Classification of real modules in monoidal categorifications of cluster algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914232574935040 |
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| author | Duan, Bing Schiffler, Ralf |
| author_facet | Duan, Bing Schiffler, Ralf |
| contents | In this paper, we propose a conjectural formula for the highest $\ell$-weight monomial of an arbitrary real module over a simply-laced quantum affine algebra. We verify the conjecture under a multiplicative reachability condition, answering the Hernandez--Leclerc classification problem in monoidal categorifications of cluster algebras under this condition. Moreover, we introduce the notion of cluster modules, generalizing Kirillov--Reshetikhin modules and Hernandez--Leclerc modules as special cases. We prove that cluster modules are reachable real modules, and obtain a system of equations governing $q$-characters of the prime cluster modules, providing a natural generalization of both the classical T-system relations for Kirillov--Reshetikhin modules and the exchange relations for Hernandez--Leclerc modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16107 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification of real modules in monoidal categorifications of cluster algebras Duan, Bing Schiffler, Ralf Representation Theory Quantum Algebra Rings and Algebras 13F60, 16G20, 17B37 In this paper, we propose a conjectural formula for the highest $\ell$-weight monomial of an arbitrary real module over a simply-laced quantum affine algebra. We verify the conjecture under a multiplicative reachability condition, answering the Hernandez--Leclerc classification problem in monoidal categorifications of cluster algebras under this condition. Moreover, we introduce the notion of cluster modules, generalizing Kirillov--Reshetikhin modules and Hernandez--Leclerc modules as special cases. We prove that cluster modules are reachable real modules, and obtain a system of equations governing $q$-characters of the prime cluster modules, providing a natural generalization of both the classical T-system relations for Kirillov--Reshetikhin modules and the exchange relations for Hernandez--Leclerc modules. |
| title | Classification of real modules in monoidal categorifications of cluster algebras |
| topic | Representation Theory Quantum Algebra Rings and Algebras 13F60, 16G20, 17B37 |
| url | https://arxiv.org/abs/2512.16107 |