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Symplectic Branching through Crystals

Bárbara Muniz

TL;DR

The paper addresses the symplectic branching problem: decomposing $gl_{2n}(\mathbb{C})$-representations when restricted to $sp_{2n}(\mathbb{C})$, as captured by the Naito–Sagaki conjecture. It provides an elementary, self-contained crystal-theoretic bijection between $sp_{2n}$-highest weight tableaux and Sundaram’s $n$-symplectic Littlewood–Richardson tableaux inside the $gl_{2n}$ crystal, avoiding heavy external machinery. The construction uses two simple maps, $\iota_{sp}$ and $\iota_{LR}$, together with a cascading operation to define $F = \iota_{sp}^{-1} \circ F \circ \iota_{LR}$, first in the stable regime $\ell(\lambda)$ small relative to $n$ and then extended to general $n$ via an $n$-symplectic compatibility condition. Consequently, the paper establishes the claimed equality between the multiplicities and Sundaram’s combinatorial counts, providing a transparent crystal-based proof of the conjecture and linking type A and type C combinatorics through a concrete bijection.

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.

Symplectic Branching through Crystals

TL;DR

The paper addresses the symplectic branching problem: decomposing -representations when restricted to , as captured by the Naito–Sagaki conjecture. It provides an elementary, self-contained crystal-theoretic bijection between -highest weight tableaux and Sundaram’s -symplectic Littlewood–Richardson tableaux inside the crystal, avoiding heavy external machinery. The construction uses two simple maps, and , together with a cascading operation to define , first in the stable regime small relative to and then extended to general via an -symplectic compatibility condition. Consequently, the paper establishes the claimed equality between the multiplicities and Sundaram’s combinatorial counts, providing a transparent crystal-based proof of the conjecture and linking type A and type C combinatorics through a concrete bijection.

Abstract

We give an alternative proof of Naito--Sagaki's conjecture, which states that the restriction of -representations to 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.

Paper Structure

This paper contains 8 sections, 18 theorems, 43 equations.

Table of Contents

  1. Preliminaries
  2. The general and symplectic Lie algebras
  3. Young tableaux
  4. Quantum groups and crystals
  5. Restricting crystals
  6. Cascading operation
  7. Symplectic branching bijection
  8. For l(lambda)

Key Result

Theorem 1

Let $\lambda$ and $\mu$ be two partitions with at most $2n$ and $n$ parts, respectively. The multiplicity of the irreducible component $V_{sp_{2n}}(\mu)$ in $V_{gl_{2n}}(\lambda)$, as the restricted $sp_{2n}(\mathbb{C})$-representation, is given by where $(sp_nc)^{\lambda}_{\mu, \nu}$ is the number of $n$-symplectic Littlewood--Richardson tableaux of shape $\lambda\setminus \mu$ and weight $\nu$.

Theorems & Definitions (54)

  • Theorem : Sundaram
  • Remark 1.1
  • Example 1.1
  • Remark 1.2
  • Definition 1.1
  • Proposition 1.1
  • Proposition 1.2
  • Definition 1.2
  • Remark 1.3
  • Definition 1.3
  • ...and 44 more