Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 2 / 19F / b

The classes of $A_4$ are the identity, the three double transpositions, and two classes of four three-cycles. Its quotient by $V_4$ is cyclic of order three, giving characters $1,\chi ,{\chi }^2$, with $\chi$ taking values $1,1,\omega ,{\omega }^2$ on these classes. The restriction of $V$ has character $(3,-1,0,0)$ and norm $(9+3)/12=1$, so it is the three-dimensional irreducible representation $T$. These four representations are complete since $1+1+1+9=12$.

The sign is trivial on $A_4$, soFor the last equality the restricted character is $(2,2,-1,-1)$, the sum of the two indicated one-dimensional characters. Equality of characters proves the decomposition.