The symplectic left companion of a Littlewood-Richardson-Sundaram tableau and the Kwon property

Abstract: As a consequence of the Littlewood-Richardson (LR) commuters coincidence and the Kumar-Torres branching model via Kushwaha-Raghavan-Viswanath flagged hives, we have solved the Lecouvey-Lenart conjecture on the bijections between the Kwon and Sundaram branching models for the pair $({GL}{2n}(\mathbb{C}), {Sp}{2n}(\mathbb{C})) $ consisting of the general linear group ${GL}{2n}(\mathbb{C})$ and the symplectic group ${Sp}{2n}(\mathbb{C})$. In particular, thanks to the Henriques-Kamnitzer $gl_n$-crystal commuter, we have recognized that the left companion of an LR-Sundaram tableau is characterized by the Kwon symplectic condition. We now show that the construction of the left companion tableau of an LR-Sundaram tableau exhibits in fact the Kwon symplectic property.