Wedgie/Archived version: Difference between revisions

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Given an ''n''-map ''f'' and an ''m''-map ''g'' we may define a new ({{nowrap|''n'' + ''m''}})-map, the [[Wikipedia: Exterior algebra|wedge product]] of ''f'' and ''g'', written {{nowrap|''f'' ∧ ''g''}}, as follows:

<math>\displaystyle f\wedge g = \sum_s sgn(s)f\left(x_s(1),x_s(2),~~...~~,x_s\left(n\right)\right)g\left(x_s\left(n+1\right),~~...~~,x_s\left(n+m\right)\right)</math>

<math>\displaystyle f\wedge g = \sum_s \operatorname{sgn}\left(s\right)f\left(x_s(1), x_s(2), \ldots, x_s\left(n\right)\right)g\left(x_s\left(n +1\right), \ldots, x_s\left(n + m\right)\right)</math>

where the sum is taken over {{nowrap|S(''n'', ''m'')}}, the set of all [[Wikipedia:Permutation|permutations]] of the first {{nowrap|''n'' + ''m''}} integers which are an [[Wikipedia:(p,q)_shuffle|{{nowrap|(''n'', ''m'')}} shuffle]], and sgn(''t'') is the [[Wikipedia: Parity of a permutation|parity of the permutation]] ''t'', which is +1 if ''t'' is even meaning an even number of transpositions of two numbers will get to ''t'', and −1 if ''t'' is odd.

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 Given an ''n''-map ''f'' and an ''m''-map ''g'' we may define a new ({{nowrap|''n'' + ''m''}})-map, the [[Wikipedia: Exterior algebra|wedge product]] of ''f'' and ''g'', written {{nowrap|''f'' ∧ ''g''}}, as follows:
-<math>\displaystyle f\wedge g = \sum_s sgn(s)f\left(x_s(1),x_s(2),...,x_s\left(n\right)\right)g\left(x_s\left(n+1\right),...,x_s\left(n+m\right)\right)</math>
+<math>\displaystyle f\wedge g = \sum_s \operatorname{sgn}\left(s\right)f\left(x_s(1), x_s(2), \ldots, x_s\left(n\right)\right)g\left(x_s\left(n  +1\right), \ldots, x_s\left(n + m\right)\right)</math>
 where the sum is taken over {{nowrap|S(''n'', ''m'')}}, the set of all [[Wikipedia:Permutation|permutations]] of the first {{nowrap|''n'' + ''m''}} integers which are an [[Wikipedia:(p,q)_shuffle|{{nowrap|(''n'', ''m'')}} shuffle]], and sgn(''t'') is the [[Wikipedia: Parity of a permutation|parity of the permutation]] ''t'', which is +1 if ''t'' is even meaning an even number of transpositions of two numbers will get to ''t'', and &minus;1 if ''t'' is odd.