Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 160 / 4 / e / Solution

If the two displayed simple quotients $D^{\alpha }$ and $D^{\beta }$ are isomorphic, compose the quotient map $S^{\alpha }\to D^{\alpha }$ with an isomorphism to $D^{\beta }\subseteq M^{\beta }/(S^{\beta }\cap (S^{\beta }{)}^{\perp })$. Part d(ii) gives $\alpha ⪰\beta$. Applying the inverse isomorphism gives $\beta ⪰\alpha$. Antisymmetry of the dominance order on partitions forces $\alpha =\beta$. Conversely, equality of the partitions makes the quotients identical, hence isomorphic.