Alternating snake modules and a determinantal formula
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915134071373824 |
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| author | Brito, Matheus Chari, Vyjayanthi |
| author_facet | Brito, Matheus Chari, Vyjayanthi |
| contents | We introduce a family of modules for the quantum affine algebra which include as very special cases both the snake modules and the modules arising from a monoidal categorification of cluster algebras. We give necessary and sufficient conditions for these modules to be prime and prove a unique factorization result. We also give an explicit formula expressing the module as an alternating sum of Weyl modules. Finally, we give an application of our results to classical questions in the category $\mathcal{ O}(\mathfrak{gl}_r)$. Specifically we apply our results to show that there are a large family of non-regular, non-dominant weights $μ$ for which the non-zero Kazhdan-Lusztig coefficients $c_{μ, ν}$ are $\pm 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03750 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Alternating snake modules and a determinantal formula Brito, Matheus Chari, Vyjayanthi Representation Theory Quantum Algebra 17B10, 81R10 We introduce a family of modules for the quantum affine algebra which include as very special cases both the snake modules and the modules arising from a monoidal categorification of cluster algebras. We give necessary and sufficient conditions for these modules to be prime and prove a unique factorization result. We also give an explicit formula expressing the module as an alternating sum of Weyl modules. Finally, we give an application of our results to classical questions in the category $\mathcal{ O}(\mathfrak{gl}_r)$. Specifically we apply our results to show that there are a large family of non-regular, non-dominant weights $μ$ for which the non-zero Kazhdan-Lusztig coefficients $c_{μ, ν}$ are $\pm 1$. |
| title | Alternating snake modules and a determinantal formula |
| topic | Representation Theory Quantum Algebra 17B10, 81R10 |
| url | https://arxiv.org/abs/2412.03750 |