= Solution
Use the two complementary <projection operators>
$$
P_+=\frac12\begin{pmatrix}1&1\\1&1\end{pmatrix},\qquad
P_-=\frac12\begin{pmatrix}1&-1\\-1&1\end{pmatrix}.
$$
They obey $P_++P_-=I_2$, $P_+P_-=0$ and $P_\pm^2=P_\pm$. The given <matrix> is $A=P_++\alpha P_-$, and consequently
$$
A^g=P_++\alpha^gP_-,\qquad A^3=I_2.
$$
It follows that multiplication of these <matrices> respects addition modulo three:
$$
R(g)R(h)=A^{g+h}=R(g+h\bmod3),\qquad R(0)=I_2.
$$
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 <1/c>, so
$$
\boxed{R\cong\rho_0\oplus\rho_1,\qquad \chi_R=(2,1+\alpha,1+\alpha^2).}
$$
The other nontrivial <irreducible representation>, $\rho_2$, is absent.
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