Past exam of the groups and representations course of the University of Oxford / 2023 / 2 / b / Solution

The trivial representation and sign of a permutation giveThey are one-dimensional representations, hence irreducible representations. Their characters, in the conjugacy class order of solution, areThese are the only one-dimensional representations. All transpositions are conjugate, so they have the same scalar image $t$ in any one-dimensional representation; since a transposition squares to the identity, $t=\pm 1$. Transpositions generate the symmetric group, fixing the whole representation once $t$ is chosen.

By the sum of squares of irreducible representation dimensions, the three remaining dimensions satisfy $d_2^2+d_3^2+d_4^2=24-1-1=22$, with each $d_i\ge 2$. No $d_i$ can be at least $4$, since $16+4+4>22$. Thus the only possible dimensions are