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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.06321 |
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| _version_ | 1866910405451841536 |
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| author | Dinkins, Erica S. Misra, Kailash C. |
| author_facet | Dinkins, Erica S. Misra, Kailash C. |
| contents | Let $\mathfrak{g}$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$ and $\mathfrak{g}^L$ be its Langlands dual. It is conjectured that for each Dynkin node $i \in I \setminus \{0\}$ the affine Lie algebra $\mathfrak{g}$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of a certain coherent family of perfect crystals for the Langland dual $\mathfrak{g}^L$. In this paper we construct positive geometric crystals for $\mathcal{V}(C_n^{(1)})$ in the level zero fundamental spin $C_n^{(1)}$- module $W(\varpi_n)$ for $n = 2, 3,4$ and show that its ultra-discretization is isomorphic to the limit $B^{n, \infty}$ of a coherent family $\{B^{n, l}\}_{l \geq 1}$ of perfect crystals for the Langland dual $D_n^{(2)}$ which proves the conjecture in these cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06321 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $C_n^{(1)}$-Geometric Crystal and its Ultradiscretization Dinkins, Erica S. Misra, Kailash C. Representation Theory Let $\mathfrak{g}$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$ and $\mathfrak{g}^L$ be its Langlands dual. It is conjectured that for each Dynkin node $i \in I \setminus \{0\}$ the affine Lie algebra $\mathfrak{g}$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of a certain coherent family of perfect crystals for the Langland dual $\mathfrak{g}^L$. In this paper we construct positive geometric crystals for $\mathcal{V}(C_n^{(1)})$ in the level zero fundamental spin $C_n^{(1)}$- module $W(\varpi_n)$ for $n = 2, 3,4$ and show that its ultra-discretization is isomorphic to the limit $B^{n, \infty}$ of a coherent family $\{B^{n, l}\}_{l \geq 1}$ of perfect crystals for the Langland dual $D_n^{(2)}$ which proves the conjecture in these cases. |
| title | On $C_n^{(1)}$-Geometric Crystal and its Ultradiscretization |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2404.06321 |