Charge functions for odd dimensional partitions

TL;DR

This paper introduces a formula for charge functions of odd-dimensional partitions, extending known results for lower dimensions, and verifies it for 5D with numerical tests for higher odd dimensions.

Contribution

It proposes a general expression for charge functions of odd-dimensional partitions and proves it for the 5D case, expanding the mathematical framework for BPS algebra representations.

Findings

Charge function formula for 5D partitions is proven.

Numerical tests support the validity of the formula for 7D and 9D.

Poles of charge functions correspond to legal box additions/removals in partitions.

Abstract

To construct a BPS algebra with representations furnished by n-dimensional partitions, the first step is to find the eigenvalues of the Cartan operators acting on them. The generating function of the eigenvalues is called the charge function. It has an important property that for each partition, the poles of the function correspond to the projection of the boxes which can be added to or removed from the partition legally. The charge functions of lower dimensional partitions, i.e., Young diagrams for 2D, plane partitions for 3D and solid partitions for 4D, are already given in the literature. In this paper, we propose an expression of the charge function for arbitrary odd dimensional partitions and have it proved for 5D case. Some explicit numerical tests for 7D and 9D case are also conducted to confirm our formula.