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

For permutations $\sigma ,\tau$, the prescribed linear maps satisfySince this holds on a basis, $R(\sigma )R(\tau )=R(\sigma \tau )$. Also $R(e)=I$ and $R(\sigma {)}^{-1}=R({\sigma }^{-1})$. Hence $R$ is a representation.

The trace of a permutation matrix counts its fixed coordinate basis vectors. There are respectively $3,1,0$ fixed vectors on the three conjugacy classes, so ${\chi }_R=(3,1,0)={\chi }_1+{\chi }_D$. ConsequentlyIn particular the sign representation is absent.