Tensor powers of vector representation of $U_q(\mathfrak{sl}_2)$ at even roots of unity
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914890300522496 |
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| author | Lachowska, Anna Postnova, Olga Reshetikhin, Nicolai Solovyev, Dmitry |
| author_facet | Lachowska, Anna Postnova, Olga Reshetikhin, Nicolai Solovyev, Dmitry |
| contents | We study the decomposition of tensor powers of two dimensional irreducible representations of quantum $\mathfrak{sl}_2$ at even roots of unity into direct sums of tilting modules. We derive a combinatorial formula for multiplicity of tilting modules in the $N$-th tensor power of two dimensional irreducible representations, interpret it in terms of lattice paths and find its asymptotic behavior when $N\to\infty$. We also describe the limit of character and Plancherel measures when $N\to\infty$. We consider both $U_q(\mathfrak{sl}_2)$ with divided powers and the small quantum $sl_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03933 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tensor powers of vector representation of $U_q(\mathfrak{sl}_2)$ at even roots of unity Lachowska, Anna Postnova, Olga Reshetikhin, Nicolai Solovyev, Dmitry Representation Theory We study the decomposition of tensor powers of two dimensional irreducible representations of quantum $\mathfrak{sl}_2$ at even roots of unity into direct sums of tilting modules. We derive a combinatorial formula for multiplicity of tilting modules in the $N$-th tensor power of two dimensional irreducible representations, interpret it in terms of lattice paths and find its asymptotic behavior when $N\to\infty$. We also describe the limit of character and Plancherel measures when $N\to\infty$. We consider both $U_q(\mathfrak{sl}_2)$ with divided powers and the small quantum $sl_2$. |
| title | Tensor powers of vector representation of $U_q(\mathfrak{sl}_2)$ at even roots of unity |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2404.03933 |