Multi-grounded partitions and character formulas

TL;DR

This paper introduces multi-grounded partitions to express characters of affine Lie algebra modules, generalizing previous methods to all ground state paths and computing characters for several affine Lie algebra modules.

Contribution

It develops multi-grounded partitions and applies them to compute characters of a broad class of affine Lie algebra modules, extending prior work.

Findings

Characters of level 1 modules for various affine Lie algebras computed.

Generalization from constant to all ground state paths achieved.

New combinatorial framework for affine Lie algebra representations introduced.

Abstract

We introduce a new generalisation of partitions, multi-grounded partitions, related to ground state paths indexed by dominant weights of Lie algebras. We use these to express characters of irreducible highest weight modules of Kac-Moody algebras of affine type as generating functions for multi-grounded partitions. This generalises the approach of our previous paper, where only irreducible highest weight modules with constant ground state paths were considered, to all ground state paths. As an application, we compute the characters of the level $1$ modules of the affine Lie algebras $A_{2n}^{(2)}(n\geq 2)$, $D_{n+1}^{(2)}(n\geq 2)$, $A_{2n-1}^{(1)}(n\geq 3)$, $B_{n}^{(1)}(n\geq 3)$, and $D_{n}^{(1)}(n\geq 4)$.