To specify, I have the following equation and I am trying to solve for $D_{\ell m}$ in terms of K:
$$K_{ijk}=\gamma_{ijk} + (\varepsilon_{ik\ell} \delta_{jm} + \varepsilon_{jk\ell}\delta_{im})D_{lm} + (\delta_{ik}\delta_{j\ell}+\delta_{jk}\delta_{i\ell}-4\delta_{ij}\delta_{k\ell})v_{\ell}$$
Some important properties are:
- $K_{ijk}$ is symmetric in the first two indices ($K_{ijk}=K_{jik}$)
- $K_{ijk}$ is traceless in the last two indices ($K_{ikk} = 0$)
- $\gamma_{ijk}$ is symmetric and traceless in all indices. I don't think its value is necessary in solving this problem but it is equal to $\frac{1}{3}(K_{ijk}+K_{kij}+K_{jki})-\frac{1}{15}(\delta_{ij}K_{ppk}+\delta_{ik}K_{ppj}+\delta_{jk}K_{ppi})$.
- $v_\ell$ is equal to $-\frac{1}{10}K_{pp\ell}$
Specifically with $D_{lm}$, it is symmetric and traceless in both indices. I know the solution is$$D_{lm} = -\frac{1}{6}(K_{\ell pq}\varepsilon_{qpm}+K_{mpq}\varepsilon_{qp\ell})$$
but I am lost in the how. Any advice is appreciated in setting the problem up or which direction I need to go in. Thank you in advance.