$\mathrm{S}\mathrm{U}(3)$ Fermi-Hubbard gas with three-body losses: symmetries and dark states

We study an $\mathrm{S}\mathrm{U}(3)$ invariant Fermi-Hubbard gas undergoing on-site three-body losses. The model presents eight independent strong symmetries preventing the complete depletion of the gas. By making use of a basis of semi-standard Young tableaux states, we reveal the presence of a rich phenomenology of stationary states. We classify the latter according to the irreducible representation of $\mathrm{S}\mathrm{U}(3)$ to which they belong. We finally discuss the presence of three-particle stationary states that are not protected by the $\mathrm{S}\mathrm{U}(3)$ symmetry.