Generalization of Exterior Product's relation to Tensor Product

[AllCombinations]

Hi.

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

 

[latex]\bigwedge_{i=1}^{n}v_i[/latex]

 

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

 

Thanks!

Edited November 19, 2015 by AllCombinations

[ajb]

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 November 20, 2015 by ajb