On the polynomiality conjecture of cluster realization of quantum groups

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.

26 pages, 4 figures