Mutations of group species with potentials and their representations. Applications to cluster algebras

Abstract

This article tries to generalize former works of Derksen, Weyman and Zelevinsky about skew-symmetric cluster algebras to the skew-symmetrizable case. We introduce the notion of group species with potentials and their decorated representations. In good cases, we can define mutations of these objects in such a way that these mutations mimic the mutations of seeds defined by Fomin and Zelevinsky for a skew-symmetrizable exchange matrix defined from the group species. These good cases are called non-degenerate. Thus, when an exchange matrix can be associated to a non-degenerate group species with potential, we give an interpretation of the $F$-polynomials and the $\g$-vectors of Fomin and Zelevinsky in terms of the mutation of group species with potentials and their decorated representations. Hence, we can deduce a proof of a serie of combinatorial conjectures of Fomin and Zelevinsky in these cases. Moreover, we give, for certain skew-symmetrizable matrices a proof of the existance of a non-degenerate group species with potential realizing this matrix. On the other hand, we prove that certain skew-symmetrizable matrices can not be realized in this way.

New version : a few small modification

References in corpus (1)

• Quivers with potentials and their representations II: Applications to cluster algebras

Cited by in corpus (12)

• Cluster algebras and triangulated orbifolds
• Universal geometric cluster algebras
• Combinatorial frameworks for cluster algebras
• Polytopal realizations of finite type $\mathbf{g}$-vector fans
• C-vectors via $\tau$-tilting theory
• A survey on maximal green sequences
• Dual canonical bases and quantum cluster algebras
• Cambrian frameworks for cluster algebras of affine type
• On Maximal Green Sequences
• Some Consequences of Categorification
• Strongly primitive species with potentials I: Mutations
• Modulation and natural valued quiver of an algebra