Structure Constants of Diagonal Reduction Algebras of $\mathfrak{gl}$ Type

We describe, in terms of generators and relations, the reduction algebra, related to the diagonal embedding of the Lie algebra $\mathfrak{gl}_n$ into $\mathfrak{gl}_n\oplus\mathfrak{gl}_n$. Its representation theory is related to the theory of decompositions of tensor products of $\mathfrak{gl}$-modules.