The Gelfand-Tsetlin algebra is the diagonal algebra of the Young basis

The idempotents $e_{\lambda }^{(m)}$ of [F1] are central in $\mathbb{C}[S_m]$, pairwise orthogonal, sum to $1$, and project each $\mathbb{C}[S_m]$-module onto its $\lambda$-isotypic part.

The idempotents attached to different levels commute: if $m\le k$, then $\mathbb{C}[S_m]\subseteq \mathbb{C}[S_k]$ and $e_{\mu }^{(k)}$ is central in $\mathbb{C}[S_k]$, hence commutes with every element of $\mathbb{C}[S_m]$, in particular with $e_{\lambda }^{(m)}$.

Let $m\ge 2$ and $\lambda ⊢m$. The restriction of $V^{\lambda }$ to $S_{m-1}$ is the direct sum of the distinct irreducible modules $V^{\lambda -x}$ over the removable nodes $x\in \mathrm{Rem}(\lambda )$; consequently $e_{\mu }^{(m-1)}$ acts on $V^{\lambda }$ as the projection onto the summand $V^{\mu }$ when $\mu =\lambda -x$ for some $x\in \mathrm{Rem}(\lambda )$, and as $0$ otherwise.

For every standard tableau $T$ of size $n$ the product $P_T=e_{{\lambda }^{(1)}}^{(1)}\cdots e_{{\lambda }^{(n)}}^{(n)}$ is an idempotent, and $P_T{P}_{T^{\mathrm{\prime }}}=0$ whenever $T\mathrm{\ne }{T}^{\mathrm{\prime }}$: distinct standard tableaux of size $n$ differ at some level $m\le n$, where their entries are distinct partitions, and the corresponding factors are orthogonal by [F1] after all factors are commuted past one another using step 1.2.

Every central element of $\mathbb{C}[S_m]$ is a linear combination of the $e_{\lambda }^{(m)}$: under the isomorphism $\mathbb{C}[S_m]\cong {\prod }_{\lambda }\mathrm{End}(V^{\lambda })$ of [F1] the centre corresponds to the product of the centres of the factors, and the centre of the matrix algebra $\mathrm{End}(V^{\lambda })$ consists of the scalars, that is, of $\mathbb{C}{e}_{\lambda }^{(m)}$. Hence $z={\sum }_{\lambda ⊢m}{\omega }_{\lambda }(z)e_{\lambda }^{(m)}$ for every $z\in Z(\mathbb{C}[S_m])$, where ${\omega }_{\lambda }(z)$ is the scalar by which $z$ acts on $V^{\lambda }$.

Rank one and the sum over paths, by induction on $m$: for every $\mu ⊢m$ and every path $T$ of length $m$ ending at $\mu$, the product $E_T:=e_{{\lambda }^{(1)}}^{(1)}\cdots e_{\mu }^{(m)}$ acts on $V^{\mu }$ as a rank-one idempotent with image a line $L_T\mathrm{\ne }0$, kills every $V^{\nu }$ with $\nu ⊢m$, $\nu \mathrm{\ne }\mu$, and ${\sum }_{T\text{ ending at }\mu }{E}_T$ acts as the identity on $V^{\mu }$, that is, ${\sum }_{T\text{ ending at }\mu }{E}_T=e_{\mu }^{(m)}$ in $\mathbb{C}[S_m]$. For $m=1$ the unique path gives $E=e_{(1)}^{(1)}=1$ acting as the identity on $V^{(1)}=\mathbb{C}$ by [F2], which is the rank-one projection onto the whole line. For the induction step write $T^{\mathrm{\prime }}=T\downarrow [m-1]$ and $\mu ={\lambda }^{(m-1)}+x$; by [F2] and step 2.1 the operator $e_{{\lambda }^{(m-1)}}^{(m-1)}$ projects $V^{\mu }$ onto the summand $V^{{\lambda }^{(m-1)}}$, on which $E_{T^{\mathrm{\prime }}}$ acts as the rank-one projection onto $L_{T^{\mathrm{\prime }}}$ by the induction hypothesis, while the remaining summands of the restriction are killed; multiplying by $e_{\mu }^{(m)}$, which is the identity on $V^{\mu }$ and kills the other $V^{\nu }$, gives the claim for $E_T$. Summing over all paths ending at $\mu$ and using the induction hypothesis at level $m-1$ together with [F1] gives ${\sum }_{T\text{ ending at }\mu }{E}_T={\sum }_{\nu ⊢m-1}{\sum }_{T^{\mathrm{\prime }}\text{ ending at }\nu }{E}_{T^{\mathrm{\prime }}}{e}_{\mu }^{(m)}=\left({\sum }_{\nu ⊢m-1}{e}_{\nu }^{(m-1)}\right)e_{\mu }^{(m)}=e_{\mu }^{(m)}$. Distinct paths give distinct lines: if two paths end at $\mu$ through different removable nodes their lines lie in different summands of the restriction, and if they end through the same node the induction hypothesis separates their prefixes.

The algebra $\mathrm{G}\mathrm{Z}(n)$ equals ${⨁}_T\mathbb{C}{P}_T$. For the inclusion $\supseteq$: each factor $e_{{\lambda }^{(m)}}^{(m)}$ of $P_T$ lies in $Z(\mathbb{C}[S_m])$, so $P_T\in \mathrm{G}\mathrm{Z}(n)$. For the inclusion $\subseteq$: step 2.3 writes every element of every generating centre as ${\sum }_{\lambda }{\omega }_{\lambda }{e}_{\lambda }^{(m)}$, and step 3.1 writes $e_{\lambda }^{(m)}={\sum }_{T\text{ ending at }\lambda }{E}_T$; substituting gives a linear combination of the $P_T=e_{{\lambda }^{(1)}}^{(1)}\cdots e_{{\lambda }^{(n)}}^{(n)}$ for paths of length $n$, so every generating element lies in the span, and the span is a subalgebra because $P_T{P}_{T^{\mathrm{\prime }}}={\delta }_{T{T}^{\mathrm{\prime }}}{P}_T$ by step 2.2; hence $\mathrm{G}\mathrm{Z}(n)\subseteq {⨁}_T\mathbb{C}{P}_T$. The $P_T$ are linearly independent because they are nonzero and pairwise orthogonal, so the sum is direct and $\mathrm{G}\mathrm{Z}(n)$ is commutative.

The idempotents add up to $1$: summing the identity of step 3.1 over all $\lambda ⊢n$ and using [F1] gives ${\sum }_T{P}_T={\sum }_{\lambda ⊢n}{e}_{\lambda }^{(n)}=1$.

The Jucys-Murphy elements lie in $\mathrm{G}\mathrm{Z}(n)$ and act diagonally. For $k\ge 2$ one has $X_k=T_k-T_{k-1}$, where $T_m={\sum }_{1\le i<j\le m}(ij)$ is the sum of the transpositions of $S_m$ (The Jucys-Murphy elements of the symmetric group algebra); any two transpositions are conjugate, since for transpositions $(ab)$ and $(cd)$ a permutation $g$ with $g(a)=c$, $g(b)=d$ satisfies $g(ab)g^{-1}=(cd)$ by Conjugating a cycle relabels each entry: $g(a_1\dots a_k)g^{-1}=(g(a_1)\dots g(a_k))$, so $T_m$ is a class sum and hence central in $\mathbb{C}[S_m]$ by For a finite group, the class sums form a basis of $Z(k[G])$ for $m\ge 2$, while $T_1=0$. Thus each $T_m$ lies in $Z(\mathbb{C}[S_m])\subseteq \mathrm{G}\mathrm{Z}(n)$ and each $X_k$ lies in $\mathrm{G}\mathrm{Z}(n)$; also $X_1=0$. By step 4.1 we may write $X_k={\sum }_T{c}_{T,k}{P}_T$, and then $X_k$ acts on the line $\mathbb{C}{v}_T=\mathrm{im}{P}_T$ by the scalar $c_{T,k}$, because $P_T$ is the identity on its own image. Hence $X_1,\dots ,X_n$ act diagonally in the Young basis.

Maximal commutativity. Under the isomorphism $\mathbb{C}[S_n]\cong {\prod }_{\lambda ⊢n}\mathrm{End}(V^{\lambda })$ of [F1], step 3.1 shows that $P_T$ corresponds to the tuple whose entry in the factor $\mathrm{End}(V^{\lambda })$, $\lambda ={\lambda }^{(n)}$, is the rank-one projection $p_T$ onto the line $L_T\subseteq V^{\lambda }$, and whose other entries are $0$; since the lines $L_T$ for $T$ of shape $\lambda$ are independent and number ${\dim }_{\mathbb{C}}{V}^{\lambda }$, they form a basis of $V^{\lambda }$. Therefore $\mathrm{G}\mathrm{Z}(n)={⨁}_T\mathbb{C}{P}_T$ corresponds to the tuples $(a_{\lambda })$ with $a_{\lambda }$ in the algebra $D_{\lambda }$ of all operators on $V^{\lambda }$ diagonal in that basis. An element $b=(b_{\lambda })$ commutes with every $P_T$ if and only if each $b_{\lambda }$ commutes with the full diagonal algebra $D_{\lambda }$; and the commutant of $D_{\lambda }$ in $\mathrm{End}(V^{\lambda })$ is $D_{\lambda }$ itself, because a matrix commuting with every diagonal matrix is diagonal. Hence the commutant of $\mathrm{G}\mathrm{Z}(n)$ in $\mathbb{C}[S_n]$ is $\mathrm{G}\mathrm{Z}(n)$, and if $B\subseteq \mathbb{C}[S_n]$ is any commutative subalgebra containing $\mathrm{G}\mathrm{Z}(n)$, then $B$ commutes with $\mathrm{G}\mathrm{Z}(n)$, so $B\subseteq \mathrm{G}\mathrm{Z}(n)$ and $B=\mathrm{G}\mathrm{Z}(n)$. Thus $\mathrm{G}\mathrm{Z}(n)$ is a maximal commutative subalgebra, and with steps 4.1-5.1 all four assertions are proved.