On the c-vectors of an acyclic cluster algebra

TL;DR

This paper establishes a precise correspondence between the c-vectors of an acyclic cluster algebra and the real Schur roots of its associated root system, deepening the understanding of the algebra's combinatorial and representation-theoretic structure.

Contribution

It proves that c-vectors in an acyclic cluster algebra exactly match the real Schur roots and their opposites, clarifying their algebraic and geometric relationships.

Findings

C-vectors coincide with real Schur roots and their opposites.

Provides a complete characterization of c-vectors in acyclic cluster algebras.

Enhances understanding of the link between cluster algebras and root systems.

Abstract

We prove that the set of c-vectors of the cluster algebra associated to an acyclic quiver Q coincides with the set of real Schur roots and their opposites in the root system associated to Q.