Unipotent quantum coordinate ring and cominuscule prefundamental representations
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914176416350208 |
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| author | Jang, Il-Seung Kwon, Jae-Hoon Park, Euiyong |
| author_facet | Jang, Il-Seung Kwon, Jae-Hoon Park, Euiyong |
| contents | We continue the study of realization of the prefundamental modules $L_{r,a}^{\pm}$, introduced by Hernandez and Jimbo, in terms of unipotent quantum coordinate rings as in [J-Kwon-Park, Int. Math. Res. Not., 2023]. We show that the ordinary character of $L_{r,a}^{\pm}$ is equal to that of the unipotent quantum coordinate ring $U_q^-(w_r)$ associated to fundamental $r$-th coweight. When $r$ is cominuscule, we prove that there exists a $U_q(\mathfrak{b})$-module structure on $U_q^-(w_r)$, which is isomorphic to $L_{r,aη_r}^\pm$ for some $η_r \in \mathbb{C}^\times$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02870 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unipotent quantum coordinate ring and cominuscule prefundamental representations Jang, Il-Seung Kwon, Jae-Hoon Park, Euiyong Representation Theory Quantum Algebra 17B37, 22E46, 05E10 We continue the study of realization of the prefundamental modules $L_{r,a}^{\pm}$, introduced by Hernandez and Jimbo, in terms of unipotent quantum coordinate rings as in [J-Kwon-Park, Int. Math. Res. Not., 2023]. We show that the ordinary character of $L_{r,a}^{\pm}$ is equal to that of the unipotent quantum coordinate ring $U_q^-(w_r)$ associated to fundamental $r$-th coweight. When $r$ is cominuscule, we prove that there exists a $U_q(\mathfrak{b})$-module structure on $U_q^-(w_r)$, which is isomorphic to $L_{r,aη_r}^\pm$ for some $η_r \in \mathbb{C}^\times$. |
| title | Unipotent quantum coordinate ring and cominuscule prefundamental representations |
| topic | Representation Theory Quantum Algebra 17B37, 22E46, 05E10 |
| url | https://arxiv.org/abs/2406.02870 |