The branching models of Kwon and Sundaram via flagged hives

TL;DR

This work settles a conjectured link between two combinatorial branching rules for $\mathfrak{sl}(2n)$ restricted to $\mathfrak{sp}(2n)$ by establishing an explicit bijection between Sundaram's $\operatorname{LRS}(\nu/\mu,\lambda)$ and Kwon's $\operatorname{LRK}^{\nu}_{\lambda,\mu}$ via the hive model. It leverages the symmetry of Littlewood–Richardson coefficients in hive form (KRV) and introduces a flagged hive framework to encode the Sundaram and Kwon conditions uniformly. The main result shows that the natural composition $\operatorname{LR}(\nu/\mu,\lambda) \to \operatorname{LR}^{\nu}_{\mu,\lambda} \to \operatorname{LR}^{\nu}_{\lambda,\mu}$, when restricted to $\operatorname{LRS}$ and $\operatorname{LRK}$, yields a bijection $\operatorname{LRS}(\nu/\mu,\lambda) \tilde{\rightarrow} \operatorname{LRK}^{\nu}_{\lambda,\mu}$, realized by $U=\mathrm{rect} \circ C \circ \hat{P} \circ \varphi$. As a byproduct, a new flagged-hive branching model emerges, enriching the combinatorial toolbox for type $C$ branching and linking LR symmetry with crystal-theoretic constructions.

Abstract

We prove a bijection between the branching models of Kwon and Sundaram, conjectured previously by Lenart-Lecouvey. To do so, we use a symmetry of Littlewood-Richardson coefficients in terms of the hive model. Along the way, we obtain a new branching model in terms of flagged hives.

Theorems & Definitions (2)

• Theorem 2.1
• Example 3.1