Some conjectures on generalized cluster algebras via the cluster formula and $D$-matrix pattern

TL;DR

This paper introduces the cluster formula and D-matrix pattern in generalized cluster algebras, leading to solutions for key conjectures and advancing understanding of their structure.

Contribution

It develops the cluster formula and D-matrix pattern, providing new tools to resolve fundamental conjectures in generalized cluster algebra theory.

Findings

Fundamental conjectures of generalized cluster algebras are affirmatively solved.

Introduction of the cluster formula and D-matrix pattern as new frameworks.

Enhanced understanding of the structure and properties of generalized cluster algebras.

Abstract

In the theory of generalized cluster algebras, we build the so-called cluster formula and $D$-matrix pattern. Then as applications, some fundamental conjectures of generalized cluster algebras are solved affirmatively.