Isomorphisms among quantum Grothendieck rings and cluster algebras

TL;DR

This paper interprets isomorphisms of quantum Grothendieck rings via cluster theory, leading to solutions for positivity and conjectures in quantum loop algebras, and revealing new relations among characters.

Contribution

It provides a cluster-theoretic framework for quantum Grothendieck ring isomorphisms, enabling quantization of monoidal categorification and solving longstanding problems.

Findings

Proves positivity of $(q,t)$-characters for all simple modules.

Establishes quantum $T$-systems for Kirillov-Reshetikhin modules.

Reveals explicit birational transformations linking characters.

Abstract

We establish a cluster theoretical interpretation of the isomorphisms of [F.-H.-O.-O., J. Reine Angew. Math., 2022] among quantum Grothendieck rings of representations of quantum loop algebras. Consequently, we obtain a quantization of the monoidal categorification theorem of [Kashiwara-Kim-Oh-Park, arXiv:2103.10067]. We establish applications of these new ingredients. First we solve long-standing problems for any non-simply-laced quantum loop algebras: the positivity of $(q,t)$-characters of all simple modules, and the analog of Kazhdan-Lusztig conjecture for all reachable modules (in the cluster monoidal categorification). We also establish the conjectural quantum $T$-systems for the $(q,t)$-characters of Kirillov-Reshetikhin modules. Eventually, we show that our isomorphisms arise from explicit birational transformations of variables, which we call substitution formulas. This reveals new non-trivial relations among $(q, t)$-characters of simple modules.