrangements (permutations) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120608.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120609.png" />, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206010.png" /> being <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206011.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206012.png" />, depending on whether the respective rearrangement is even or odd. Alternation over a group of subscripts is defined in a similar manner.

Alternation over a group of indices is denoted by enclosing the indices between square brackets. Secondary indices inside the square brackets are separated by vertical strokes. For instance:

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Successive alternation over groups of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206016.png" />, coincides with alternation over the group of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206017.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206018.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206019.png" /> is the dimension of the vector space on which the tensor is defined, alternation by a group of indices the number of which is larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206020.png" /> will always produce the zero tensor. Alternation over a given group of indices of a tensor which is symmetric with respect to this group (cf. [[Symmetrization (of tensors)|Symmetrization (of tensors)]]) also yields the zero tensor. A tensor that remains unchanged under alternation over a given group of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206021.png" /> is called skew-symmetric or alternating over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206022.png" />. Interchanging any pair of such indices changes the sign of the component of the tensor.

The operation of tensor alternation, together with the operation of symmetrization, is employed to decompose a tensor into simpler tensors.

The product of two tensors with subsequent alternation over all indices is called an alternated product (exterior product).

Alternation is also employed to produce sign-alternating sums of the form (*) with multi-indexed terms. For instance, a determinant with elements which commute under multiplication can be computed by the formulas

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206024.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.A. Shirokov,   "Tensor calculus. Tensor algebra" , Kazan'  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.V. Beklemishev,   "A course of analytical geometry and linear algebra" , Moscow  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Schouten,   "Tensor analysis for physicists" , Cambridge Univ. Press  (1951)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.V. Efimov,   E.R. Rozendorn,   "Linear algebra and multi-dimensional geometry" , Moscow  (1970)  (In Russian)</TD></TR></table>