Exchange matrices of I-boxes
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914172768354304 |
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| author | Kashiwara, Masaki Kim, Myungho |
| author_facet | Kashiwara, Masaki Kim, Myungho |
| contents | Admissible chains of i-boxes are important combinatorial tools in the monoidal categorification of cluster algebras, as they provide seeds of the cluster algebra. In this paper, we explore the properties of maximal commuting families of i-boxes in a more general setting, and define a certain matrix associated with such a family, which we call the exchange matrix. It turns out that, when considering the cluster algebra structure on the Grothendieck rings, this matrix is indeed the exchange matrix of the seed associated with the family, both in certain categories of modules over quantum affine algebras and over quiver Hecke algebras. We prove this by constructing explicit short exact sequences that represent the mutation relations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14359 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exchange matrices of I-boxes Kashiwara, Masaki Kim, Myungho Representation Theory Quantum Algebra 17B37, 13F60, 18D10 Admissible chains of i-boxes are important combinatorial tools in the monoidal categorification of cluster algebras, as they provide seeds of the cluster algebra. In this paper, we explore the properties of maximal commuting families of i-boxes in a more general setting, and define a certain matrix associated with such a family, which we call the exchange matrix. It turns out that, when considering the cluster algebra structure on the Grothendieck rings, this matrix is indeed the exchange matrix of the seed associated with the family, both in certain categories of modules over quantum affine algebras and over quiver Hecke algebras. We prove this by constructing explicit short exact sequences that represent the mutation relations. |
| title | Exchange matrices of I-boxes |
| topic | Representation Theory Quantum Algebra 17B37, 13F60, 18D10 |
| url | https://arxiv.org/abs/2409.14359 |