# Formula for the matrix units in the Gelfand-Tsetlin basis of the symmetric group algebra?

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 $$\mathbb{C}[S_n]$$.

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.