A path description for $\varepsilon$-characters of representations of type $A$ restricted quantum loop algebras at roots of unity
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| Format: | Preprint |
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2025
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| _version_ | 1866912495727280128 |
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| author | An, Xiao-Juan Li, Jian-Rong Luo, Yan-Feng Zhang, Wen-Ting |
| author_facet | An, Xiao-Juan Li, Jian-Rong Luo, Yan-Feng Zhang, Wen-Ting |
| contents | Fix $\varepsilon^{2\ell}=1$ with $\ell \geq 2$. In this paper, we show that all finite-dimensional simple modules of any restricted quantum loop algebra $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ in a certain category can be transformed into snake modules. We obtain an effective and concrete path description for $\varepsilon$-characters of any simple module with highest $l$-weight of degree two and any Kirillov-Reshetikhin module of $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$. As an application of our path description, we obtain a necessary and sufficient condition for the tensor product of two fundamental representations of $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ to be irreducible. Additionally, we obtain a necessary condition for the tensor product of two or more fundamental representations of $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ to be irreducible. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_05440 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A path description for $\varepsilon$-characters of representations of type $A$ restricted quantum loop algebras at roots of unity An, Xiao-Juan Li, Jian-Rong Luo, Yan-Feng Zhang, Wen-Ting Quantum Algebra 17B37 Fix $\varepsilon^{2\ell}=1$ with $\ell \geq 2$. In this paper, we show that all finite-dimensional simple modules of any restricted quantum loop algebra $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ in a certain category can be transformed into snake modules. We obtain an effective and concrete path description for $\varepsilon$-characters of any simple module with highest $l$-weight of degree two and any Kirillov-Reshetikhin module of $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$. As an application of our path description, we obtain a necessary and sufficient condition for the tensor product of two fundamental representations of $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ to be irreducible. Additionally, we obtain a necessary condition for the tensor product of two or more fundamental representations of $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ to be irreducible. |
| title | A path description for $\varepsilon$-characters of representations of type $A$ restricted quantum loop algebras at roots of unity |
| topic | Quantum Algebra 17B37 |
| url | https://arxiv.org/abs/2503.05440 |