The symplectic left companion of a Littlewood-Richardson-Sundaram tableau and the Kwon property
Olga Azenhas
TL;DR
The work tackles the branching problem for the pair $({GL}_{2n}(\\mathbb{C}), {Sp}_{2n}(\\mathbb{C}))$ by linking Sundaram’s Littlewood-Richardson-Sundaram tableaux with Kwon’s symplectic model through the left companion construction. By leveraging the Henriques–Kamnitzer $gl_n$ crystal commuter and flagged hive techniques, it proves that the left companion of an LR–Sundaram tableau satisfies the Kwon symplectic property, thus realizing a direct left-companion bijection that confirms the Lecouvey–Lenart conjecture. The central result equates Sundaram property violations with non-symplecticity of the left companion $G_ackslash(T)$ and pinpoints the exact location of violations in the first column, providing a concrete combinatorial bridge between the Sundaram and Kwon branching rules. This work thus clarifies the structure of $c^ackslash{ackslash{ackslash{ackslash{lambda}}}}$-type branching and strengthens the connection between LR theory, symplectic tableaux, and crystal/combinatorial models for classical types.
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.