Geometric Realizations of the Accordion Complex of a Dissection

Consider 2n points on the unit circle and a reference dissection D◦ of the convex hull of the odd points. The accordion complex of D◦ is the simplicial complex of non-crossing subsets of the diagonals with even endpoints that cross an accordion of the dissection D◦. In particular, this complex is an associahedron when D◦ is a triangulation and a Stokes complex when D◦ is a quadrangulation. In this paper, we provide geometric realizations (by polytopes and fans) of the accordion complex of any reference dissection D◦, generalizing known constructions arising from cluster algebras. keywords. Permutahedra · Zonotopes · Associahedra · g-, cand d-vectors. The (n− 3)-dimensional associahedron is a polytope whose boundary complex is isomorphic to the reverse inclusion poset of non-crossing subsets of diagonals of a convex n-gon. Introduced in early works of D. Tamari [Tam51] and J. Stasheff [Sta63], it was first realized as a convex polytope by M. Haiman [Hai84] and C. Lee [Lee89], and later constructed by more systematic methods developed by several authors, in particular [GKZ08, Lod04, HL07, CSZ15]. Various relevant generalizations of the associahedron were introduced and studied, in particular secondary polytopes and fiber polytopes [GKZ08, BFS90], generalized associahedra [FZ03b, CFZ02, HLT11, Ste13, Hoh] in connection to cluster algebras [FZ02, FZ03a], graph associahedra [CD06, Pos09, FS05, Zel06, Pil13, MP16], or brick polytopes [PS12, PS15]. In a different context, Y. Baryshnikov [Bar01] introduced the simplicial complex of crossing-free subsets of the set of diagonals of a polygon that are in some sense compatible with a reference quadrangulation Q◦. Although the precise definition of compatibility is a bit technical in [Bar01], it turns out that a diagonal is compatible with Q◦ if and only if it crosses a connected subset of diagonals of Q◦ that we call accordion of Q◦. We thus call Y. Baryshnikov’s simplicial complex the accordion complex AC(Q◦). A polytopal realization of AC(Q◦) was announced in [Bar01], but the explicit construction and its proof were never published as far as we know. Revisiting some combinatorial and algebraic properties of AC(Q◦), F. Chapoton [Cha16] raised three explicit challenges: first prove that the oriented dual graph of AC(Q◦) has a lattice structure extending the Tamari and Cambrian lattices [MHPS12, Rea06]; second construct geometric realizations of AC(Q◦) as fans and polytopes generalizing the known constructions of the associahedron; third show that the facets of AC(Q◦) are in bijection with other combinatorial objects called serpent nests [Cha16]. In [GM16], A. Garver and T. McConville defined and studied the accordion complex AC(D◦) of any reference dissection D◦ (their presentation slightly differs as they use a compatibility condition on the dual tree of the dissection D◦, but the simplicial complex is the same). In this context, they settled F. Chapoton’s lattice question, using lattice quotients of a lattice of biclosed sets. In this paper, we present geometric realizations of AC(D◦) for any reference dissection D◦, providing in particular an answer to F. Chapoton’s geometric question. In fact, we present three methods to realize AC(D◦) based on constructions of the classical associahedron. Our first method is based on the g-vector fan. It belongs to a series of constructions of the (generalized) associahedra initiated by S. Shnider and S. Sternberg [SS93], popularised by J.-L. Loday [Lod04], developed by C. Hohlweg, C. Lange and H. Thomas [HL07, HLT11] using works of N. Reading and D. Speyer [Rea06, Rea07, RS09], and revisited by S. Stella [Ste13] and by V. Pilaud, F. Santos, and C. Stump [PS12, PS15]. It was recently extended by C. Hohlweg, V. Pilaud, and S. Stella [HPS17] 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 S. Fomin and A. Zelevinski [FZ07]. When D◦ is a triangulation, our definitions coincide with those given in terms of triangulations and laminations for cluster algebras from surfaces by S. Fomin and D. Thurston [FT12]. We then show that the g-vectors with respect to the dissection D◦ support a complete simplicial fan F(D◦) realizing the D◦-accordion Partially supported by the French ANR grant SC3A (15 CE40 0004 01). 1 2 THIBAULT MANNEVILLE AND VINCENT PILAUD complex AC(D◦). Finally, we construct a D◦-accordiohedron Acco(D◦) realizing the g-vector fan F(D◦) by deleting inequalities from the facet description of the D◦-zonotope Zono(D◦) obtained as the Minkowski sum of all c-vectors. See Figure 6 for an illustration of D◦-accordiohedra. Our second method is based on the d-vector fan. This construction is inspired from the original cluster fan of S. Fomin and A. Zelevinsky [FZ03a] later realized as a polytope by F. Chapoton, S. Fomin and A. Zelevinsky [CFZ02], and from the generalization of F. Santos [CSZ15] to construct a compatibility fan and an associahedron from any initial triangulation. For any reference dissection D◦, we associate to each diagonal a d-vector which records the crossings of this diagonal with those of D◦. We show that the d-vectors support a complete simplicial fan realizing the D◦-accordion complex AC(D◦) if and only if D◦ contains no even interior cell. The polytopality of the resulting fan remains open in general, but was shown for arbitrary triangulations in [CSZ15]. Finally, our third method is based on projections of associahedra. Namely, for any dissection D◦ and triangulation T◦ such that D◦ ⊆ T◦, the accordion complex AC(D◦) is a subcomplex of the simplicial associahedron AC(T◦). It turns out that the g-vector fan F(D◦) is then a section of the g-vector fan F(T◦) by a coordinate subspace. Therefore, the accordion complex AC(D◦) is realized by a projection of the associahedron Asso(T◦) of [HPS17]. 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. The paper is organized as follows. Section 1 introduces the accordion complex and accordion lattice of a dissection D◦. We essentially follow the definitions and arguments of A. Garver and T. McConville [GM16], except that we prefer to work on the dissection D◦ rather than on its dual graph. Section 2 is devoted to the generalization of the g-vector fan and the associahedra of [HL07, HPS17]. Section 3 discusses the generalization of the construction of the d-vector fan and associahedra of [FZ03a, CSZ15]. Finally, Section 4 shows that the accordion complex is realized by a projection of a well-chosen associahedron and presents related conjectures on cluster algebras, subcomplexes of the cluster complex, and sections of the g-vector fan. 1. The accordion complex and the accordion lattice In this section, we define the accordion complex AC(D◦) of a dissection D◦, show that it is a pseudo-manifold, and define an orientation of its dual graph. Our definitions and proofs are essentially translations of the arguments of A. Garver and T. McConville [GM16] given in terms of the dual tree of the dissection D◦. However our presentation in terms of dissections is more convenient for our latter purposes. 1.1. The accordion complex. Let P be a convex polygon. We call diagonals of P the segments connecting two vertices of P. This includes both the internal diagonals and the external diagonals (or boundary edges) of P. A dissection of P is a set D of non-crossing internal diagonals of P. The cells of D are the closures of the connected components of P minus the diagonals of D. We denote by D̄ the dissection D together with all boundary edges of P. An accordion of D is a subset of D̄ which contains either no or two consecutive diagonals in each cell of D. A subaccordion of D is a subset of D formed by the diagonals between two given internal diagonals in an accordion of D. A zigzag of D is a subset {δ0, . . . , δp+1} of D where δi shares distinct endpoints with and Figure 1. A dissection D (left) and three accordions whose zigzags are bolded (middle and right). GEOMETRIC REALIZATIONS OF THE ACCORDION COMPLEX OF A DISSECTION 3 Figure 2. A hollow dissection Dex , a solid D ex -accordion diagonal whose corresponding hollow accordion is bolded, and two maximal solid Dex -accordion dissections. separates i 1 and i +1 for any i 2 [p]. The zigzagof an accordion A is the subset of the diagonals of A that disconnect A. Note that we include boundary edges of P in the accordions of D, but not in the subaccordions nor in the zigzags of D. See Figure 1. We consider 2n points on the unit circle labeled clockwise by 1 , 2 , 3 , 4 , . . . , (2n 1) , (2n) . We say that 1 ; : : : ; (2n 1) are the hollow vertices while 2 ; : : : ; (2n) are the solid vertices. The hollow polygon is the convex hull P of 1 ; : : : ; (2n 1) while the solid polygon is the convex hull P of 2 ; : : : ; (2n) . We simultaneously considerhollow diagonals (with two hollow vertices) and solid diagonals (with two solid vertices), but we never consider diagonals with one hollow vertex and one solid vertex. Similarly, we considerhollow dissectionsD (of the hollow polygon, with only hollow diagonals) and solid dissectionsD (of the solid polygon, with only solid diagonals), but never mix hollow and solid diagonals in a dissection. To help distinguishing them, hollow (resp. solid) vertices and diagonals appear red (resp. blue) in all pictures. We x an arbitrary reference hollow dissection D . A solid diagonal is a D -accordion diagonal if the hollow diagonals of D crossed by form an accordion of D . In other words, cannot enter and exit a cell of D using two non-incident diagonals. For example, note that for any hollow diagonal i j 2 D , the solid diagonals (i 1) (j 1) and (i + 1) (j + 1) are D -accordion diagonals (here and throughout, labels are considered modulo 2 n). In particular, all boundary edges of the solid polygon are D -accordion diagonals. A D -accordion dissectionis a set of non-crossing internal D -accordion diagonals. We call D -accordion complexthe simplicial complex AC(D ) of D -accordion dissections. Example 1. As a running example, we consider the reference dissection D ex of Figure 2 (left). Examples of maximal Dex -accordion dissections are given in Figure 2 (right). The D ex -accordion complex is illustrated in Figure 3 (left). Remark 2. Special reference hollow dissections D give rise to special accordion complexes AC(D ): If D is the empty dissection with the whole hollow polygon as unique cell, then the D -accordion complexAC(D ) is reduced to the empty D -accordion dissection. If D has a unique internal diagonal, then the D -accordion complexAC(D ) is a segment. For a hollow triangulation T , all solid diagonals are T -accordions, so that the T -accordion complex AC(T ) is the simplicial associahedron. For a hollow quadrangulation Q , a solid diagonal is a Q -accordion if and only if it does not cross two opposite edges of a quadrangle of Q . The Q -accordion complexAC(Q ) is thus the Stokes complex de ned by Y. Baryshnikov [Bar01] and studied by F. Chapoton [Cha16]. Remark 3. Following the original de nition of the non-crossing complex of A. Garver and T. McConville [GM16], the accordion complex could equivalently be de ned in terms of the dual tree D? of D (with one node in each cell of D and one edge connecting two adjacent cells). For example, a diagonalu v is a D -accordion diagonal if and only if any two consecutive edges of the (unique) path between the leavesu? and v ? in D ? belong to the boundary of a face of the complement of D ? in the unit disk. The g-, cand d-vectors de ned in Section 2.1 could as well be de ned in terms of D? , but we nd more convenient to work directly with dissections, in particular in Section 3. 4 THIBAULT MANNEVILLE AND VINCENT PILAUD Figure 3. The D ◦ -accordion complex (left) and the D ex ◦ -accordion lattice (right), oriented from bottom to top, for the reference hollow dissection D ◦ of Figure 2 (left). Remark 4. Assume that D◦ has a cell C◦ containing p boundary edges of the hollow polygon P◦. Let C 1 ◦, . . . ,C p ◦ denote the p (possibly empty) connected components of the hollow polygon minus C◦. For i ∈ [p], let D◦ denote the dissection formed by the cell C◦ together with the cells of D◦ in C i ◦. Since no D◦-accordion can contain internal diagonals from distinct dissections D◦ and D j ◦ (with i 6= j), the D◦-accordion complex is the join of the D◦-accordion complexes: AC(D◦) = AC(D◦) ∗ · · · ∗ AC(D p ◦). In particular, we can do the following reductions: (i) If a non-triangular cell of D◦ has two consecutive boundary edges γ◦, δ◦ of the hollow polygon, then contracting γ◦ and δ◦ to a single boundary edge preserves the D◦-accordion complex. (ii) If a cell of D◦ has two non-consecutive boundary edges of the hollow polygon, then the D◦-accordion complex is a join of smaller accordion complexes. In all the examples of the paper, we therefore only consider dissections where any non-triangular cell of D◦ has at most one boundary edge. All our constructions work in general, but are just obtained as products or joins of the non-degenerate situation. Remark 5. The links in an accordion complex are joins of accordion complexes. Namely, consider a D◦-accordion dissection D• with cells C 1 •, . . . ,C p •. Let D i ◦ denote the hollow dissection obtained from D◦ by contracting all hollow boundary edges which do not cross C i •. Then the link of D• in AC(D◦) is isomorphic to the join AC(D◦) ∗ · · · ∗ AC(D p ◦). 1.2. Pseudo-manifold. We now prove that the accordion complex AC(D◦) is a pseudo-manifold, i.e. that it is: (i) pure: all maximal D◦-accordion dissections have as many diagonals as D◦, and (ii) thin: any codimension 1 simplex of AC(D◦) is contained in exactly two maximal D◦-accordion dissections. We follow the arguments of A. Garver and T. McConville [GM16] (except that they work on the dual tree of the dissection D◦). A much more concise but less instructive proof of the pseudomanifold property will be derived from geometric considerations in Remark 56. Recall that we denote by D̄◦ the set formed by D◦ together with all boundary edges of the hollow polygon. An angle u◦v◦w◦ of D̄◦ is a pair {u◦v◦, v◦w◦} of two consecutive diagonals of D̄◦ around a common vertex v◦, called apex. Note that D̄◦ has 2|D◦|+ n = 2|D̄◦| − n angles. We say that a solid vertex p• belongs to a hollow angle u◦v◦w◦ if it lies in the cone generated by the edges v◦u◦ and v◦w◦ of the angle. The main observation is given in the following statement. GEOMETRIC REALIZATIONS OF THE ACCORDION COMPLEX OF A DISSECTION 5