Past exam of the groups and representations course of the University of Oxford / 2021 / 2 / c

The two invariant subspaces areCoordinate permutations preserve both equal coordinates and the coordinate sum. Their dimensions add to three and their intersection is zero, so ${\mathbb{C}}^3=U_0\oplus U_2$. On $U_0$, the basis $u_0=(1,1,1{)}^T$ gives $R_{U_0}(\tau )=(1)$ for every transposition.

On $U_2$ choose the basis $u=e_1-e_3$, $v=e_2-e_3$. Directly permuting their entries givesThus, taking images as columns, the matrices of the matrix representation areEach has trace zero and squares to $I$. Their three-cycle products have trace $-1$, so this is exactly the two-dimensional irreducible representation of solution. These are all nonzero invariant subspaces affording individual irreducible representations, since each occurs once in solution.