= 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 $\alpha\succeq\beta$. Applying the inverse isomorphism gives $\beta\succeq\alpha$. Antisymmetry of the <dominance order on partitions> forces $\alpha=\beta$. Conversely, equality of the partitions makes the quotients identical, hence isomorphic.
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