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Generalization of Exterior Product's relation to Tensor Product

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I came across the following relation. Given two vectors v1 and v2, their exterior product is related to their tensor product by the relation


[latex]v_1 \wedge v_2 = v_1 \otimes v_2 - v_2 \otimes v_1[/latex]


which expands for three vectors [latex]v_1,v_2,v_3[/latex] as


[latex]v_1 \wedge v_2 \wedge v_3 \wedge=v_1\otimes v_2\otimes v_3-v_2\otimes v_1\otimes v_3 +v_3\otimes v_1\otimes v_2 - v_3\otimes v_2\otimes v_1+v_2\otimes v_3\otimes v_1 - v_1\otimes v_3\otimes v_2[/latex]


I get the basic idea of the exterior and tensor products but I don't know the notation for the right hand side permutation sum/product/whatever. The left side of the equation will be




for [latex]v_1 \wedge v_2 \wedge ... \wedge v_n[/latex]



Edited by AllCombinations

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That is not great notation and I am not sure what would be the best to use. I don't recall anything very nice in the literature either.


EDIT: Sometimes I see notation like


[math]v_{[i_{1}} v_{i_{2}} \cdots v_{i_{n}]} [/math]


for antisymmeterising over the indices. Some authors include a factor of 1/n! and others do not.

Edited by ajb

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