 Gelfand–Tsetlin algebra

For the chain $S_0\subset \cdots \subset S_n$, the Gelfand–Tsetlin algebra in $\mathbb{C}{S}_n$ is generated by the centers of $\mathbb{C}{S}_r$ for $0\le r\le n$. Multiplicity-free restriction makes its joint eigenspaces one-dimensional in each irreducible representation. Products of the central primitive idempotents along restriction paths give its minimal projections. Thus it is a maximal commutative subalgebra and a semisimple algebra, isomorphic to a finite product of copies of $\mathbb{C}$.