A combinatorial realization of Kirillov-Reshetikhin crystals for type E arising from translations
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866929736542846976 |
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| author | Jang, Il-Seung |
| author_facet | Jang, Il-Seung |
| contents | The main purpose of this paper is to give a combinatorial realization of Kirillov-Reshetikhin (KR simply) crystals $B^{r, s}$ for type $\text{E}_n^{(1)}$ with a minuscule node $r$ and $s \ge 1$. To do this, we describe explicitly the crystal of the quantum nilpotent subalgebra associated with the translation by the negative of the $r$-th fundamental weight. Then the crystal can be extended as an affine crystal, in which a certain subcrystal characterized by the $\varepsilon_r^*$-statistic is isomorphic to $B^{r,s}$ as an affine crystal, where $\varepsilon_r^*$ is also realized precisely in terms of triple and quadruple paths. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_11510 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A combinatorial realization of Kirillov-Reshetikhin crystals for type E arising from translations Jang, Il-Seung Quantum Algebra Combinatorics 17B37, 22E46, 05E10 The main purpose of this paper is to give a combinatorial realization of Kirillov-Reshetikhin (KR simply) crystals $B^{r, s}$ for type $\text{E}_n^{(1)}$ with a minuscule node $r$ and $s \ge 1$. To do this, we describe explicitly the crystal of the quantum nilpotent subalgebra associated with the translation by the negative of the $r$-th fundamental weight. Then the crystal can be extended as an affine crystal, in which a certain subcrystal characterized by the $\varepsilon_r^*$-statistic is isomorphic to $B^{r,s}$ as an affine crystal, where $\varepsilon_r^*$ is also realized precisely in terms of triple and quadruple paths. |
| title | A combinatorial realization of Kirillov-Reshetikhin crystals for type E arising from translations |
| topic | Quantum Algebra Combinatorics 17B37, 22E46, 05E10 |
| url | https://arxiv.org/abs/2103.11510 |