Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors

Abstract

We show that the mesh mutations are the minimal relations among the g-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 g-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 N. Arkani-Hamed, Y. Bai, S. He, and G. Yan in type A and of V. Bazier-Matte, G. Douville, K. Mousavand, H. Thomas and E. Yıldırım for acyclic initial seeds.Moreover, we use a similar approach to study the space of polytopal realizations of the g-vector fans of other generalizations of the associahedron. For non-kissing complexes (a.k.a. support τ -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. For graphical nested complexes, we show that the space of realizations of the nested fan is simplicial only in the case of the associahedron, and we describe its facet-defining inequalities in general.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 M. Auslander on minimal relations for Grothendieck groups of module categories.The type cone of the cluster fan of type A 2 .

Towards a categorification of scattering amplitudes

“…This furnishes two further sets, namely, [1,3], [4,6] and [1,4], [4,6] of projectives and hence two more terms to the scattering amplitude. The repeated set [1,3], [1,4] is not to be counted again. As a third example, let us consider the thirteenth diagram in Fig.…”

“…• is obtained again from the first one, by replacing [1,4] by [3,5]. The choice of projectives for the pseudo-periodic quiver is…”

“…This yields yet two more sets of projectives, namely, [1,3], [3,5] and [3,5], [1,5]. Proceeding in this manner we obtain all the sets of projectives of hearts of intermediate t-structures of the pseudo-periodic orbit category.…”

“…The ABHY programme provided a geometric way of writing scattering amplitudes of a biadjoint cubic scalar field theory in four dimensions as the volume form of the dual to the associahedron, a polyhedron associated to the combinatorics of grouping the scattering particles according to the vertices of Feynman diagrams they give rise to. This relationship between the scattering amplitudes and the associahedron led to the connection of the former with the algebraic geometry of Grassmannians, tropical geometry [2], cluster algebras [3,4] and representation theory of quivers and related subjects [5]. The ABHY scheme has been generalized in various directions.…”

“…have three sets of projectives {[1,3],[1,4]}, {[1,3],[1,5]} and {[1,4],[1,5]} arising from this. They contribute in Fig.12.…”