Finite-dimensional representations of Yangians in complex rank

We classify the “finite-dimensional” irreducible representations of the Yangians $Y({\mathfrak{g}\mathfrak{l}}_t)$ and $Y({\mathfrak{s}\mathfrak{l}}_t)$. These are associative ind-algebras in the Deligne category $\mathbf{\text{Rep}}(G{L}_t)$, which generalize the regular Yangians $Y({\mathfrak{g}\mathfrak{l}}_n)$ and $Y({\mathfrak{s}\mathfrak{l}}_n)$ to complex rank. They were first defined by Etingof in his paper “Representation theory in complex rank”. Here we solve Problem 7.2 from this paper. We work with the Deligne category $\mathbf{\text{Rep}}(G{L}_t)$ using the ultraproduct approach introduced by Deligne and discussed by Nate Harman in arXiv:1601.03426.