Non-Abelian R-symmetries in $\mathcal{N}=1$ supersymmetry

We investigate non-Abelian R-symmetries in $\mathcal{N}=1$ supersymmetric theory, where fields may transform under the R-symmetry in representations with dimension higher than one. While a continuous non-Abelian R-symmetry can always be decomposed to a $U(1)$ R-symmetry and non-R symmetries, there are non-trivial discrete non-Abelian R-symmetries that do not admit such a decomposition, and effective R-charges cannot be defined in such models. Previous results on sufficient conditions for R-symmetric supersymmetric vacua in Wess-Zumino models still hold, and do not depend on fields in representations of dimension greater than one. However, fields in higher-dimensional representations enter the sufficient conditions for supersymmetric vacua that break R-symmetry, but it is difficult to identify the independent variables which can be used to solve the F-flatness equation in this case, unless other conditions are fulfilled. We present examples with discrete non-Abelian R-symmetries of the lowest order in this case.