Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 4 / 15F / b / iii / Solution

Write $V_r={\mathrm{Sym}}^r{\mathbb{C}}^2$ for the irreducible SU(2) representation of dimension $r+1$, with character of a representation $z^r+z^{r-2}+\cdots +z^{-r}$. Here $V=V_3$, with weights $3,1,-1,-3$. Squaring its character of a representation and successively removing the highest-weight irreducible characters givesThe dimensions $7+5+3+1=16$ check the decomposition; complete reducibility follows from the invariant inner product.

For the exterior square, add distinct pairs of weights. Its character of a representation is $z^4+z^2+2+z^{-2}+z^{-4}={\chi }_{V_4}+{\chi }_{V_0}$. Equivalently it is $({\chi }_V(z{)}^2-{\chi }_V(z^2))/2$. Hence $\boxed{{\Lambda }^{2}V\cong V_4\oplus V_0}$, of dimensions five and one. The complementary symmetric square is $V_6\oplus V_2$.