On polyhedral formulas for Kirillov-Reshetikhin modules

We propose a method to prove a polyhedral branching formula for Kirillov-Reshetikhin (KR) modules over a quantum affine algebra. When the underlying simple Lie algebra is of exceptional type, such a formula remains conjectural in many cases. Using a linear recurrence relation satisfied by the characters of KR modules, we convert the verification of a polyhedral formula into an identity between two rational functions of a single variable with only simple poles at known locations. It is then sufficient to compare the residues at those poles, which are explicitly computable quantities. By applying this strategy, we obtain new, computer-assisted and easily verifiable proofs of known polyhedral formulas in types $F_4$ and $G_2$ within a uniform framework.