Can Two Levi-Civita Symbols be Reduced to One with Indices?

[Safinaz]

Hi there,

Can two Levi-Civia symbols $\epsilon^{ckl} \epsilon_{ibj}$ reduced to one with indices $\epsilon_{kij}$ ?

WHERE b and c run from 4 to 5, the other indices run from 1 to 3 and both symbols are multiplied by the following matrices:

$A_{ib}^c ~ B^{kl} ~ C_j ~\epsilon^{ckl} \epsilon_{ibj} ~ [1]$,

$B^{kl}$ is antisymmetric in k and l, and A is antisymmetric in b and c ..

So I hope my question is clear enough .. in summary, can expression [1] written in the forum : $A_i ~ B_k ~ C_j ~ \epsilon_{ikj}$ ?

I think doing this needs with using Levi-Civita symbol properties , to use some direct products of matrices, as: $3 \times 3 = 3^*_{Antisymm}+6_{Symm}$, and $2 \times 2 = 1_A +3_S$ .

Any ideas ?
Thanx.

 

[ChrisVer]

Are you sure that this is even possible? you are dropping away free indices,..

 

[Safinaz]

mmmm .. which one ? in [1] all the indices (i,b,j,k,l,c) are contacted .

Can we use here Levi-Civita symbol contraction with second or third rank tensor ? if there any relation like in the metric tensor $A^\alpha = g^{\alpha\beta} A_\beta$ ..