Yangian for ${\mathfrak{g}\mathfrak{l}}_2$ and Gelfand-Tsetlin basis for ${\mathfrak{g}\mathfrak{l}}_n$

We construct an explicit realization of finite-dimensional irreducible representations $V_{\lambda }$ of ${\mathfrak{g}\mathfrak{l}}_n$ using the representation theory of $\mathrm{Y}({\mathfrak{g}\mathfrak{l}}_2)$. We consider $V_{\lambda }$ as the evaluation representation of $\mathrm{Y}({\mathfrak{g}\mathfrak{l}}_n)$ and restrict it to the subalgebras $\mathrm{Y}({\mathfrak{g}\mathfrak{l}}_2{)}^{(i)}$. Then $V_{\lambda }$ is completely reducible as a representation of $\mathrm{Y}({\mathfrak{g}\mathfrak{l}}_2{)}^{(i)}$. It allows us to use the explicit formulas for irreducible representations of $\mathrm{Y}({\mathfrak{g}\mathfrak{l}}_2)$.