- 219

- 2

## Main Question or Discussion Point

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

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.

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.

Last edited: