# Generalization of Exterior Product's relation to Tensor Product

## Recommended Posts

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

$v_1 \wedge v_2 = v_1 \otimes v_2 - v_2 \otimes v_1$

which expands for three vectors $v_1,v_2,v_3$ as

$v_1 \wedge v_2 \wedge v_3 \wedge=v_1\otimes v_2\otimes v_3-v_2\otimes v_1\otimes v_3 +v_3\otimes v_1\otimes v_2 - v_3\otimes v_2\otimes v_1+v_2\otimes v_3\otimes v_1 - v_1\otimes v_3\otimes v_2$

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

$\bigwedge_{i=1}^{n}v_i$

for $v_1 \wedge v_2 \wedge ... \wedge v_n$

Thanks!

Edited by AllCombinations

##### Share on other sites

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

$v_{[i_{1}} v_{i_{2}} \cdots v_{i_{n}]}$

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

Edited by ajb

## Create an account

Register a new account