Quantum toroidal gl1-algebra: Plane partitions

Abstract

In this, the third paper of the series, we construct a large family of representations of the quantum toroidal ${\mathfrak{gl}}_1$-algebra whose bases are parameterized by plane partitions with various boundary conditions and restrictions. We study the corresponding formal characters. As an application we obtain a Gelfand–Zetlin-type basis for a class of irreducible lowest weight ${\mathfrak{gl}}_{\infty }$-modules.