Totally ordered pseudo q-factorization graphs and prime factorization

TL;DR

This paper explores the structure of modules over quantum affine algebras of type A, focusing on totally ordered pseudo q-factorization graphs, and establishes their role in prime factorization and module decomposition.

Contribution

It introduces modules with snake support, proves their unique decomposition, and characterizes prime snake modules via totally ordered pseudo q-factorization graphs.

Findings

Modules with snake support have unique decompositions into prime factors.

Prime snake modules are characterized by totally ordered pseudo q-factorization graphs.

The study advances understanding of monoidal structures in quantum affine algebra representations.

Abstract

In an earlier publication, the last two authors showed that a finite-dimensional module for a quantum affine algebra of type $A$ whose $q$-factorization graph is totally ordered is prime. In this paper, we continue the investigation of the role of totally ordered pseudo $q$-factorization graphs in the study of the monoidal structure of the underlying abelian category. We introduce the notions of modules with (prime) snake support and of maximal totally ordered subgraphs decompositions. Our main result shows that modules with snake support have unique such decomposition and that it determines the corresponding prime factorization. Along the way, we also prove that prime snake modules (for type $A$) can be characterized as the modules for which every pseudo $q$-factorization graph is totally ordered.