Simplification of multiple Levi-Civita Epsilons

Question

Given the expression: $$\epsilon_{ijk}\epsilon_{klm}\epsilon_{mni}=c\epsilon_{jln}$$ where $c$ is a constant I have to determine. Given the fact: $$\epsilon_{jln}\epsilon_{jln}=6$$ I have said $$c=\frac{\epsilon_{ijk}\epsilon_{klm}\epsilon_{mni}\epsilon_{jln}}6$$ I am wondering whether it is possible to condense these epsilons further and, if so, how would one do so.(I'm not sure how the sifting property of $\delta$ works for more than $\delta\delta$. Any pointers or clarifications would be really helpful!

Answer 1

There is an expression for the product of two Levi-Civita symbols in terms of the Kronecker delta, namely $$\varepsilon_{ijk} \varepsilon_{imn} = \delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km}$$ Applying this to the left-hand side of your equation (and noting that $\varepsilon_{mni} = \varepsilon_{imn}$, you get $$\begin{align} \varepsilon_{ijk} \varepsilon_{k \ell m} \varepsilon_{mni} &= (\delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km}) \varepsilon_{k \ell m} \\ &= \varepsilon_{n \ell j} - \delta_{jn} \varepsilon_{k \ell k} \end{align}$$

Most of the work is done now, so see if you can take it from here.