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Symplectic tableaux and quantum symmetric pairs

Hideya Watanabe

TL;DR

The paper establishes a new, explicit branching rule from $GL_{2n}(\mathbb{C})$ to $Sp_{2n}(\mathbb{C})$ by constructing a bijection that maps each semistandard tableau of shape $\lambda$ to a pair consisting of a symplectic tableau of some shape $\nu\subseteq\lambda$ and a recording tableau of skew shape $\lambda/\nu$. This bijection, denoted $\mathrm{LR}^{A\mathrm{II}}$, is derived from a representation-theoretic framework involving a quantum symmetric pair of type $A\mathrm{II}_{2n-1}$, and its $q$-analogue intertwines $V(\lambda)$ with a direct sum of $V^\imath(\nu)\otimes\mathrm{Rec}_{2n}(\lambda/\nu)$ summands. The work provides a detailed factorization of the reduction map, proves injectivity, and uses quantum Littlewood–Richardson theory to establish surjectivity, thereby showing that the multiplicities $m_{\lambda,\nu}$ equal the counts of recording tableaux $|\mathrm{Rec}_{2n}(\lambda/\nu)|$. The results give a concrete combinatorial realization of the branching rule and reveal deep connections between crystal bases for quantum symmetric pairs, recording tableaux, and symplectic Littlewood–Richardson tableaux, with potential implications for crystal theory and q-deformations of classical branching phenomena.

Abstract

We provide a new branching rule from the general linear group $GL_{2n}(\mathbb{C})$ to the symplectic group $Sp_{2n}(\mathbb{C})$ by establishing a simple algorithm which gives rise to a bijection from the set of semistandard tableaux of a fixed shape to a disjoint union of several copies of sets of symplectic tableaux of various shapes. The algorithm arises from representation theory of a quantum symmetric pair of type $A\mathrm{II}_{2n-1}$, which is a $q$-analogue of the classical symmetric pair $(\mathfrak{gl}_{2n}(\mathbb{C}), \mathfrak{sp}_{2n}(\mathbb{C}))$.

Symplectic tableaux and quantum symmetric pairs

TL;DR

The paper establishes a new, explicit branching rule from to by constructing a bijection that maps each semistandard tableau of shape to a pair consisting of a symplectic tableau of some shape and a recording tableau of skew shape . This bijection, denoted , is derived from a representation-theoretic framework involving a quantum symmetric pair of type , and its -analogue intertwines with a direct sum of summands. The work provides a detailed factorization of the reduction map, proves injectivity, and uses quantum Littlewood–Richardson theory to establish surjectivity, thereby showing that the multiplicities equal the counts of recording tableaux . The results give a concrete combinatorial realization of the branching rule and reveal deep connections between crystal bases for quantum symmetric pairs, recording tableaux, and symplectic Littlewood–Richardson tableaux, with potential implications for crystal theory and q-deformations of classical branching phenomena.

Abstract

We provide a new branching rule from the general linear group to the symplectic group by establishing a simple algorithm which gives rise to a bijection from the set of semistandard tableaux of a fixed shape to a disjoint union of several copies of sets of symplectic tableaux of various shapes. The algorithm arises from representation theory of a quantum symmetric pair of type , which is a -analogue of the classical symmetric pair .
Paper Structure (37 sections, 58 theorems, 316 equations)

This paper contains 37 sections, 58 theorems, 316 equations.

Table of Contents

  1. Introduction
  2. Branching rules
  3. Results
  4. Quantum symmetric pairs
  5. Organization
  6. Preliminaries from combinatorics
  7. Partitions
  8. Semistandard tableaux
  9. Plactic monoid
  10. Symplectic tableaux
  11. Reduction map
  12. Successor map
  13. Main result
  14. Statement
  15. Outline of the proof
  16. ...and 22 more sections

Key Result

Lemma 2.1.3

Let $\lambda'$ denote the partition $(\lambda_1-1,\dots,\lambda_{\ell(\lambda)}-1)$, and $\mu$ be a partition such that $\lambda' \underset{\text{vert}}{\subseteq} \mu$. Then, we have $\mu \underset{\text{vert}}{\subseteq} \lambda$ if and only if $\ell(\mu) \leq \ell(\lambda)$.

Theorems & Definitions (124)

  • Example 1.2.1
  • Definition 2.1.1
  • Definition 2.1.2
  • Lemma 2.1.3
  • Example 2.3.1
  • Proposition 2.3.2
  • Proposition 2.3.3: Ful97
  • Proposition 2.3.4
  • Definition 2.4.1: Ki76
  • Proposition 2.4.2
  • ...and 114 more