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

TL;DR

This paper constructs a broad class of representations for the quantum toroidal gl_1 algebra using plane partitions, analyzes their characters, and derives a Gelfand-Zetlin basis for certain irreducible modules.

Contribution

It introduces new representations parameterized by plane partitions and provides a Gelfand-Zetlin basis for specific irreducible modules, advancing the understanding of quantum toroidal algebras.

Findings

Constructed a large family of representations with plane partition bases.

Analyzed the formal characters of these representations.

Derived a Gelfand-Zetlin basis for irreducible lowest weight modules.

Abstract

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.