Are there any formulas for the irreducible off-diagonal elements $E^{\lambda}\_{ij}$ in the Gelfand-Tsetlin basis of the symmetric group algebra $\mathbb{C}[S\_n]$?


Here is the context for my question. There exists well-known formula for *minimal idempotents* (sometimes also called *primitive idempotents*) in the Gelfand-Tsetlin basis (in the sense of Okounkov-Vershik approach) of the symmetric group algebra $\mathbb{C}[S\_n]$ originally due to Murphy ["A new construction of Young's seminormal representation of the symmetric group", J. Algebra, 69 (1981), 287-297]:
$$
E\_T=\prod\_{k=1}^n \prod\_{c \neq c\_k(T)} \frac{J\_k-c}{c\_k(T)-c},
$$
where $T$ is a path in the Bratteli diagram of $\mathbb{C}[S\_n]$ (or equivalently, *standard Young tableaux* - SYT), $J\_k$ is the $k$-th Jucys-Murphy element of $\mathbb{C}[S\_n]$, $c\_k(T)$ is the content of the box $k$ in SYT $T$, and $c$ runs over all possible contents in all SYTs of size $n$.


If we denote by $\lambda$ the Young diagram of the SYT $T$ and use the index $i$ to refer to the SYT $T$ as a basis vector in the Gelfand-Tsetlin basis of the irreducible representation $\lambda$, then we can think about $E\_T$ as being an orthogonal projector $E^{\lambda}\_{ii}$ onto diagonal entry $i$ in the block $\lambda$ under Artin-Wedderburn isomorphism $\mathbb{C}[S\_n] \cong \bigoplus\_{\lambda} \text{End}(V^\lambda)$, where $V^\lambda$ is the irreducible representation labeled by $\lambda$. This naturally raises the question of the existence of a formula for the *off-diagonal* "elementary matrices" $E^{\lambda}\_{ij}$ as elements of $\mathbb{C}[S\_n]$.



Formula for the off-diagonal "elementary matrices" in the Gelfand-Tsetlin basis of the symmetric group algebra?
Yes, these exist and are known as Young's orthogonal matrix units.


To construct $E\_{T,T'}$ note that $E\_T \,\mathbb{C}[S\_n] \, E\_{T'}$ is one dimensional. Thus it suffices to pick any $x\in \mathbb{C}[S\_n]$ such that $E\_T \, x\, E\_{T'}$ is nonzero, for example $x \in S\_n$ the unique permutation that sends $T'$ to $T$. (One can also use the Baxterised elements to exchange boxes labelled $i,i+1$ in the tableaux successively to get from $T'$ to $T$.)


For the $q$-case (Hecke algebra) their construction is given in Prop 2.1 of [Ram and Wenzl, "Matrix units for centralizer algebras", J Algebra
145 (1992) 378-395](https://www.sciencedirect.com/science/article/pii/002186939290109Y).


