Categorification of the Heisenberg algebra and MacMahon function

Starting from a one dimensional vector space, we construct a categorification ${}^{\mathrm{\prime }}\mathcal{H}$ of a deformed Heiserberg algebra ${}^{\mathrm{\prime }}H$ by Cautis and Licata’s method. The Grothendieck ring of ${}^{\mathrm{\prime }}\mathcal{H}$ is ${}^{\mathrm{\prime }}H$. 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.