Totally ordered pseudo q-factorization graphs and prime factorization
Matheus Brito, Adriano Moura, Clayton Silva
TL;DR
The paper develops a combinatorial framework using totally ordered pseudo $q$-factorization graphs to study prime factorizations in the monoidal category of finite-dimensional $U_q(\tilde{\mathfrak{g}})$-modules of type $A$. It introduces snake modules and the notion of snake support, and proves that modules with snake support admit a unique mtos-decomposition that yields the prime factorization; it also characterizes prime snake modules as those for which every pseudo $q$-factorization graph is totally ordered. A precise dictionary is established between $(i,n)$-segments and prime snakes in monochromatic cases, generalizing earlier results for modules supported on a single Dynkin node. The work connects graph-theoretic decompositions with Drinfeld polynomials and KR-type factors, providing structural insight into prime factorizations and practical methods for their computation in the quantum affine setting.
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.