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

Abstract

For each simple Lie algebra $\mathfrak{g}$ of simply-laced type, Hernandez and Leclerc introduced a certain category ${\mathcal{C}}_{\mathbb{Z}}$ of finite-dimensional representations of the quantum affine algebra of $\mathfrak{g}$, as well as certain subcategories ${\mathcal{C}}_{\mathbb{Z}}^{\le \xi }$ depending on a choice of height function adapted to an orientation of the Dynkin graph of $\mathfrak{g}$. In our previous work we constructed an algebra homomorphism ${\tilde{D}}_{\xi }$ whose domain contains the image of the Grothendieck ring of ${\mathcal{C}}_{\mathbb{Z}}^{\le \xi }$ under the truncated $q$-character morphism ${\tilde{\chi }}_q$ corresponding to $\xi$. We exhibited a close relationship between the composition of ${\tilde{D}}_{\xi }$ with ${\tilde{\chi }}_q$ and the morphism $\overline{D}$ recently introduced by Baumann, Kamnitzer and Knutson in their study of the equivariant homology of Mirković-Vilonen cycles. In this paper, we extend ${\tilde{D}}_{\xi }$ 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 ${\mathcal{C}}_{\mathbb{Z}}$ lie in the kernel of ${\tilde{D}}_{\xi }$. This provides a large family of new non-trivial rational identities suggesting possible geometric interpretations.

Revision following referee's suggestions. 32 pages