On the extended T-system of type C3C_3

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.Comment: arXiv admin note: substantial text overlap with arXiv:1208.482

Extended TT-System of Type G2G_2

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