Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Langlands branching rule for type B snake modules

Jingmin Guo, Jian-Rong Li, Keyu Wang

TL;DR

This work proves that snake modules of the affine type $B_n^{(1)}$ admit Langlands dual representations to the twisted type $A_{2n-1}^{(2)}$, and it provides an explicit Langlands branching rule: the folded character $\Pi(\chi(V))$ decomposes positively as a sum of $\chi^{\sigma}(W)$ over twisted-type snake modules $W$, with a betweenness constraint and unit multiplicities. The approach hinges on folding relations between non-twisted and twisted $q$-characters via explicit path descriptions, along with a determinant formula that translates to a combinatorial identity for type $A$ snakes; these ingredients yield the duality and the branching rule for a broad class of representations, including all snake modules and Kirillov–Reshetikhin modules as special cases. The results extend Frenkel–Hernandez-type Langlands duality to affine types, providing both existence of Langlands duals and an explicit, combinatorial decomposition at the level of characters. The paper also sketches the dual direction from $A_{2n-1}^{(2)}$ to $B_n^{(1)}$, including a conjectural stabilization for general snake modules and connections to generalized Kirillov–Reshetikhin modules.

Abstract

We prove that each snake module of the quantum Kac-Moody algebra of type $B_n^{(1)}$ admits a Langlands dual representation, as conjectured by Frenkel and Hernandez (Lett. Math. Phys. (2011) 96:217-261). Furthermore, we establish an explicit formula, called the Langlands branching rule, which gives the multiplicities in the decomposition of the character of a snake module of the quantum Kac-Moody algebra of type $B_n^{(1)}$ into a sum of characters of irreducible representations of its Langlands dual algebra.

Langlands branching rule for type B snake modules

TL;DR

This work proves that snake modules of the affine type admit Langlands dual representations to the twisted type , and it provides an explicit Langlands branching rule: the folded character decomposes positively as a sum of over twisted-type snake modules , with a betweenness constraint and unit multiplicities. The approach hinges on folding relations between non-twisted and twisted -characters via explicit path descriptions, along with a determinant formula that translates to a combinatorial identity for type snakes; these ingredients yield the duality and the branching rule for a broad class of representations, including all snake modules and Kirillov–Reshetikhin modules as special cases. The results extend Frenkel–Hernandez-type Langlands duality to affine types, providing both existence of Langlands duals and an explicit, combinatorial decomposition at the level of characters. The paper also sketches the dual direction from to , including a conjectural stabilization for general snake modules and connections to generalized Kirillov–Reshetikhin modules.

Abstract

We prove that each snake module of the quantum Kac-Moody algebra of type admits a Langlands dual representation, as conjectured by Frenkel and Hernandez (Lett. Math. Phys. (2011) 96:217-261). Furthermore, we establish an explicit formula, called the Langlands branching rule, which gives the multiplicities in the decomposition of the character of a snake module of the quantum Kac-Moody algebra of type into a sum of characters of irreducible representations of its Langlands dual algebra.

Paper Structure

This paper contains 31 sections, 37 theorems, 183 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Frenkel--Hernandez conjecture
  4. Main results
  5. Organization
  6. Acknowledgments
  7. Preliminaries
  8. Quantum Kac--Moody algebras
  9. Non-twisted type
  10. Twisted type
  11. Folding characters
  12. Langlands duality
  13. Path description
  14. Paths
  15. Snake positions
  16. ...and 16 more sections

Key Result

Theorem 1

Let $V$ be a snake module of type $B_n^{(1)}$. Then $V$ has a Langlands dual representation ${}^L{V}$, which is a snake module of type $A_{2n-1}^{(2)}$.

Figures (5)

  • Figure 1: Two paths of type $B_4$ in $\mathscr{P}^B_{2,k}$. Two consecutive points are connected by a blue edge. Note that the vertical coordinates increase from top to bottom.
  • Figure 2: A path $p \in \mathscr{P}^A_{2,k}$ of type $A_7$ (left), and its image $F(p) \in \mathscr{P}^B_{2,2k}$ of type $B_4$ (right). Horizontal coordinates relabeled by $\tau^{-1}$ are marked at the bottom.
  • Figure 3: Two examples of $2$-tuple of NOP of gap $1$.
  • Figure 4: A triple of NOP in $A_{k,k+4,k+8; k+1,k+3,k+5;1}$ (left) and a triple of NOP in $B_{k,k+4,k+8; k+1,k+3,k+5;1}$ (right). The points $(0,x_0^{(t)})$ and $(n,x_n^{(t)})$ are marked by squares.
  • Figure 5: Two triples of NOP $\overline{s}$ (left) and $\overline{p}$ (right) in correspondence. Note that $\overline{s}|_{[0,n]}$ corresponds to $L(\overline{p})$ as in Figure \ref{['fig: AB correspondence']} and $\overline{s}|_{[n,2n]} = R(\overline{p})$.

Theorems & Definitions (95)

  • Theorem 1: = Theorem \ref{['thm: langlands dual rep A to B']}
  • Theorem 2: = Theorem \ref{['thm: langlands branching rule']}
  • Definition 2.1
  • Theorem 2.2: chari1991quantumchari1995quantum
  • Conjecture 2.3: hernandez2010kirillov,wang2023qq
  • Remark 2.4
  • Definition 2.5
  • Conjecture 2.6: frenkel2011langlandsfinite
  • Conjecture 2.7: frenkel2011langlandsfinite
  • Definition 3.1
  • ...and 85 more