发表机构
University of Coimbra（科英布拉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对n≤4的情形，扩展了量子Littlewood-Richardson规则中𝔨最高权表的刻画，通过Naito-Suzuki-Watanabe双射得到对应的ĝ优表，还建立了相关表与标记蜂巢格点的双射。
AI 中文摘要
此前，针对给定正整数n，我们已通过特定线性不等式明确刻画了量子Littlewood-Richardson（LR）规则中由1-0松弛记录表生成的、形状长度为2n或2n-1的𝔨最高权表。当n为3或4时，我们将该刻画扩展至形状长度小于2n或2n-1的情形。随后，利用Naito-Suzuki-Watanabe双射（该双射由提升算子的复合定义，用于连接𝔨最高权表与ĝ优表），我们还针对n=3和n=4的情形，通过特定线性不等式明确刻画了对应的ĝ优表。对于n=4，当形状长度为6或5时，我们的结果在两种情形下均为部分结果。由于量子Littlewood-Richardson规则中的记录表与Littlewood-Richardson-Sundaram（LRS）表存在自然双射，我们将关于量子LR规则逆的结果与Naito-Sagaki猜想的其他双射相关联，并建立了𝔨最高权表、ĝ优表与标记蜂巢的格点之间的双射。
英文摘要
We have previously, for a given positive integer $n$, explicitly characterized by certain linear inequalities the $\mathfrak{k}$-highest weight tableaux of shape length $2n$ or $2n-1$ produced by $1$-$0$-slack recording tableaux in the quantum Littlewood-Richardson (LR) rule. We now extend that characterization for lower shape lengths. Then using the composition of promotion operators defining the Naito-Suzuki-Watanabe bijection between $\mathfrak{k}$-highest weight tableaux and $\widehat{\mathfrak{g}}$-dominant tableaux, we also explicitly characterize by certain linear inequalities the corresponding $\widehat{\mathfrak{g}}$-dominant tableaux when the shape length is $2n$ or $2n-1$. In addition, when the given $n$ is $3$ or $ 4$ the $\mathfrak{k}$-highest weight and the $\widehat{\mathfrak{g}}$-dominant tableaux via $1$-$0$-slack recording tableaux are characterized by linear inequalities for lower shape lengths. Since recording tableaux in the quantum Littlewood-Richardson rule are in natural bijection with Littlewood-Richardson-Sundaram (LRS) tableaux, we relate our results on the inverse quantum LR rule with other two bijections for the Naito-Sagaki conjecture and establish bijections between $\mathfrak{k}$-highest weight tableaux, $\widehat{\mathfrak{g}}$-dominant tableaux and the lattice points in a (disjoint) union of flagged hive polytopes.
CommentsThe case $n$ even is solved for $\mathfrak{k}$-highest weight tableaux. 50 pages and many figures
