Generalized cluster complexes via quiver representations

We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. Using d-cluster categories defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a d-compatibility degree (−∥−) on any pair of “colored” almost positive real Schur roots which generalizes previous definitions on the noncolored case and call two such roots compatible, provided that their d-compatibility degree is zero. Associated to the root system Φ corresponding to the valued quiver, using this compatibility relation, we define a simplicial complex which has colored almost positive real Schur roots as vertices and d-compatible subsets as simplices. If the valued quiver is an alternating quiver of a Dynkin diagram, then this complex is the generalized cluster complex defined by Fomin and Reading. Supported by the NSF of China (Grants 10471071) and by the Leverhulme Trust through the network ‘Algebras, Representations and Applications’.