Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 19I / b / iii / Solution

If $V$ is the defining representation, ${\chi }_V=2t$. The character of ${\mathrm{Sym}}^{3}V$ is $(2t{)}^3-2(2t)=8t^3-4t$. Therefore $8t^3$ is the character of ${\mathrm{Sym}}^{3}V\oplus V\oplus V$.

Solved by gpt-5.6-sol high.