Overview
This work builds upon prior research that explicitly characterized $\mathfrak{k}$-highest weight tableaux of shape length $2n$ or $2n-1$ within the quantum Littlewood-Richardson (LR) rule, specifically those produced by $1$-$0$-slack recording tableaux. The current study extends this characterization to lower shape lengths for the specific cases where the positive integer $n$ is $3$ or $4$. Furthermore, employing the composition of promotion operators that define the Naito-Suzuki-Watanabe bijection, the research explicitly characterizes $\widehat{\mathfrak{g}}$-dominant tableaux for $n=3$ and $n=4$ through specified linear inequalities.
The study also investigates the relationship between its findings on the inverse of the quantum LR rule and other bijections pertinent to the Naito-Sagaki conjecture. By leveraging the natural bijection between recording tableaux in the quantum Littlewood-Richardson rule and Littlewood-Richardson-Sundaram (LRS) tableaux, the paper establishes bijections connecting $\mathfrak{k}$-highest weight tableaux, $\widehat{\mathfrak{g}}$-dominant tableaux, and the lattice points found within a disjoint union of flagged hive polytopes.
Research Context
Previous investigations had established an explicit characterization of $\mathfrak{k}$-highest weight tableaux. This characterization was defined by certain linear inequalities and applied to tableaux of shape length $2n$ or $2n-1$, generated within the quantum Littlewood-Richardson rule via $1$-$0$-slack recording tableaux. This earlier work provided a foundation for the current extensions.
The Naito-Suzuki-Watanabe bijection plays a role in the broader context of relating different types of tableaux. This bijection maps $\mathfrak{k}$-highest weight tableaux to $\widehat{\mathfrak{g}}$-dominant tableaux through a composition of promotion operators. The Naito-Sagaki conjecture also forms part of the underlying theoretical framework, with the current research relating its results to bijections relevant to this conjecture.
The connection between recording tableaux in the quantum Littlewood-Richardson rule and Littlewood-Richardson-Sundaram (LRS) tableaux is recognized as a natural bijection. This equivalence allows for the translation of findings between these combinatorial objects.
Approach
The methodology involved extending a previously established characterization. Initially, $\mathfrak{k}$-highest weight tableaux of shape length $2n$ or $2n-1$ were characterized using linear inequalities. This study focused on extending this characterization to encompass lower shape lengths. This extension was specifically undertaken for cases where the positive integer $n$ is equal to $3$ or $4$.
To characterize $\widehat{\mathfrak{g}}$-dominant tableaux, the researchers utilized the Naito-Suzuki-Watanabe bijection. This bijection is defined by the composition of promotion operators, which map $\mathfrak{k}$-highest weight tableaux to $\widehat{\mathfrak{g}}$-dominant tableaux. By applying this bijection, explicit characterizations for $\widehat{\mathfrak{g}}$-dominant tableaux were derived, also expressed through linear inequalities, for the specific values of $n=3$ and $n=4$.
The research also involved establishing relationships between these characterizations and other combinatorial structures. This was achieved by leveraging the natural bijection that exists between recording tableaux in the quantum Littlewood-Richardson rule and Littlewood-Richardson-Sundaram (LRS) tableaux. Through these relationships, the inverse of the quantum LR rule was connected to other bijections relevant to the Naito-Sagaki conjecture. This approach culminated in establishing bijections between $\mathfrak{k}$-highest weight tableaux, $\widehat{\mathfrak{g}}$-dominant tableaux, and the lattice points contained within a disjoint union of flagged hive polytopes.
Findings
- The explicit characterization of $\mathfrak{k}$-highest weight tableaux, previously established for shape lengths $2n$ or $2n-1$ in the quantum LR rule from $1$-$0$-slack recording tableaux, was extended to lower shape lengths for $n=3$ and $n=4$. This extension is defined by specific linear inequalities.
- $\widehat{\mathfrak{g}}$-dominant tableaux for $n=3$ and $n=4$ were explicitly characterized by certain linear inequalities. This characterization was derived through the application of the composition of promotion operators that define the Naito-Suzuki-Watanabe bijection.
- Bijections were established connecting $\mathfrak{k}$-highest weight tableaux, $\widehat{\mathfrak{g}}$-dominant tableaux, and the lattice points found within a (disjoint) union of flagged hive polytopes. This was achieved by relating the inverse of the quantum LR rule to other bijections for the Naito-Sagaki conjecture, utilizing the natural bijection between recording tableaux in the quantum LR rule and Littlewood-Richardson-Sundaram (LRS) tableaux.