On the polynomiality conjecture of cluster realization of quantum groups

TL;DR

This paper establishes a criterion for when elements in quantum cluster algebras are universally polynomial, resolving conjectures about the polynomiality of quantum group generators in positive representations.

Contribution

It provides a necessary and sufficient condition for polynomiality in quantum cluster algebras, advancing understanding of quantum group realizations.

Findings

Characterization of universally polynomial elements

Resolution of conjectures on polynomiality of quantum group generators

Enhanced understanding of quantum cluster algebra structures

Abstract

In this paper, we give a sufficient and necessary condition for a regular element of a quantum cluster algebra $\mathcal{O}_q(\mathcal{X})$ to be universally polynomial. This resolves several conjectures by the first author on the polynomiality of the cluster realization of quantum group generators in different families of positive representations.