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A Gabriel‐type theorem for cluster tilting

We study the relationship between n‐cluster tilting modules over n representation finite algebras and the Euler forms. We show that the dimension vectors of cluster‐indecomposable modules give the roots of the Euler form. Moreover, we show that cluster‐indecomposable modules are uniquely determined by their dimension vectors. This is a generalization of Gabriel's theorem by cluster tilting theory. We call the above roots cluster‐roots and investigate their properties. Furthermore, we provide the description of quivers with relations of n‐Auslander‐Platzeck‐Reiten tilts. Using this, we provide a generalization of Bernstein‐Gelfand‐Ponomarev reflection functors.