Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 302 / 2 / a / Solution

Define the ladder operators $J_{\pm }=J_1\pm i{J}_2$. The SU(2) Lie algebra relations implyThus $d(J_{\pm })v_m$ has weight $m\pm 1$ when nonzero. Starting from a maximum-weight vector $v_I$, repeated lowering gives weightsThe norm formula derived from the Casimir element,shows that lowering stops precisely at $m=-I$. Therefore $2I$ is a nonnegative integer and the irreducible representation has dimension $2I+1$.