# On polyhedral formulas for Kirillov-Reshetikhin modules

## TL;DR

This paper introduces a new method to verify polyhedral formulas for Kirillov-Reshetikhin modules, especially in exceptional Lie types, using rational function identities and residue comparisons, with applications to types F4 and G2.

## Contribution

It develops a novel, computer-assisted approach to prove polyhedral formulas for KR modules, simplifying verification through rational function residue analysis.

## Key findings

- Verified polyhedral formulas for KR modules in types F4 and G2.
- Provided a uniform, computational framework for checking these formulas.
- Extended the method to cases where formulas are conjectural, aiding future proofs.

## Abstract

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.