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For $\alpha ,\beta \in \mathrm{Spec}(n)$, define $\alpha \sim \beta$ when their Gelfand–Tsetlin basis lines occur in the same irreducible representation. This is well defined: $G{Z}_n$ contains the centre of $\mathbb{C}{S}_n$, whose primitive central idempotents separate the irreducible modules, and part 1(ii) says these central elements are polynomials in the $X_j$. Consequently the same weight cannot label two different irreducible modules.

For content vectors of standard Young tableaux, an admissible adjacent swap of a standard Young tableau exchanges coordinates $i,i+1$ whose difference has absolute value greater than $1$. Define $\alpha \approx \beta$ as the equivalence relation generated by such swaps within $\mathrm{Cont}(n)$.

Equivalently, after reconstructing the tableaux, $\alpha \approx \beta$ means that the two tableaux have the same final Young diagram. Indeed, an admissible swap exchanges consecutive entries in incomparable cells, preserves the diagram and leaves a standard Young tableau. Conversely, tableaux of a fixed shape are linear extensions of a partially ordered set, with the cells ordered by their row and column requirements; any two linear extensions are connected by adjacent swaps of incomparable elements. We explain this connectivity and its spectral consequence in part (v).