$(q,t)$-characters of Kirillov-Reshetikhin modules of type $A_r$ as quantum cluster variables

TL;DR

This paper demonstrates that Nakajima's $t$-deformed $T$-system for type $A_r$ corresponds to a quantum mutation relation within a specific quantized cluster algebra structure, linking geometric and algebraic frameworks.

Contribution

It establishes a precise connection between Nakajima's $t$-deformed $T$-system and quantum cluster algebra mutations for type $A_r$, clarifying their algebraic structure.

Findings

The $t$-deformed $T$-system satisfies a quantum mutation relation.

This relation is realized in a specific quantization of cluster algebra structures.

The work links geometric $t$-deformations with algebraic quantum cluster theory.

Abstract

Nakajima introduced a $t$-deformation of $q$-characters, $(q,t)$-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima $(q,t)$-characters of Kirillov-Reshetikhin modules satisfy a $t$-deformed $T$-system. The $T$-system is a discrete dynamical system that can be interpreted as a mutation relation in a cluster algebra in two different ways, depending on the choice of direction of evolution. In this paper, we show that the Nakajima $t$-deformed $T$-system of type $A_r$ forms a quantum mutation relation in a quantization of exactly one of the cluster algebra structures attached to the $T$-system.