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)