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Reality determining subgraphs and strongly real modules

Matheus Brito, Adriano Moura, Clayton Silva

TL;DR

The paper develops a combinatorial framework of pseudo $q$-factorization graphs to study the reality and primality of tensor products in the category of finite-dimensional $U_q(\tilde{\mathfrak{g}})$-modules, focusing on Drinfeld polynomials and KR-type factors. It introduces reality-determining subgraphs (rds) and quochains, culminating in the notions of strongly real modules, with prime snakes and snake trees as central examples. The authors establish criteria for reality via KKOP invariants, classify 3-vertex cases, and demonstrate that snake trees and broader generalized trees yield strongly real modules, including multicuts and $\mathcal{G}$-tree constructions. Applications include constructing new strongly real modules, linking to cluster algebras, and providing a framework to stratify real modules by the rds-quochain index. The work offers a versatile combinatorial toolkit to analyze Drinfeld polynomials, tensor products, and the real/prime structure in type $A$ and beyond, with potential extensions to cluster theory and geometric representation theory.

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.

Reality determining subgraphs and strongly real modules

TL;DR

The paper develops a combinatorial framework of pseudo -factorization graphs to study the reality and primality of tensor products in the category of finite-dimensional -modules, focusing on Drinfeld polynomials and KR-type factors. It introduces reality-determining subgraphs (rds) and quochains, culminating in the notions of strongly real modules, with prime snakes and snake trees as central examples. The authors establish criteria for reality via KKOP invariants, classify 3-vertex cases, and demonstrate that snake trees and broader generalized trees yield strongly real modules, including multicuts and -tree constructions. Applications include constructing new strongly real modules, linking to cluster algebras, and providing a framework to stratify real modules by the rds-quochain index. The work offers a versatile combinatorial toolkit to analyze Drinfeld polynomials, tensor products, and the real/prime structure in type and beyond, with potential extensions to cluster theory and geometric representation theory.

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.
Paper Structure (19 sections, 24 theorems, 143 equations)

This paper contains 19 sections, 24 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Basic Notation and Background
  3. Cartan Data and Quantum Algebras
  4. Finite-Dimensional Modules
  5. Tensor Products
  6. KKOP Invariants
  7. Digraph Notation and Terminology
  8. A Reality Test via Graphs
  9. Pseudo $q$-Factorization Graphs
  10. On the Prime Factorization of 3-Vertex Non Totally Ordered Graphs
  11. Strongly Real Modules
  12. Valence-1 Vertices
  13. Prime Snakes as Adjacency Subgraphs
  14. Applications
  15. Multicuts of Tree Type
  16. ...and 4 more sections

Key Result

Proposition 2.3.1

Let $\boldsymbol{\pi}, \boldsymbol{\varpi}\in\mathcal{P}^+$. Then, $V(\boldsymbol{\pi})\otimes V(\boldsymbol{\varpi})$ is simple if and only if $V(\boldsymbol{\varpi})\otimes V(\boldsymbol{\pi})$ is simple and, in that case, $V(\boldsymbol{\pi})\otimes V(\boldsymbol{\varpi}) \cong V(\boldsymbol{\pi}

Theorems & Definitions (53)

  • Proposition 2.3.1
  • Proposition 2.3.2: ms:to
  • Proposition 2.3.3: ms:to
  • Proposition 2.3.4: ms:to
  • Theorem 2.3.5
  • Corollary 2.3.6
  • Corollary 2.3.7
  • proof
  • Lemma 2.3.8
  • proof
  • ...and 43 more