Langlands branching rule for type B snake modules

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.

💡 Research Summary

The paper addresses two conjectures of Frenkel and Hernandez concerning Langlands duality for quantum affine algebras, focusing on the pair of types (B_n^{(1)}) (non‑twisted) and its Langlands dual (A_{2n-1}^{(2)}) (twisted). The first conjecture predicts that every finite‑dimensional representation (V) of (U_q(\widehat{\mathfrak{so}}{2n+1})) possesses a Langlands dual representation (L V) of (U_q(\widehat{\mathfrak{sl}}{2n}^{(2)})) whose highest weight is the image of the highest weight of (V) under the natural lattice map (\Pi). The second conjecture asserts a positive branching rule: the character (\chi(V)) can be expressed as a sum of characters of irreducible representations of the dual algebra with non‑negative integer coefficients.

The authors work within the class of snake modules, a broad family that includes all minimal affinizations and, in particular, all Kirillov–Reshetikhin (KR) modules. Snake modules are parametrised by two strictly increasing integer sequences ((l_1,\dots,l_T)) and ((r_1,\dots,r_T)) satisfying (r_t\ge l_t). They admit a combinatorial description via non‑overlapping lattice paths introduced by Mukhin and Young. This path model translates the (q)-character of a snake module into a set of monomials indexed by such paths.

The paper proceeds in three main steps:

1. Folding between twisted and untwisted type A – The authors prove that the folding map, previously known for KR modules, extends to all snake modules (Proposition 3.14). This yields a correspondence between characters of type (A_{2n-1}^{(1)}) and type (A_{2n-1}^{(2)}) modules.

2. Path description and bijection – Using the Mukhin–Young path model, they construct explicit bijections:

   • Between paths of type (B_n^{(1)}) and non‑overlapping paths of type (A_{2n-1}^{(1)}) (Section 6.2, Theorem 6.6).
   • Between non‑overlapping paths of type (A_{2n-1}^{(1)}) and those of the twisted type (A_{2n-1}^{(2)}) (Section 4).

   These bijections preserve the monomial weights, allowing the transfer of character identities across the three algebras.

3. Determinant identity for type (A_{n-1}^{(1)}) snake modules – The authors prove a new determinant formula (Theorem 5.2) expressing the character of a type (A_{n-1}^{(1)}) snake module as a determinant of elementary monomials. This identity, built on earlier works on Jacobi–Trudi‑type formulas, is the key technical ingredient that enables the final branching rule.

With these tools, they establish Theorem 1 (Theorem 4.8): every snake module of type (B_n^{(1)}) has a Langlands dual snake module of type (A_{2n-1}^{(2)}). This confirms the first Frenkel–Hernandez conjecture for the whole snake family.

The central result, Theorem 2 (Theorem 6.7), provides the Langlands branching rule: \