The Gelfand-Tsetlin algebra of the symmetric group chain

For $n\ge 1$ and $1\le m\le n$ let $Z(\mathbb{C}[S_m])$ be the centre of the complex group algebra $\mathbb{C}[S_m]$ (The center $Z(k[G])$ of the group algebra, The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements). We use the standard chain of symmetric groups $S_1\subset S_2\subset \cdots \subset S_n,S_m=\mathrm{Sym}(\{1,\dots ,m\}),$ in which $S_m$ is the subgroup of $S_n$ fixing every element of $\{m+1,\dots ,n\}$ pointwise; thus $\mathbb{C}[S_m]$ is the subalgebra of $\mathbb{C}[S_n]$ spanned by $S_m$, and $Z(\mathbb{C}[S_1])=\mathbb{C}\cdot 1$. Each centre $Z(\mathbb{C}[S_m])$ is thereby regarded as a subalgebra of $\mathbb{C}[S_n]$.

The Gelfand-Tsetlin algebra of the chain is $\mathrm{G}\mathrm{Z}(n):=⟨Z(\mathbb{C}[S_1]),Z(\mathbb{C}[S_2]),\dots ,Z(\mathbb{C}[S_n])⟩\subseteq \mathbb{C}[S_n],$ the unital subalgebra of $\mathbb{C}[S_n]$ generated by all these centres.

• Well-definedness. Each $Z(\mathbb{C}[S_m])$ is a $\mathbb{C}$-subalgebra of $\mathbb{C}[S_m]$, hence, via the inclusion $\mathbb{C}[S_m]\subseteq \mathbb{C}[S_n]$, a subset of $\mathbb{C}[S_n]$ closed under addition, multiplication and scalar multiplication; the generated subalgebra is therefore a unital subalgebra of $\mathbb{C}[S_n]$. It is finite-dimensional because $\mathbb{C}[S_n]$ is: each $\mathbb{C}[S_m]$ is a finite-dimensional semisimple algebra (If $\mathrm{char}k\nmid \mathrm{\mid }G\mathrm{\mid }$, then $k[G]$ is a semisimple ring), so its centre is finite-dimensional as well.

• Commutativity. Every element of $Z(\mathbb{C}[S_m])$ is central in $\mathbb{C}[S_m]$. If $m\le k$ and $z\in Z(\mathbb{C}[S_m])$, $w\in Z(\mathbb{C}[S_k])$, then $\mathbb{C}[S_m]\subseteq \mathbb{C}[S_k]$ and $w$ commutes with every element of $\mathbb{C}[S_k]$, in particular with $z$; hence $zw=wz$. Thus all the generating centres commute with one another, and $\mathrm{G}\mathrm{Z}(n)$ is a commutative subalgebra of $\mathbb{C}[S_n]$. Only this centrality, not semisimplicity, is used.

• The top centre is included. The term $m=n$ shows $Z(\mathbb{C}[S_n])\subseteq \mathrm{G}\mathrm{Z}(n)$; in fact the centre of $\mathbb{C}[S_n]$ is the algebra generated by its class sums, and the other centres are generated by the class sums of the subgroups, so $\mathrm{G}\mathrm{Z}(n)$ is generated by class sums of the chain together with $\mathbb{C}[S_1]=\mathbb{C}$.

• The algebra used below. $\mathrm{G}\mathrm{Z}(n)$ is the diagonal algebra of the Young basis: The Gelfand-Tsetlin algebra is the diagonal algebra of the Young basis identifies it with ${⨁}_T\mathbb{C}{P}_T$ over the standard tableaux $T$ of size $n$, and The Jucys-Murphy elements generate the Gelfand-Tsetlin algebra shows that it is also the algebra generated by the Jucys-Murphy elements.