Extended $T$-System of Type $G_2$

TL;DR

This paper extends the $T$-system relations for the affine quantum algebra of type $G_2$, enabling computation of module classes and explicit dimension formulas for irreducible modules.

Contribution

It introduces a family of 3-term relations extending known $T$-systems for type $G_2$, facilitating module class calculations and dimension formulas.

Findings

Derived new 3-term relations in the Grothendieck ring

Computed classes of all minimal affinizations of type $G_2$

Provided explicit dimension formulas for modules

Abstract

We prove a family of 3-term relations in the Grothendieck ring of the category of finite-dimensional modules over the affine quantum algebra of type $G_2$ extending the celebrated $T$-system relations of type $G_2$. We show that these relations can be used to compute classes of certain irreducible modules, including classes of all minimal affinizations of type $G_2$. We use this result to obtain explicit formulas for dimensions of all participating modules.