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''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)|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====
<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>