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

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 $\hat{\mathfrak{g}}$-dominant tableaux, we also explicitly characterize by certain linear inequalities $\hat{\mathfrak{g}}$-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, $\hat{\mathfrak{g}}$-dominant tableaux and the lattice points in a (disjoint) union of flagged hive polytopes.

Case n=4 has been substantially extended. 28 pages and many figures