This has to do with the wedge product ^ on alternating tensors. I cant seem to prove w ^ n = (-1)^kl * n ^ w. where w is a k alternating tensor and n is an l alternating tensor.(adsbygoogle = window.adsbygoogle || []).push({});

I know w ^ n = (k+l)!/(k! l!) * Alt(w x n) where x is the tensor product.

Now, Alt(wxn) (v_1,...,v_k+l) = 1/(k+l)! * (sum over all s in Sk) ( sgn(s) * (w x n) (v_s(1),...,v_s(k+l) )

= 1/(k+l)! * (sum over all s in Sk) ( sgn(s) * (n x w) (v_s(k+1),...,v_s(k+l),v_s(1),...,v_s(k)) )

Note Sk is the set of all permutations of the numbers 1,...,k.

Now if I let s' be the permutation such that

Alt(w x n) = 1/(k+l)! * (sum over all s in Sk) ( sgn(s) * (n x w) (v_s'(1),...,v_s'(k+l)) )

then shouldnt Alt(w x n) = 1/(k+l)! * (sum over all s' in Sk) ( (-1)^m * sgn(s') * (n x w) (v_s'(1),...,v_s'(k+l)) )

where m is the number of transpositions needed to transform (k+1,...,k+l,1,...,k) to (1,....,k+l). Because if this is true then doesnt the property imply that m = l*k. And this isnt true. So what am i doing wrong here?

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# Wedge product property

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