Quasi-particles and the Kanade-Russell and Kur\c{s}ung\"{o}z formula for Capparelli's identity

TL;DR

This paper constructs a basis for a specific affine Lie algebra module using quasi-particles and relates it to Capparelli's identities through advanced series representations.

Contribution

It introduces polychromatic quasi-particles and establishes their relations, providing a new basis construction for the module and connecting it to Capparelli's identities.

Findings

Constructed a quasi-particle basis for the module

Established relations among quasi-particles

Linked the basis to Capparelli's identities via series representations

Abstract

We construct a quasi-particle basis of the integrable highest weight module of highest weight $3\Lambda_0$ for the twisted affine Lie algebra of type $A_2^{(2)}$ in the principal realization. More specifically, by introducing the concept of polychromatic quasi-particle and finding relations among quasi-particles, we construct the spanning set of the standard module. Finally, its linear independence is proved by using Kanade-Russell and Kur\c{s}ung\"{o}z's Andrews-Gordon type series of Capparelli's identities.