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arXiv · 2605.30806

Novel energy preserving bijections between affine crystals for $U_q(\widehat{\mathfrak{sl}}_2)$ and integer partitions

Sota Miyazawa·Taichiro Takagi
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Abstract

Let $B(\Lambda_a) \, (a=0,1)$ 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 $B(\Lambda_0)$ (resp. $B(\Lambda_1)$). 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.

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Sota Miyazawa, Taichiro Takagi. 2026-05-29. Novel energy preserving bijections between affine crystals for $U_q(\widehat{\mathfrak{sl}}_2)$ and integer partitions. https://arxiv.org/abs/2605.30806

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