The $({\mathfrak{gl}}_m,{\mathfrak{gl}}_n$) duality in the quantum toroidal setting

B. Feigin, M. Jimbo, E. Mukhin

On a Fock space constructed from $mn$ free bosons and lattice ${\mathbb{Z}}^{mn}$, we give a level $n$ action of the quantum toroidal algebra ${\mathcal{E}}_m$ associated to $\mathfra…