Some symmetry properties of renormalization-group transformations

For a Landau-Ginzburg-Wilson Hamiltonian of any given symmetry we show how one can find a group $G_T$ of orthogonal transformations in parameter space, which commute with renormalization-group transformations. Then a renormalization-group transformation may be expanded into covariants of $G_T$. We also present a systematic procedure for finding fixed points; they are most likely to decouple the Hamiltonian or to increase its symmetry. The merit of the conclusions obtained is illustrated using an example of a system with $C_4$ symmetry. Agreement with the results of $\epsilon$-expansion calculations has been found.