Past exam of the groups and representations course of the University of Oxford / 2024 / 1 / d

All three displayed transposition matrices have trace zero. Multiplying two of them givessince ${\alpha }^3=1$. Their traces are $\alpha +{\alpha }^{-1}=-1$, whereas $\mathrm{tr}R(e)=2$. ThereforeBy character orthogonality, $R$ is the irreducible representation $2$ from c.

There is also a direct invariant subspace argument. The two distinct eigenvalues of $R({\tau }_1{\tau }_2)$ force any invariant line to be one of the two coordinate lines. But $R({\tau }_3)$ exchanges those lines. Neither is invariant under the whole group, so this two-dimensional representation is irreducible.