Editing Exterior algebra (section)

=== Alternating multilinear forms ===
<!-- Alternating form redirects here -->
{{See also|Alternating multilinear map}}

[[File:N-form.svg|thumb|upright=0.6|Geometric interpretation for the '''exterior product''' of {{math|''n''}} [[1-form]]s ({{math|''ε''}}, {{math|''η''}}, {{math|''ω''}}) to obtain an {{math|''n''}}-form ("mesh" of [[coordinate surface]]s, here planes),<ref name=Penrose07/> for {{math|1=''n'' = 1, 2, 3}}.  The "circulations" show [[Orientation (vector space)|orientation]].<ref>''Note'': The orientations shown here are not correct; the diagram simply gives a sense that an orientation is defined for every {{math|''k''}}-form.</ref><ref>{{cite book|title=Gravitation |first1=J.A. |last1=Wheeler |first2=C. |last2=Misner |first3=K.S. |last3=Thorne |publisher=W.H. Freeman & Co|year=1973|pages=58–60, 83, 100–9, 115–9|isbn=0-7167-0344-0}}</ref>]]

The above discussion specializes to the case when {{tmath|1=X = K}}, the base field.  In this case an alternating multilinear function
: <math> f : V^k \to K</math>
is called an '''alternating multilinear form'''.  The set of all [[Alternating map|alternating]] [[multilinear form]]s is a vector space, as the sum of two such maps, or the product of such a map with a scalar, is again alternating.  By the universal property of the exterior power, the space of alternating forms of degree <math>k</math> on <math>V</math> is [[natural transformation|naturally]] isomorphic with the [[dual vector space]] {{tmath|\bigl({\textstyle\bigwedge}^{\!k}(V)\bigr)^*}}.  If <math>V</math> is finite-dimensional, then the latter is {{Clarify|date=December 2021|reason=By what isomorphism? See, for example, https://mathoverflow.net/q/68004|text=naturally isomorphic}} to {{tmath|{\textstyle\bigwedge}^{\!k}\left(V^*\right)}}.  In particular, if <math>V</math> is <math>n</math>-dimensional, the dimension of the space of alternating maps from <math>V^k</math> to <math>K</math> is the [[binomial coefficient]] {{tmath|1=\textstyle\binom{n}{k} }}.

Under such identification, the exterior product takes a concrete form: it produces a new anti-symmetric map from two given ones.  Suppose {{nowrap|''ω'' : ''V''<sup>''k''</sup> → ''K''}} and {{nowrap|''η'' : ''V''<sup>''m''</sup> → ''K''}} are two anti-symmetric maps.  As in the case of [[tensor product]]s of multilinear maps, the number of variables of their exterior product is the sum of the numbers of their variables.  Depending on the choice of identification of elements of exterior power with multilinear forms, the exterior product is defined as
: <math> \omega \wedge \eta = \operatorname{Alt}(\omega \otimes \eta) </math>
or as
: <math>
\omega \dot{\wedge} \eta
= \frac{(k+m)!}{k!\,m!}\operatorname{Alt}(\omega \otimes \eta),
</math>
where, if the characteristic of the base field <math>K</math> is 0, the alternation Alt of a multilinear map is defined to be the average of the sign-adjusted values over all the [[permutation]]s of its variables:
: <math>
\operatorname{Alt}(\omega)(x_1,\ldots,x_k)
= \frac{1}{k!}\sum_{\sigma \in S_k}\operatorname{sgn}(\sigma)\, \omega(x_{\sigma(1)}, \ldots, x_{\sigma(k)}).
</math>

When the [[field (mathematics)|field]] <math>K</math> has [[characteristic of a field|finite characteristic]], an equivalent version of the second expression without any factorials or any constants is well-defined:
: <math>
{\omega \dot{\wedge} \eta(x_1,\ldots,x_{k+m})}
= \sum_{\sigma \in \mathrm{Sh}_{k,m}} \operatorname{sgn}(\sigma)\, \omega(x_{\sigma(1)}, \ldots, x_{\sigma(k)})\, \eta(x_{\sigma(k+1)}, \ldots, x_{\sigma(k+m)}),
</math>
where here {{nowrap|Sh<sub>''k'',''m''</sub> ⊂ ''S''<sub>''k''+''m''</sub>}} is the subset of [[(p,q) shuffle|{{math|(''k'', ''m'')}} shuffles]]: [[permutation]]s ''σ'' of the set {{nowrap|{{mset|1, 2, ..., ''k'' + ''m''}}}} such that {{nowrap|''σ''(1) < ''σ''(2) < ⋯ < ''σ''(''k'')}}, and {{nowrap|''σ''(''k'' + 1) < ''σ''(''k'' + 2) < ... < ''σ''(''k'' + ''m'')}}. As this might look very specific and fine tuned, an equivalent raw version is to sum in the above formula over permutations in left cosets of {{nowrap|''S''<sub>''k''+''m''</sub> / (''S''<sub>''k''</sub> × ''S''<sub>''m''</sub>)}}.