Categorification of the Heisenberg algebra and MacMahon function

TL;DR

This paper constructs a categorification of a deformed Heisenberg algebra using Cautis and Licata's method, linking it to partition functions and the MacMahon function, with potential for further applications.

Contribution

It introduces a new categorification of a deformed Heisenberg algebra and connects it to partition functions and 3D Young diagrams.

Findings

Grothendieck ring of the categorification equals the deformed Heisenberg algebra

Establishes a connection between the categorification and the MacMahon function

Lays groundwork for future applications in related mathematical fields

Abstract

Starting from a one dimensional vector space, we construct a categorification $'\mathcal H$ of a deformed Heiserberg algebra $'H$ by Cautis and Licata's method. The Grothendieck ring of $'\mathcal H$ is $'H$. As an application, we discuss some related partition functions related to the MacMahon function of 3D Young diagram. We expect further applications of the results of this paper.