Past exam of the groups and representations course of the University of Oxford / 2018 / 1 / d / Solution

Use the two complementary projection operatorsThey obey $P_{+}+P_{-}=I_2$, $P_{+}{P}_{-}=0$ and $P_{\pm }^2=P_{\pm }$. The given matrix is $A=P_{+}+\alpha P_{-}$, and consequentlyIt follows that multiplication of these matrices respects addition modulo three:Thus this is a group representation. The eigenvectors $(1,1{)}^T$ and $(1,-1{)}^T$ have eigenvalues $1$ and $\alpha$, respectively. They span invariant lines carrying ${\chi }_0$ and ${\chi }_1$ in the convention of c, soThe other nontrivial irreducible representation, ${\rho }_2$, is absent.