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A proof of the Naito--Sagaki conjecture via the branching rule for $\imath$quantum groups

Satoshi Naito, Yujin Suzuki, Hideya Watanabe

TL;DR

The Naito--Sagaki conjecture describes how irreducible polynomial representations of $GL_{2n}(\mathbb{C})$ decompose upon restriction to $Sp_{2n}(\mathbb{C})$ via rational-path or tableau models. The authors give a new, independent proof by leveraging the branching rules of $\imath$quantum groups of type $A\mathrm{II}$ and crystal theory, together with promotion techniques and a local-global analysis. They formulate two weight-preserving bijections between $\widehat{\mathfrak{g}}$-dominant tableaux and $\mathfrak{k}$-highest/lowest weight tableaux, establish them for $n=2$ explicitly and then extend to general $n$ using a local-global principle and an induction on ranks. The result provides a fresh algebraic-combinatorial route to NS, with potential to yield further branching phenomena via quantum symmetric pairs and $\imath$-crystals.

Abstract

The Naito--Sagaki conjecture asserts that the branching rule for the restriction of finite-dimensional, irreducible polynomial representations of $GL_{2n}(\mathbb{C})$ to $Sp_{2n}(\mathbb{C})$ amounts to the enumeration of certain ``rational paths'' satisfying specific conditions. This conjecture can be thought of as a non-Levi type analog of the Levi type branching rule, stated in terms of the path model due to Littelmann, and was proved combinatorially in 2018 by Schumann--Torres. In this paper, we give a new proof of the Naito--Sagaki conjecture independently of Schumann--Torres, using the branching rule based on the crystal basis theory for $\imath$quantum groups of type $\mathrm{AII}_{2n-1}$. Here, note that $\imath$quantum groups are certain coideal subalgebras of a quantized universal enveloping algebra obtained by $q$-deforming symmetric pairs, and also regarded as a generalization of quantized universal enveloping algebras; these were defined by Letzter in 1999, and since then their representation theory has become an active area of research. The main ingredients of our approach are certain combinatorial operations, such as promotion operators and Kashiwara operators, which are well-suited to the representation theory of complex semisimple Lie algebras.

A proof of the Naito--Sagaki conjecture via the branching rule for $\imath$quantum groups

TL;DR

The Naito--Sagaki conjecture describes how irreducible polynomial representations of decompose upon restriction to via rational-path or tableau models. The authors give a new, independent proof by leveraging the branching rules of quantum groups of type and crystal theory, together with promotion techniques and a local-global analysis. They formulate two weight-preserving bijections between -dominant tableaux and -highest/lowest weight tableaux, establish them for explicitly and then extend to general using a local-global principle and an induction on ranks. The result provides a fresh algebraic-combinatorial route to NS, with potential to yield further branching phenomena via quantum symmetric pairs and -crystals.

Abstract

The Naito--Sagaki conjecture asserts that the branching rule for the restriction of finite-dimensional, irreducible polynomial representations of to amounts to the enumeration of certain ``rational paths'' satisfying specific conditions. This conjecture can be thought of as a non-Levi type analog of the Levi type branching rule, stated in terms of the path model due to Littelmann, and was proved combinatorially in 2018 by Schumann--Torres. In this paper, we give a new proof of the Naito--Sagaki conjecture independently of Schumann--Torres, using the branching rule based on the crystal basis theory for quantum groups of type . Here, note that quantum groups are certain coideal subalgebras of a quantized universal enveloping algebra obtained by -deforming symmetric pairs, and also regarded as a generalization of quantized universal enveloping algebras; these were defined by Letzter in 1999, and since then their representation theory has become an active area of research. The main ingredients of our approach are certain combinatorial operations, such as promotion operators and Kashiwara operators, which are well-suited to the representation theory of complex semisimple Lie algebras.

Paper Structure

This paper contains 25 sections, 29 theorems, 104 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Some combinatorial objects
  3. Partitions and Young diagrams
  4. Tableaux
  5. Words
  6. Naito--Sagaki conjecture
  7. General linear Lie algebras
  8. Fixed point subalgebra $\widehat{\mathfrak{g}}$
  9. Statement of the Naito--Sagaki conjecture
  10. Quantized universal enveloping algebras and Kashiwara crystals
  11. Quantized universal enveloping algebra of $\mathfrak{g}$
  12. Crystal bases
  13. Kashiwara crystals
  14. Branching rule for $\imath$quantum groups of type $A\mathrm{II}$
  15. $\imath$quantum groups of type $A\mathrm{II}$
  16. ...and 10 more sections

Key Result

Theorem 3.2

We have the following decomposition of $\mathsf{Res}^{\mathfrak{g}}_{\widehat{\mathfrak{g}}}\,L(\lambda)$ as a $\widehat{\mathfrak{g}}$-module: Equivalently, for all $\mu \in \widehat{P}^+$, the following equality holds:

Figures (12)

  • Figure 1: The relation between (universal enveloping) algebras and their modules that have appeared so far.
  • Figure 2:
  • Figure 3:
  • Figure 4: A $\widehat{\mathfrak{g}}$-dominant tableau
  • Figure 5:
  • ...and 7 more figures

Theorems & Definitions (75)

  • Remark 3.1
  • Theorem 3.2: NS $=$ ST
  • Remark 4.1
  • Example 4.2: Kas
  • Example 4.3: BS
  • Example 4.4: BS
  • Remark 4.5
  • Proposition 4.6: HK
  • Proposition 4.7: BS
  • Remark 4.8
  • ...and 65 more