Kirillov–Reshetikhin Conjecture: The General Case

By D. Hernandez. 2007

119 citations recorded by the source corpus.

We prove the Kirillov-Reshetikhin (KR) conjecture in the general case : for all twisted quantum affine algebras we prove that the characters of KR modules solve the twisted Q-system and we get explicit formulas for the character of their tensor products (the untwisted simply-laced case was treated by Najakima, and the untwisted case by the author). The proof is uniform and provides several new developments for the representation theory of twisted quantum affine algebras, including twisted Frenkel-Reshetikhin q-characters (expected by Frenkel-Reshetikhin and Frenkel-Mukhin). We also prove the twisted T-system. As an application we get explicit formulas for the twisted q-characters of fundamental representations for all types, including the formulas for types D_4^{(3)}, E_6^{(2)} conjectured by Reshetikhin. We prove the formulas for KR modules in types A_n^{(2)} and D_4^{(3)} conjectured by Kuniba-Suzuki. Eventually our results imply the conjectural branching rules [HKOTT] to the quantum subalgebra of finite type.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f12dc37a980154f1041566134a3aae237141c991

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Citations in the corpus

Existence of Kirillov–Reshetikhin crystals for nonexceptional types (2007, background): “Q A ] 1 1 N ov that the crystals Br,s of type B (1) n , D (1) n , and A (2) 2n−1 are isomorphic to the combinatorial crystals of [31] for r not a spin node.”
Drinfeld Realization of Affine Quantum Algebras: the Relations (2012, background): “ there areseveral partialresults: in[A] the author studies the case A(2) 2 , constructing ψfollowing [Be], but the proof that it is well deﬁned is incomplete; a contribution to this proof is given in [H]. In [Jn], [JZ2] and [JZ1], the authors construct a homomorphism from UDJ q (X (k) n˜ ) to U Dr q”
Quantum toroidal algebras and their representations (2008, methodology): “ch is polynomial in the monomials mγ with positive coeﬃcients : this is analog to the Fomin-Zelevinsky Laurent Phenomena positivity properties for analog systems (Y -systems) [FZ1, FZ2] as noticed in [He9]. In the case of the quantum aﬃne algebra Uq(slˆn+1)-modules (n ≥ 2) we have : 0 → W(2) 1,aq → ”
Kirillov–Reshetikhin crystals for nonexceptional types (2008, background): “We also show that for types C (1) n and D (2) n+1 the analog of the Dynkin diagram automorphism exists on the level of crystals.”