Characterization of k-highest weight and ĝ-dominant tableaux via $1−0$-slack recording tableaux in the quantum Littlewood-Richardson rule and lattice points in flagged hive polytopes
Abstract
We have previously, for a given positive integer n, explicitly characterized by certain linear inequalities the 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 k-highest weight tableaux and ĝ-dominant tableaux, we also explicitly characterize by certain linear inequalities the corresponding ĝ-dominant tableaux when the shape length is 2n or 2n−1. In addition, when the given n is $3$ or $ 4$ the k-highest weight and the ĝ-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 k-highest weight tableaux, ĝ-dominant tableaux and the lattice points in a (disjoint) union of flagged hive polytopes.
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BibTeX
@misc{azenhas2026characterization,
title = {{Characterization of $\mathfrakk$-highest weight and $\widehat\mathfrakg$-dominant tableaux via $1$-$0$-slack recording tableaux in the quantum Littlewood-Richardson rule and lattice points in flagged hive polytopes}},
author = {Olga Azenhas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.16800},
eprint = {2608.16800},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16800},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Characterization of k-highest weight and ĝ-dominant tableaux via $1−0$-slack recording tableaux in the quantum Littlewood-Richardson rule and lattice points in flagged hive polytopes
AU - Olga Azenhas
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2608.16800
UR - https://arxiv.org/abs/2608.16800
AB - We have previously, for a given positive integer n, explicitly characterized by certain linear inequalities the 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 k-highest weight tableaux and ĝ-dominant tableaux, we also explicitly characterize by certain linear inequalities the corresponding ĝ-dominant tableaux when the shape length is 2n or 2n−1. In addition, when the given n is $3$ or $ 4$ the k-highest weight and the ĝ-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 k-highest weight tableaux, ĝ-dominant tableaux and the lattice points in a (disjoint) union of flagged hive polytopes.
ER -