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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.23833 |
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| _version_ | 1866913768166916096 |
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| author | Kanakubo, Yuki Koshevoy, Gleb Nakashima, Toshiki |
| author_facet | Kanakubo, Yuki Koshevoy, Gleb Nakashima, Toshiki |
| contents | We show that a $q$-character of a Kirillov-Reshetikhin module (KR modules) for untwisted quantum affine algebras of simply laced types $A_n^{(1)}$, $D_n^{(1)}$, $E_6^{(1)}$, $E_7^{(1)}$, $E_8^{(1)}$ might be obtained from a specific cluster variable of a seed obtained by applying a maximal green sequence to the initial (infinite) quiver of the Hernandez-Leclerc cluster algebra. For a collection of KR-modules with nested supports, we show an explicit construction of a cluster seed, which has cluster variables corresponding to the $q$-characters of KR-modules of such a collection. We prove that the product of KR-modules of such a collection is a simple module. We also construct cluster seeds with cluster variables corresponding to $q$-characters of KR-modules of some non-nested collections. We make a conjecture that tensor products of KR-modules for such non-nested collections are simple. We show that the cluster Donaldson-Thomas transformations for double Bruhat cells for $ADE$ types can be computed using $q$-characters of KR-modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23833 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Products of Kirillov-Reshetikhin modules and maximal green sequences Kanakubo, Yuki Koshevoy, Gleb Nakashima, Toshiki Representation Theory Quantum Algebra We show that a $q$-character of a Kirillov-Reshetikhin module (KR modules) for untwisted quantum affine algebras of simply laced types $A_n^{(1)}$, $D_n^{(1)}$, $E_6^{(1)}$, $E_7^{(1)}$, $E_8^{(1)}$ might be obtained from a specific cluster variable of a seed obtained by applying a maximal green sequence to the initial (infinite) quiver of the Hernandez-Leclerc cluster algebra. For a collection of KR-modules with nested supports, we show an explicit construction of a cluster seed, which has cluster variables corresponding to the $q$-characters of KR-modules of such a collection. We prove that the product of KR-modules of such a collection is a simple module. We also construct cluster seeds with cluster variables corresponding to $q$-characters of KR-modules of some non-nested collections. We make a conjecture that tensor products of KR-modules for such non-nested collections are simple. We show that the cluster Donaldson-Thomas transformations for double Bruhat cells for $ADE$ types can be computed using $q$-characters of KR-modules. |
| title | Products of Kirillov-Reshetikhin modules and maximal green sequences |
| topic | Representation Theory Quantum Algebra |
| url | https://arxiv.org/abs/2503.23833 |