On $q$-series for principal characters of standard $A_2^{(2)}$-modules

TL;DR

This paper derives explicit sum-sides for principal characters of all standard modules of the affine Lie algebra A2^{(2)}, using Bailey pairs and lattice techniques, confirming a conjecture about their modular structure.

Contribution

It introduces a novel application of Bailey lattice with extended parameters to obtain sum-sides for A2^{(2)} modules, confirming a conjecture on their modular families.

Findings

Sum-sides for all standard A2^{(2)} modules derived.

Sum-sides split into six families based on module level modulo 6.

Extension of Bailey lattice technique to include out-of-bounds parameters.

Abstract

We present sum-sides for principal characters of all standard (i.e., integrable and highest-weight) irreducible modules for the affine Lie algebra $A_2^{(2)}$. We use modifications of five known Bailey pairs; three of these are sufficient to obtain all the necessary principal characters. We then use the technique of Bailey lattice appropriately extended to include "out-of-bounds" values of one of the parameters, namely, $i$. We demonstrate how the sum-sides break into six families depending on the level of the modules modulo 6, confirming a conjecture of McLaughlin--Sills.