The enough $g$-pairs property and denominator vectors of cluster algebras

Peigen Cao, Fang Li

Published 2018-03-14Version 1

In this artcile, we firstly prove that any skew-symmetrizable cluster algebra with principal coefficients at $t_0$ has the enough $g$-pairs property. The proof depends on a result by Muller on scattering diagrams. Based on this property of cluster algebras, we give a new explanation of the sign-coherence of $G$-matrices, and a positive answer to the conjecture on denominator vectors proposed by Fomin and Zelevinsky. Moreover, we give two applications of the positive answer. As the first application, any skew-symmetrizable cluster algebra $\mathcal A(\mathcal S)$ is proved to have the proper Laurent monomial property. In the viewpoint of a theorem by Irelli and Labardini-Fragoso, we actually provide an unified method to prove the linear independence of cluster monomials of $\mathcal A(\mathcal S)$. Secondly, by the positive answer said above, a function which is called compatibility degree can be well-defined on the set of cluster variables of $\mathcal A(\mathcal S)$, and then a characterization for several cluster variables to be contained or not in some cluster of $\mathcal A(\mathcal S)$ is given via compatibility degree, which is the other application of the positive answer.