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

TL;DR

This paper explores the relationship between combinatorics of Auslander-Reiten theory and $q$-characters of representations of affine quantum groups, revealing new identities and potential geometric interpretations.

Contribution

It extends the algebra homomorphism $ ilde{D}_\xi$ to analyze its interaction with the original $q$-character morphism, showing that standard modules' $q$-characters lie in its kernel.

Findings

All $q$-characters of standard modules are in the kernel of $ ilde{D}_\xi$.

Identifies new rational identities related to $q$-characters.

Suggests possible geometric interpretations of these identities.

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 $\mathcal{C}_{\mathbb{Z}}^{\leq \xi}$ 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 $\widetilde{D}_{\xi}$ whose domain contains the image of the Grothendieck ring of $\mathcal{C}_{\mathbb{Z}}^{\leq \xi}$ under the truncated $q$-character morphism $\widetilde{\chi}_q$ corresponding to $\xi$. We exhibited a close relationship between the composition of $\widetilde{D}_{\xi}$ with $\widetilde{\chi}_q$ and the morphism $\overline{D}$ recently introduced by Baumann, Kamnitzer and Knutson in their study of the equivariant homology of Mirkovi\'c-Vilonen cycles. In this paper, we extend $\widetilde{D}_{\xi}$ 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 $\widetilde{D}_{\xi}$. This provides a large family of new non-trivial rational identities suggesting possible geometric interpretations.