On the extended T-system of type $C_3$

TL;DR

This paper explores a specific sub-system of extended T-systems in quantum affine algebras of type C3, providing methods to compute characters and conjectural decompositions of minimal affinizations.

Contribution

It identifies a closed sub-system of the extended T-system for type C3 and uses it to compute characters and propose decomposition formulas for minimal affinizations.

Findings

Identified a closed sub-system of the extended T-system for type C3.

Developed methods to compute characters of restricted minimal affinizations.

Proposed conjectural formulas for decompositions of minimal affinizations.

Abstract

We continue the study of extended T-systems of quantum affine algebras. We find a sub-system of the extended T-system of the quantum affine algebra $U_q \hat{\mathfrak{g}}$ of type $C_3$. The sub-system consisting of four systems which are denoted by I, II, III, and IV. Each of the systems I, II, III, IV is closed. The systems I-IV can be used to compute minimal affinizations with weights of the form $\lambda_1 \omega_1 + \lambda_2 \omega_2 + \lambda_3 \omega_3$, where at least one of $\lambda_1$, $\lambda_2$, $\lambda_3$ are zero. Using the systems I-IV, we compute the characters of the restrictions of the minimal affinizations in the systems to $ U_q \mathfrak{g} $ and obtain some conjectural decomposition formulas for the restrictions of some minimal affinizations.