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In the group algebra of $S_n$, the product of the Young–Jucys–Murphy elements $X_2,\dots ,X_n$ is the sum of all $n$-cycles, each with coefficient one. To prove it, multiply the sum of all $m$-cycles by $X_{m+1}$. Right multiplication by $(jm+1)$ inserts $m+1$ immediately after $j$ in the cycle. Every $(m+1)$-cycle has a unique predecessor of $m+1$, so deletion inverts this insertion bijectively. Induction starts at $X_2=(12)$. The identity turns a product of cell contents into a central character value of a conjugacy-class sum.

Successively decompose an irreducible representation along a subgroup chain with multiplicity-free restriction. A complete path selects a one-dimensional subspace; choosing one nonzero vector on each path gives a Gelfand–Tsetlin basis. For a symmetric group over the complex numbers, paths are standard Young tableaux, and the Young–Jucys–Murphy elements act diagonally with their cell contents.

For the joint spectrum of the Young–Jucys–Murphy elements, adjacent coordinates $a_i,a_{i+1}$ are distinct. If $a_{i+1}-a_i=\pm 1$, the adjacent transposition $s_i$ acts on that line by $\pm 1$. Otherwise interchanging the coordinates gives a spectral vector in the same irreducible representation. These rules follow from $s_i{X}_i{s}_i+s_i=X_{i+1}$ and its two-dimensional eigenspace calculation. Together with the braid relation in a Coxeter group, they exclude the consecutive patterns $(a,a\pm 1,a)$.

For the standard inclusion $S_{n-1}\subset S_n$, over the complex numbers,Here $X_n$ is a Young–Jucys–Murphy element. One way to see generation is to use multiplicity-free restriction: a central idempotent of $S_{n-1}$ selects a preceding shape, and the distinct contents of its addable nodes of a Young diagram distinguish all possible succeeding shapes. Polynomial interpolation in $X_n$ supplies every diagonal projection in the centralizer of a subalgebra.

The Young–Jucys–Murphy elements arewith products of permutations composed from right to left. Fix $i<j$. In $[X_i,X_j]$, all pairs of disjoint transpositions commute. The only terms left, for each $a<i$, areIndeed $(ai)(ij)=(aj)(ai)$ and $(ij)(ai)=(ai)(aj)$. Summing directly provesThus all the Young–Jucys–Murphy elements commute.

Use the Young–Jucys–Murphy elements in the group algebra $\mathbb{C}{S}_n$. Their joint spectrum isEquivalently one can use the regular representation, which contains every irreducible representation. The commuting elements are self-adjoint in a unitary representation, so there is a simultaneous eigenbasis.

A standard Young tableau has entries increasing along each row and down each column. If entry $r$ occupies cell $(i_r,j_r)$, its Content of a Young-diagram cell is $c_T(r)=j_r-i_r$. Define the set of content vectors of standard Young tableaux byThe shape here is a partition of an integer; it is not itself the vector of eigenvalues. For $n=1$ both sets consist of $(0)$.

The first Young–Jucys–Murphy element is the empty sum $X_1=0$. Applying it to the nonzero simultaneous eigenvector defining a point of the joint spectrum gives $0=a_{1}v$. HenceFor the remaining spectral arguments use the following local facts, without assuming the restriction branching rule for a symmetric group: consecutive eigenvalues are distinct; a difference $a_{r+1}-a_r=\pm 1$ makes $s_r=(rr+1)$ act on the corresponding line as $\pm 1$; every other adjacent interchange gives another point of the joint spectrum in the same irreducible representation. These are the local spectral rules for Young–Jucys–Murphy elements. They follow from $s_r{X}_r{s}_r+s_r=X_{r+1}$ and the one- or two-dimensional analysis of the two neighboring eigenspaces. They are the appropriate local results used below.

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 identityproves 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 $\mathrm{Spec}(n)=\mathrm{Cont}(n)$.

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 isThe 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.

Let $T$ be a standard Young tableau, $s_i=(ii+1)$, and $d=c_T(i+1)-c_T(i)$. A suitable Gelfand–Tsetlin basis has, for an admissible pair $R=s_{i}T$ with tableau length increasing,For a nonadmissible swap the scalar is $+1$ in a row and $-1$ in a column. The diagonal coefficient follows from the Young–Jucys–Murphy element relation $s_i{X}_i{s}_i+s_i=X_{i+1}$, and $s_i^2=1$ forces the product of off-diagonal coefficients. One global normalization is $v_T=P_T{\pi }_T{v}_{T_0}$ from the row-reading tableau: a reduced admissible path makes this vector nonzero and gives coefficient one on every length-increasing edge.