Three-vertex prime graphs and reality of trees

TL;DR

This paper investigates prime simple modules in quantum affine algebras, focusing on their $q$-factorization graphs shaped as trees, and establishes criteria for primality and reality of these modules, especially in type A.

Contribution

It provides a complete classification of prime simple modules with three $q$-factors in type A and proves that modules with tree-shaped $q$-factorization graphs are real, extending results across all types.

Findings

Classification of prime modules with three $q$-factors in type A.

Prime modules with tree-shaped $q$-factorization graphs are real.

Results hold for all types given certain criteria are established.

Abstract

We continue the study of prime simple modules for quantum affine algebras from the perspective of $q$-fatorization graphs. In this paper we establish several properties related to simple modules whose $q$-factorization graphs are afforded by trees. The two most important of them are proved for type $A$. The first completes the classification of the prime simple modules with three $q$-factors by giving a precise criterion for the primality of a $3$-vertex line which is not totally ordered. Using a very special case of this criterion, we then show that a simple module whose $q$-factorization graph is afforded by an arbitrary tree is real. Indeed, the proof of the latter works for all types, provided the aforementioned special case is settled in general.