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 $2I+1$.

Solution

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

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A ladder operator maps weight vectors to weight vectors whose weights differ by one root. Positive-root vectors raise weights and negative-root vectors lower them.


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