Young tableau descriptions for the polyhedral realizations of crystal bases in type $A_n$
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910203556921344 |
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| author | Han, Shaolong |
| author_facet | Han, Shaolong |
| contents | By utilizing the combinatorial properties of various tableau models, we establish an explicit correspondence between the polyhedral realizations of the crystal bases $\mathcal B(λ)$ (resp. $\mathcal B(\infty)$) of type $A_n$ and the reverse semi-standard Young tableaux (resp. reverse marginally large tableaux), thereby providing a combinatorial description of the corresponding polyhedral realizations. Furthermore, a crystal structure on the set of Gelfand-Tsetlin patterns is obtained via the correspondence between the polyhedral realization of $\mathcal{B}(λ)$ and the reverse tableaux. As applications of our framework, we present concrete combinatorial realizations of the crystal embedding of $\mathcal B(λ)$ into $\mathcal B(\infty)$ and the set of Lusztig data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_08602 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Young tableau descriptions for the polyhedral realizations of crystal bases in type $A_n$ Han, Shaolong Representation Theory Combinatorics By utilizing the combinatorial properties of various tableau models, we establish an explicit correspondence between the polyhedral realizations of the crystal bases $\mathcal B(λ)$ (resp. $\mathcal B(\infty)$) of type $A_n$ and the reverse semi-standard Young tableaux (resp. reverse marginally large tableaux), thereby providing a combinatorial description of the corresponding polyhedral realizations. Furthermore, a crystal structure on the set of Gelfand-Tsetlin patterns is obtained via the correspondence between the polyhedral realization of $\mathcal{B}(λ)$ and the reverse tableaux. As applications of our framework, we present concrete combinatorial realizations of the crystal embedding of $\mathcal B(λ)$ into $\mathcal B(\infty)$ and the set of Lusztig data. |
| title | Young tableau descriptions for the polyhedral realizations of crystal bases in type $A_n$ |
| topic | Representation Theory Combinatorics |
| url | https://arxiv.org/abs/2605.08602 |