Associahedra for finite‐type cluster algebras and minimal relations between g‐vectors

Abstract

AbstractWe show that the mesh mutations are the minimal relations among the ‐vectors with respect to any initial seed in any finite‐type cluster algebra. We then use this algebraic result to derive geometric properties of the ‐vector fan: we show that the space of all its polytopal realizations is a simplicial cone, and we then observe that this property implies that all its realizations can be described as the intersection of a high‐dimensional positive orthant with well‐chosen affine spaces. This sheds a new light on and extends earlier results of Arkani‐Hamed, Bai, He, and Yan in type  and of Bazier‐Matte, Chapelier‐Laget, Douville, Mousavand, Thomas, and Yıldırım for acyclic initial seeds. Moreover, we use a similar approach to study the space of polytopal realizations of the ‐vector fans of another generalization of the associahedron: nonkissing complexes (also known as support ‐tilting complexes) of gentle algebras. We show that the space of realizations of the nonkissing fan is simplicial when the gentle bound quiver is brick and 2‐acyclic, and we describe in this case its facet‐defining inequalities in terms of mesh mutations. Along the way, we prove algebraic results on 2‐Calabi–Yau triangulated categories, and on extriangulated categories that are of independent interest. In particular, we prove, in those two setups, an analogue of a result of Auslander on minimal relations for Grothendieck groups of module categories.