Quantum toroidal $mathfrak{gl}_1$ algebra : plane partitions

In third paper of the series we construct a large family of representations of the quantum toroidal $gl_1$ 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 $gl_infty$-modules.