Rotational symmetry of the multiple-relaxation-time collision model

We point out that the minimal components of tensorial moments distribution can be independently relaxed in collision without violating rotational symmetry are its irreducible representation (irrep) SO(3), and a generic multiple-relaxation-time model constructed by relaxing these components. As simplest example irreps second moment traceless deviatoric stress an isotropic tensor which is conserved monatomic gases. Applying decomposition to thermal lattice Boltzmann for polyatomic gases [Phys. Rev. E 77, 035701(R) (2008)], shear bulk viscosities decoupled two independent relaxation times. The hydrodynamic equation obtained via Chapman-Enskog calculation verified numerical simulation.