Past exam of the groups and representations course of the University of Oxford / 2020 / 2 / c / Solution

The specified linear maps form a permutation representation, sinceThe trace of a permutation matrix counts fixed coordinate basis vectors. On the identity, three-cycles and transpositions, respectively, this gives ${\chi }_R=(3,0,1)$. Comparing with solution,For an explicit direct sum,Both summands are invariant subspaces under coordinate permutations. The first is fixed pointwise and hence gives the trivial representation; the second has character $(2,-1,0)$ and hence gives $D$. The sign representation has multiplicity zero.