Cluster algebras and symmetric matrices

TL;DR

This paper explores the relationship between c-vectors in skew-symmetric cluster algebras and quasi-Cartan companions, providing new insights into the structure of cluster tilted quivers and their admissible cuts.

Contribution

It establishes that c-vectors form a quasi-Cartan companion for skew-symmetric cluster algebras and demonstrates the existence of admissible cuts in cluster tilted quivers.

Findings

C-vectors define a quasi-Cartan companion of the exchange matrix

Any cluster tilted quiver has an admissible cut of edges

Provides structural insights into skew-symmetric cluster algebras

Abstract

In this paper, we show that, for skew-symmetric cluster algebras, the c-vectors of any seed with respect to an acyclic initial seed define a quasi-Cartan companion of the corresponding exchange matrix. As an application, we show that any cluster tilted quiver has an admissible cut of edges.