Symplectic Branching through Crystals

TL;DR

This paper provides a new proof of a conjecture relating to how certain complex algebraic representations decompose when restricted to subalgebras, using crystal bases and combinatorial models.

Contribution

It offers an alternative, explicit proof of Naito--Sagaki's conjecture by constructing a bijection between crystal highest weight elements and Sundaram's branching model.

Findings

Confirmed the conjecture through explicit bijection

Connected crystal theory with combinatorial branching models

Provided a self-contained proof using tableau models

Abstract

We give an alternative proof of Naito--Sagaki's conjecture, which states that the restriction of $gl_{2n}(\mathbb{C})$-representations to $sp_{2n}(\mathbb{C})$ can be described in terms of crystals. Using the tableau model for crystals, we construct an explicit and self-contained bijection between their highest weight elements and Sundaram's branching model.