Gelfand–Tsetlin algebra ($G{Z}_n$)

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 $\mathbb{C}$.

In the group algebra $\mathbb{C}{S}_n$, setThese elements commute: for each triple $a<i<j$, the only overlapping contributions cancel as $[(ai),(ij)+(aj)]=0$. They generate the Gelfand–Tsetlin algebra, and on a standard Young tableau vector $X_r$ acts by the Content of a Young-diagram cell containing $r$.

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.

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.

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.

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.

Normalizing the tableau lines in a compatible real phase convention gives the orthonormal basis formAdmissible swaps have $|d|\ge 2$. Nonadmissible swaps act by $+1$ within a row and $-1$ within a column. The symmetric matrix is an orthogonal involution and is useful for making the unitary representation and its phases explicit.