# How to know the symmetries of a coupling in the Hamiltonian

Suppose you have an interaction term in your hamiltonian that looks like

$$$$H=\sum_{ijkl}U_{ijkl}c^\dagger_ic^\dagger_jc_kc_l$$$$

where $$U$$ is the coupling and $$c$$, $$c^\dagger$$ are fermion operators. The questions is: What can you do to figure out the properties of the tensor $$U_{ijkl}$$ under index exchange? I first thought that I can demand that $$H$$ is hermitian, but then I run into the problem of not knowing what is the adjoint of a tensor with more than two indeces. If it is what I think so, then I'd get

$$$$H^\dagger=\sum_{ijkl}U^*_{lkji}c^\dagger_lc^\dagger_kc_jc_i$$$$

But then only demanding that $$U$$ is real seems enough. However I've seen in many places that $$U$$ needs to have a specific symmetry under index exchange. My other approach was to use the commutation rules, For example switching the first two operators we get

\begin{aligned} H&=\sum_{ijkl}U_{ijkl}c^\dagger_ic^\dagger_jc_kc_l \\ &=-\sum_{ijkl}U_{ijkl}c^\dagger_jc^\dagger_ic_kc_l \\ &=-\sum_{ijkl}U_{jikl}c^\dagger_ic^\dagger_jc_kc_l \end{aligned} which would mean that the matrix is antisymmetric under exchange of $$i$$ and $$j$$. Is this reasoning correct?