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arXiv · 2608.16800

Explicit characterization of $\mathfrak{k}$-highest weight and $\widehat{\mathfrak{g}}$-dominant tableaux for $n\le 4$ via $1$-$0$-slack recording tableaux in the quantum LR rule and lattice points in flagged hive polytopes

Olga Azenhas
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Abstract

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 $2n-1$ in the quantum Littlewood-Richardson (LR) rule produced by $1$-$0$-slack recording tableaux. We now extend that characterization for lower shape lengths than $2n$ or $2n-1$ when the given $n$ is $3$ or $ 4$. 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 $\widehat{\mathfrak{g}}$-dominant tableaux for $n= 3$ and $n=4$. 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 of the 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.

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Olga Azenhas. 2026-09-13. Explicit characterization of $\mathfrak{k}$-highest weight and $\widehat{\mathfrak{g}}$-dominant tableaux for $n\le 4$ via $1$-$0$-slack recording tableaux in the quantum LR rule and lattice points in flagged hive polytopes. https://arxiv.org/abs/2608.16800

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