Normalized highest-weight lowering formula ($|I,m⟩=\sqrt{\frac{(I+m)!}{(2I)!(I-m)!}}{J}_{-}^{I-m}|I,I⟩$)

In an irreducible spin representation, the angular momentum lowering operator has squared norm factor $(I+m)(I-m+1)$. Multiplying these factors from the highest-weight vector gives the factorial normalization displayed above. All factorial arguments are integers even when $I$ is half-integral. This produces a consistent positive-coefficient phase convention for all weight states and fixes signs in subsequent isospin rotations.