= Solution
For an increasing chain of finite groups $G_1\subseteq\cdots\subseteq G_n$, regard every $\mathbb C G_r$ as a subalgebra of $\mathbb C G_n$. The <Gelfand–Tsetlin algebra> is the unital algebra generated by their centres:
$$
\boxed{GZ_n=\operatorname{alg}\bigl(Z(\mathbb C G_1),\ldots,Z(\mathbb C G_n)\bigr)}.
$$
For this paper the chain is $S_1\subseteq\cdots\subseteq S_n$, so these centres are $Z_1,\ldots,Z_n$. The generators commute: if $r\leq s$, an element of $Z_s$ commutes with all of $\mathbb C S_s$, including $Z_r$. Thus the <Gelfand–Tsetlin algebra> is commutative. Its joint eigenspaces record successive choices in the restriction of an <irreducible representation> down the chain.
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