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

Abstract

We 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.

ABHY Associahedra and Newton polytopes of F$F$‐polynomials for cluster algebras of simply laced finite type

“…After the appearance of the first version of this paper in 2018, Palu, Padrol, Pilaud, and Plamondon [26] showed that a very similar construction can be applied to construction of generalized associahedra for all seeds in all Dynkin‐type cluster algebras. The first author of the present paper also extended the techniques in the present paper in her thesis [5], giving another proof that the same construction can be applied to any seed of simply laced Dynkin type, and that this construction also gives Newton polytopes of $$F$$‐polynomials in the same generality.…”

Deformed Graphical Zonotopes

“…The search of irredundant facet descriptions of deformation cones of particular families of combinatorial polytopes has received considerable attention recently [3,7,9,10,12,32,33]. One of the motivations sparking this interest arises from the amplituhedron program to study scattering amplitudes in mathematical physics [6].…”

“…One of the motivations sparking this interest arises from the amplituhedron program to study scattering amplitudes in mathematical physics [6]. As described in [33,Sec. 1.4], the deformation cone provides canonical realizations of a polytope (seen as a positive geometry [5]) in the positive region of the kinematic space, akin to those of the associahedron in [4].…”