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

TL;DR

This paper extends the theory of cluster algebras to skew-symmetrizable cases using group species with potentials, providing new interpretations of key invariants and proving several conjectures.

Contribution

It introduces group species with potentials and their mutations, generalizing previous skew-symmetric results to skew-symmetrizable matrices, and proves the existence or non-existence of such structures for certain matrices.

Findings

Provides a framework to interpret F-polynomials and g-vectors via mutations of group species.

Proves the existence of non-degenerate group species with potentials for some skew-symmetrizable matrices.

Identifies classes of matrices that cannot be realized by these group species with potentials.

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.