= Solution
The local spectral interchanges preserve an <irreducible representation>, and <standard Young tableaux> of a fixed shape are connected by admissible adjacent interchanges. Different shapes have different multisets of <Young-diagram cell contents>. Their elementary symmetric functions in the <Young–Jucys–Murphy elements> are central: the identity
$$
\prod_{r=1}^n(u+X_r)=\sum_{\pi\in S_n}u^{\#\text{cycles}(\pi)}\pi
$$
proves this coefficient by coefficient. By the <Schur lemma>, one <irreducible representation> cannot therefore contain two different shape classes. The multiset determines a diagram because the counts on its positive and negative diagonals determine its arm and leg lengths at the diagonal cells.
Different <irreducible representations> cannot share a joint eigenvalue vector: the <Gelfand–Tsetlin algebra> is generated by the <Young–Jucys–Murphy elements>, so all its elements would act identically on that vector in both representations, whereas a central <primitive idempotent> distinguishes their two blocks. Thus distinct irreducibles give distinct shape classes.
The number of <irreducible representations> is the number $p(n)$ of <conjugacy classes>, each indexed by a <partition of an integer>. Every spectral vector gives a tableau by the preceding construction. Closure under admissible interchanges makes every shape class that occurs occur in full. The two counts $p(n)$ then force every shape to occur, with exactly one <irreducible representation> per shape. This supplies $\operatorname{Spec}(n)=\operatorname{Cont}(n)$.
On restricting to $S_{n-1}$, delete the last coordinate. For tableaux of shape $\lambda$, the entry $n$ occupies one <Removable node of a Young diagram>. Grouping by that node gives one copy of the corresponding module; the path decomposition has simple multiplicities. Hence the <restriction branching rule for a symmetric group> is
$$
\boxed{\operatorname{Res}^{S_n}_{S_{n-1}}V^\lambda\cong\bigoplus_{\mu\in\lambda^-}V^\mu.}
$$
The different removable nodes produce distinct partitions. The multiplicity-free structural input defining the <Gelfand–Tsetlin basis> is distinct from identifying this graph with the graph of <Young diagrams>.
Back to article page
