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 $\mathbb{C}[S_n]$.


1 Answer 1


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.


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