Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 103 / 1 / b / 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:For this paper the chain is $S_1\subseteq \cdots \subseteq S_n$, so these centres are $Z_1,\dots ,Z_n$. The generators commute: if $r\le s$, 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.