= Solution
Define the <ladder operators> $J_\pm=J_1\pm iJ_2$. The <SU(2) Lie algebra> relations imply
$$
[J_3,J_\pm]=\pm J_\pm,
\qquad
[J_+,J_-]=2J_3.
$$
Thus $d(J_\pm)v_m$ has weight $m\pm1$ when nonzero. Starting from a maximum-weight vector $v_I$, repeated lowering gives weights
$$
I,I-1,\ldots,-I.
$$
The norm formula derived from the <Casimir element>,
$$
\lVert J_-|I,m\rangle\rVert^2=(I+m)(I-m+1),
$$
shows that lowering stops precisely at $m=-I$. Therefore $2I$ is a nonnegative integer and the irreducible representation has dimension $2I+1$.
Back to article page
