Reality determining subgraphs and strongly real modules

TL;DR

This paper extends the understanding of real modules in quantum affine algebras by introducing strongly real modules and demonstrating that certain generalized trees of modules, including snake trees, produce these modules.

Contribution

It introduces the concept of strongly real modules and generalizes the class of trees that generate real modules, expanding the framework for studying module properties.

Findings

Modules associated to certain generalized trees are strongly real.

Snake trees and their generalizations produce strongly real modules.

Extension of known results from type A algebras to broader classes.

Abstract

The concept of pseudo q-factorization graphs was recently introduced by the last two authors as a combinatorial language which is suited for capturing certain properties of Drinfeld polynomials. Using certain known representation theoretic facts about tensor products of Kirillov Reshetikhin modules and qcharacters, combined with special topological/combinatorial properties of the underlying q-factorization graphs, the last two authors showed that, for algebras of type A, modules associated to totally ordered graphs are prime, while those associated to trees are real. In this paper, we extend the latter result. We introduce the notions of strongly real modules and that of trees of modules satisfying certain properties. In particular, we can consider snake trees, i.e., trees formed from snake modules. Among other results, we show that a certain class of such generalized trees, which properly contains the snake trees, give rise to strongly real modules.