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Levi-Civita symbol reduction

  1. May 13, 2015 #1
    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.
     
    Last edited: May 13, 2015
  2. jcsd
  3. May 13, 2015 #2

    ChrisVer

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    Gold Member

    Are you sure that this is even possible? you are dropping away free indices,..
     
  4. May 13, 2015 #3
    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 ## ..
     
    Last edited: May 13, 2015
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