Totally ordered pseudo q-factorization graphs and prime factorization
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| Format: | Preprint |
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2024
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| _version_ | 1866911274817814528 |
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| author | Brito, Matheus Moura, Adriano Silva, Clayton |
| author_facet | Brito, Matheus Moura, Adriano Silva, Clayton |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_01519 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Totally ordered pseudo q-factorization graphs and prime factorization Brito, Matheus Moura, Adriano Silva, Clayton Representation Theory Combinatorics Quantum Algebra 17B10, 17B37, 20G42, 81R10, 05C20 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. |
| title | Totally ordered pseudo q-factorization graphs and prime factorization |
| topic | Representation Theory Combinatorics Quantum Algebra 17B10, 17B37, 20G42, 81R10, 05C20 |
| url | https://arxiv.org/abs/2410.01519 |