# 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 $A$ with components $\{ a _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} }, 1 \leq i _ \nu , j _ \mu \leq n \}$ is the result of alternation of a tensor $T$ with components $\{ t _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} }, 1 \leq i _ \nu , j _ \mu \leq n \}$, for example, over superscripts, over a group of indices $I = (i _ {1} \dots i _ {m} )$ if

$$\tag{* } a _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } = \ \frac{1}{m!} \sum _ {I \rightarrow \alpha } \sigma ( I , \alpha ) t _ {j _ {1} \dots j _ {q} } ^ {\alpha _ {1} \dots \alpha _ {m} i _ {m+1} \dots i _ {p} } .$$

The summation is conducted over all $m!$ rearrangements (permutations) $\alpha = ( \alpha _ {1} \dots \alpha _ {m} )$ of $I$, the number $\sigma (I, \alpha )$ being $+1$ or $-1$, 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:

$$t _ {[ 4 | 23 | 1 ] } = \ \frac{1}{2!} [ t _ {4 2 3 1 } - t _ {1 2 3 4 } ].$$

Successive alternation over groups of indices $I _ {1}$ and $I _ {2}$, $I _ {1} \subset I _ {2}$, coincides with alternation over the group of indices $I _ {2}$:

$$t _ {[ i _ {1} \dots [ i _ {k} \dots i _ {l} ] \dots i _ {q} ] } = t _ {[ i _ {1} \dots i _ {q} ] } .$$

If $n$ 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 $n$ 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)) also yields the zero tensor. A tensor that remains unchanged under alternation over a given group of indices $I$ is called skew-symmetric or alternating over $I$. 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

$$\left | \begin{array}{ccc} a _ {1} ^ {1} &\dots &a _ {n} ^ {1} \\ . &{} & . \\ . &{} & . \\ a _ {1} ^ {n} &\dots &a _ {n} ^ {n} \\ \end{array} \ \right | = n ! a _ {1} ^ {[1{} } \dots a _ {n} ^ { {}n] } =$$

$$= \ n ! a _ {[1{} } ^ {1} \dots a _ { {}n] } ^ {n} = \ a _ {[1{} } ^ {[1{} } \dots a _ { {}n] } ^ { {}n] } .$$

#### References

 [1] P.A. Shirokov, "Tensor calculus. Tensor algebra" , Kazan' (1961) (In Russian) [2] D.V. Beklemishev, "A course of analytical geometry and linear algebra" , Moscow (1971) (In Russian) [3] J.A. Schouten, "Tensor analysis for physicists" , Cambridge Univ. Press (1951) [4] N.V. Efimov, E.R. Rozendorn, "Linear algebra and multi-dimensional geometry" , Moscow (1970) (In Russian)
How to Cite This Entry:
Alternation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alternation&oldid=45091
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article