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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.30806 |
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| _version_ | 1866914615736139776 |
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| author | Miyazawa, Sota Takagi, Taichiro |
| author_facet | Miyazawa, Sota Takagi, Taichiro |
| contents | Let $B(Λ_a) \, (a=0,1)$ be the crystal of the level 1 integrable irreducible highest weight representation of the affine quantum group $U_q(\widehat{\mathfrak{sl}}_2)$. We consider the crystal graphs of degree $n$ associated with the irreducible $(2r+1)$-dimensional (resp. $(2r+2)$-dimensional) $U_q(\mathfrak{sl}_2)$ module in $B(Λ_0)$ (resp. $B(Λ_1)$). In this paper, we construct an explicit combinatorial procedure providing a bijection between the set of highest weight paths in these graphs with respect to the action of the Kashiwara operator $\tilde{f}_{1}$, and the set of integer partitions of $n$ with sqrank (resp. rerank) $r$, which is a recently introduced partition statistic. As a byproduct, we also obtain a precise interpretation of the motif description of spinons suggested by Bernard-Pasquier-Serban in the spinon picture for Wess-Zumino-Witten conformal field theory models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30806 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Novel energy preserving bijections between affine crystals for $U_q(\widehat{\mathfrak{sl}}_2)$ and integer partitions Miyazawa, Sota Takagi, Taichiro Mathematical Physics Let $B(Λ_a) \, (a=0,1)$ be the crystal of the level 1 integrable irreducible highest weight representation of the affine quantum group $U_q(\widehat{\mathfrak{sl}}_2)$. We consider the crystal graphs of degree $n$ associated with the irreducible $(2r+1)$-dimensional (resp. $(2r+2)$-dimensional) $U_q(\mathfrak{sl}_2)$ module in $B(Λ_0)$ (resp. $B(Λ_1)$). In this paper, we construct an explicit combinatorial procedure providing a bijection between the set of highest weight paths in these graphs with respect to the action of the Kashiwara operator $\tilde{f}_{1}$, and the set of integer partitions of $n$ with sqrank (resp. rerank) $r$, which is a recently introduced partition statistic. As a byproduct, we also obtain a precise interpretation of the motif description of spinons suggested by Bernard-Pasquier-Serban in the spinon picture for Wess-Zumino-Witten conformal field theory models. |
| title | Novel energy preserving bijections between affine crystals for $U_q(\widehat{\mathfrak{sl}}_2)$ and integer partitions |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2605.30806 |