Polytopal realizations of finite type g-vector fans

Abstract

This paper shows the polytopality of any finite type g-vector fan, acyclic or not. In fact, for any finite Dynkin type Γ, we construct a universal associahedron Asso un (Γ) with the property that any g-vector fan of type Γ is the normal fan of a suitable projection of Asso un (Γ).

Minkowski decompositions for generalized associahedra of acyclic type

“…Constructing such generalized associahedra has been a fruitful area of mathematical research since the introduction of cluster algebras by S. Fomin and A. Zelevinsky in the early 2000s. We refer to [9,5,11,17,12] in this chronological order for some of the milestones and history. This paper is a continuation of [3] and builds on recent results from [2,1] and from [16].…”

Qué anidaedra son quitaedra?

“….) are easy to realize by fans that coarsen the normal fan of a zonotope (often the classical permutahedron, but also Coxeter permutahedra [6], or even other zonotopes [7,10]). Polytopal realization problems would thus be simpler if all these fans would be realized by removing the undesired facets of these zonotopes.…”

Geometric Realizations of the Accordion Complex of a Dissection

“…It belongs to a series of constructions of the (generalized) associahedra initiated by Shnider and Sternberg [41], popularised by Loday [29], developed by Hohlweg et al [25,26] using works of Reading and Speyer [38][39][40], and revisited by Stella [43] and by Pilaud et al [35,36]. It was recently extended by Hohlweg et al [27] to construct an associahedron parametrized by any initial triangulation. Here, we first extend to the D • -accordion complex AC(D • ) the g-vectors and c-vectors defined in the context of cluster algebras by Fomin and Zelevinski [19].…”

“…It turns out that the g-vector fan F g (D • ) is then a section of the g-vector fan F g (T • ) by a coordinate subspace. Therefore, the accordion complex AC(D • ) is realized by a projection of the associahedron Asso(T • ) of [27]. This point of view provides a complementary perspective on accordion complexes that leads on the one hand to more concise but less instructive proofs of combinatorial and geometric properties of the accordion complex (pseudomanifold, g-vector fan, accordiohedron), and on the other hand to natural extensions to coordinate sections of the g-vector fan in arbitrary cluster algebras.…”

“…We essentially follow the definitions and arguments of Garver and McConville [20], except that we prefer to work on the dissection D • rather than on its dual graph. Section 3 is devoted to the generalization of the g-vector fan and the associahedra of [25,27]. Section 4 discusses the generalization of the construction of the d-vector fan and associahedra of [8,17].…”