Graded limits of minimal affinizations over the quantum loop algebra of type $G_2$

TL;DR

This paper investigates the graded limits of minimal affinizations over the quantum loop algebra of type G_2, revealing their structure as generalized Demazure modules and deriving a multiplicity formula for their decomposition.

Contribution

It establishes the isomorphism of graded limits to generalized Demazure modules and provides defining relations and a multiplicity formula for type G_2 minimal affinizations.

Findings

Graded limits are isomorphic to generalized Demazure modules.

Derived defining relations for these modules.

Obtained a polyhedral multiplicity formula for module decomposition.

Abstract

The aim of this paper is to study the graded limits of minimal affinizations over the quantum loop algebra of type $G_2$. We show that the graded limits are isomorphic to multiple generalizations of Demazure modules, and obtain defining relations of them. As an application, we obtain a polyhedral multiplicity formula for the decomposition of minimal affinizations of type $G_2$ as a $U_q(\mathfrak{g})$-module, by showing the corresponding formula for the graded limits.