Three-vertex prime graphs and reality of trees
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866911912142307328 |
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| author | Moura, Adriano Silva, Clayton |
| author_facet | Moura, Adriano Silva, Clayton |
| contents | We continue the study of prime simple modules for quantum affine algebras from the perspective of $q$-fatorization graphs. In this paper we establish several properties related to simple modules whose $q$-factorization graphs are afforded by trees. The two most important of them are proved for type $A$. The first completes the classification of the prime simple modules with three $q$-factors by giving a precise criterion for the primality of a $3$-vertex line which is not totally ordered. Using a very special case of this criterion, we then show that a simple module whose $q$-factorization graph is afforded by an arbitrary tree is real. Indeed, the proof of the latter works for all types, provided the aforementioned special case is settled in general. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_10442 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Three-vertex prime graphs and reality of trees Moura, Adriano Silva, Clayton Representation Theory Quantum Algebra 17B10, 17B37, 20G42, 81R10, 05C05 We continue the study of prime simple modules for quantum affine algebras from the perspective of $q$-fatorization graphs. In this paper we establish several properties related to simple modules whose $q$-factorization graphs are afforded by trees. The two most important of them are proved for type $A$. The first completes the classification of the prime simple modules with three $q$-factors by giving a precise criterion for the primality of a $3$-vertex line which is not totally ordered. Using a very special case of this criterion, we then show that a simple module whose $q$-factorization graph is afforded by an arbitrary tree is real. Indeed, the proof of the latter works for all types, provided the aforementioned special case is settled in general. |
| title | Three-vertex prime graphs and reality of trees |
| topic | Representation Theory Quantum Algebra 17B10, 17B37, 20G42, 81R10, 05C05 |
| url | https://arxiv.org/abs/2204.10442 |