Auslander-Reiten combinatorics and q-characters of representations of affine quantum groups

2024/10/08 by Casbi, Élie, Li, Jian-Rong
#Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2410.06046

Abstract

For each simple Lie algebra \mathfrakg of simply-laced type, Hernandez and Leclerc introduced a certain category C_ℤ of finite-dimensional representations of the quantum affine algebra of \mathfrakg, as well as certain subcategories C_ℤ^{≤ ξ} depending on a choice of height function adapted to an orientation of the Dynkin graph of \mathfrakg. In our previous work we constructed an algebra homomorphism \widetildeD_ξ whose domain contains the image of the Grothendieck ring of C_ℤ^{≤ ξ} under the truncated q-character morphism \widetildeχ_q corresponding to ξ. We exhibited a close relationship between the composition of \widetildeD_ξ with \widetildeχ_q and the morphism D recently introduced by Baumann, Kamnitzer and Knutson in their study of the equivariant homology of Mirković-Vilonen cycles. In this paper, we extend \widetildeD_ξ in order to investigate its composition with Frenkel-Reshetikhin's original q-character morphism. Our main result consists in proving that the q-characters of all standard modules in C_ℤ lie in the kernel of \widetildeD_ξ. This provides a large family of new non-trivial rational identities suggesting possible geometric interpretations.

Related

• Equivariant multiplicities via representations of quantum affine algebras
• Algebraic Approach to q,t-Characters
• Weyl group symmetry of q-characters
• The t-Analogs of q-Characters at Roots of Unity for Quantum Affine Algebras and Beyound
• q,t-characters and the structure of the ℓ-weight spaces of standard modules over simply laced quantum affine algebras