GEOMETRIC REALIZATIONS OF THE ACCORDION COMPLEX OF A DISSECTION

THIBAULT MANNEVILLE AND VINCENT PILAUD Abstract. 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 a connected subset of diagonals of 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 simple polytope whose face poset 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 [START_REF] Tamari | Monoides préordonnés et chaînes de Malcev[END_REF] and J. Stasheff [START_REF] Stasheff | Homotopy associativity of H-spaces I[END_REF], it was first realized as a convex polytope by M. Haiman [START_REF] Haiman | Constructing the associahedron[END_REF] and C. Lee [START_REF] Lee | The associahedron and triangulations of the n-gon[END_REF], and later constructed by more systematic methods developed by several authors, in particular [START_REF] Gelfand | Discriminants, resultants and multidimensional determinants[END_REF][START_REF] Loday | Realization of the Stasheff polytope[END_REF][START_REF] Hohlweg | Realizations of the associahedron and cyclohedron[END_REF][START_REF] Ceballos | Many non-equivalent realizations of the associahedron[END_REF]. Various relevant generalizations of the associahedron were introduced and studied, in particular secondary polytopes and fiber polytopes [START_REF] Gelfand | Discriminants, resultants and multidimensional determinants[END_REF][START_REF] Billera | Constructions and complexity of secondary polytopes[END_REF], generalized associahedra [START_REF] Fomin | Y -systems and generalized associahedra[END_REF][START_REF] Chapoton | Polytopal realizations of generalized associahedra[END_REF][START_REF] Hohlweg | Permutahedra and generalized associahedra[END_REF][START_REF] Stella | Polyhedral models for generalized associahedra via Coxeter elements[END_REF][START_REF] Hohlweg | Permutahedra and associahedra[END_REF] in connection to cluster algebras [START_REF] Fomin | Cluster algebras. I. Foundations[END_REF][START_REF] Fomin | Cluster algebras. II. Finite type classification[END_REF], graph associahedra [CD06, Pos09, FS05, Zel06, Pil13, MP17], or brick polytopes [START_REF] Pilaud | The brick polytope of a sorting network[END_REF][START_REF] Pilaud | Brick polytopes of spherical subword complexes and generalized associahedra[END_REF].

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 [START_REF] Baryshnikov | On Stokes sets[END_REF], 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 [START_REF] Baryshnikov | On Stokes sets[END_REF], 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 [START_REF] Chapoton | Stokes posets and serpent nests[END_REF]Intro. p.4] raised three explicit challenges: first prove that the oriented dual graph of AC(Q • ) has a lattice structure extending the Tamari and Cambrian lattices [START_REF]Associahedra, Tamari Lattices and Related Structures[END_REF][START_REF] Reading | Cambrian lattices[END_REF]; 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 [START_REF] Chapoton | Stokes posets and serpent nests[END_REF]Sect. 4].

In [START_REF] Garver | Oriented flip graphs and noncrossing tree partitions[END_REF], 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 [START_REF] Shnider | Quantum groups: From coalgebras to Drinfeld algebras[END_REF], popularised by J.-L. Loday [START_REF] Loday | Realization of the Stasheff polytope[END_REF], developed by C. Hohlweg, C. Lange and H. Thomas [START_REF] Hohlweg | Realizations of the associahedron and cyclohedron[END_REF][START_REF] Hohlweg | Permutahedra and generalized associahedra[END_REF] using works of N. Reading and D. Speyer [START_REF] Reading | Cambrian lattices[END_REF][START_REF] Reading | Sortable elements and Cambrian lattices[END_REF][START_REF] Reading | Cambrian fans[END_REF], and revisited by S. Stella [START_REF] Stella | Polyhedral models for generalized associahedra via Coxeter elements[END_REF] and by V. Pilaud, F. Santos, and C. Stump [START_REF] Pilaud | The brick polytope of a sorting network[END_REF][START_REF] Pilaud | Brick polytopes of spherical subword complexes and generalized associahedra[END_REF]. It was recently extended by C. Hohlweg, V. Pilaud, and S. Stella [START_REF] Hohlweg | Polytopal realizations of finite type g-vector fans[END_REF] 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 [START_REF] Fomin | Cluster algebras[END_REF]. Note that cvectors were already implicitly considered in [START_REF] Garver | Oriented flip graphs and noncrossing tree partitions[END_REF], while g-vectors are new in this context. 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 [START_REF] Fomin | Cluster algebras and triangulated surfaces. part II: Lambda lengths[END_REF]. We then show that the g-vectors with respect to the dissection D • support a complete simplicial fan F g (D • ) realizing the D • -accordion complex AC(D • ). Finally, we construct a D •accordiohedron Acco(D • ) realizing the g-vector fan F g (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 7 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 [START_REF] Fomin | Cluster algebras. II. Finite type classification[END_REF] later realized as a polytope by F. Chapoton, S. Fomin and A. Zelevinsky [START_REF] Chapoton | Polytopal realizations of generalized associahedra[END_REF], and from the generalization of C. Ceballos, F. Santos and G. Ziegler [START_REF] Ceballos | Many non-equivalent realizations of the associahedron[END_REF] 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 [START_REF] Ceballos | Many non-equivalent realizations of the associahedron[END_REF].

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 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 [START_REF] Hohlweg | Polytopal realizations of finite type g-vector fans[END_REF]. 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.

As recently observed in [GM16, PPP17, PPS17, BDM + 17], accordion complexes are prototypes of support τ -tilting complexes introduced in [AIR14], for certain associative algebras called gentle algebras. In this context, g-vectors have a deep algebraic meaning and still define a g-vector fan. Although this fan is still polytopal for finite support τ -tilting complexes, it is not in general obtained by deleting inequalities in the facet description of a zonotope. We refer to [PPP17, Part 4] for details.

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 [START_REF] Garver | Oriented flip graphs and noncrossing tree partitions[END_REF], 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 [START_REF] Hohlweg | Realizations of the associahedron and cyclohedron[END_REF][START_REF] Hohlweg | Polytopal realizations of finite type g-vector fans[END_REF]. Section 3 discusses the generalization of the construction of the d-vector fan and associahedra of [START_REF] Fomin | Cluster algebras. II. Finite type classification[END_REF][START_REF] Ceballos | Many non-equivalent realizations of the associahedron[END_REF]. Finally, Section 4 shows that the accordion complex is realized by a projection of a well-chosen associahedron and presents related questions on cluster algebras, subcomplexes of the cluster complex, and sections of the g-vector fan.

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 pseudomanifold, and define an orientation of its dual graph. Our definitions and proofs are essentially translations of the arguments of A. Garver and T. McConville [START_REF] Garver | Oriented flip graphs and noncrossing tree partitions[END_REF] given in terms of the dual tree of the dissection D • . However our presentation in terms of dissections is more convenient for our 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. A triangulation (resp. quadrangulation) is a dissection whose cells are all triangles (resp. quadrangles). We denote by D the dissection D together with all boundary edges of P. A cut of D is the subset of D intersected by a line crossing two boundary edges of P. An accordion is a connected cut. By definition, an accordion is a tree and contains two boundary edges of P. The zigzag of an accordion A is the chain obtained by deleting all degree 1 vertices of A. A subaccordion of D is a connected subset of D intersected by a segment in the interior of P. Note that any subaccordion of an accordion A consists of the diagonals of A between two internal diagonals of 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 consider hollow diagonals δ • (with two hollow vertices) and solid diagonals δ • (with two solid vertices), but we never consider diagonals with one hollow vertex and one solid vertex. Similarly, we consider hollow dissections D • (of the hollow polygon, with only hollow diagonals) and solid dissections D • (of the solid polygon, with only solid diagonals), but never mix hollow and solid diagonals in a dissection. To help distinguish them, hollow (resp. solid) vertices and diagonals appear red (resp. blue) in all pictures.

We fix an arbitrary reference hollow dissection D • . A solid diagonal δ • is a D • -accordion diagonal if the hollow diagonals of D • crossed by δ • form an accordion of D • . In other words, δ • cannot enter and exit a cell of D • using two non-incident diagonals. For example, note that for any hollow diagonal i • j • ∈ D • , the solid diagonals (i -1) • (j -1) • and (i + 1) • (j + 1) • are D • -accordion diagonals (here and throughout, labels are considered modulo 2n). In particular, all boundary edges of the solid polygon are D • -accordion diagonals. A D • -accordion dissection is a set of non-crossing internal D • -accordion diagonals. We define the D • -accordion complex to be the simplicial complex AC(D • ) of D • -accordion dissections. 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 complex AC(Q • ) is thus the Stokes complex defined by Y. Baryshnikov [Bar01] and studied by F. Chapoton [START_REF] Chapoton | Stokes posets and serpent nests[END_REF].

Remark 3. Following the original definition of the non-crossing complex of A. Garver and T. Mc-Conville [START_REF] Garver | Oriented flip graphs and noncrossing tree partitions[END_REF], the accordion complex could equivalently be defined in terms of the dual tree D • of D • (with one node in each cell of D and one edge connecting two adjacent cells). More precisely, the duality provides the following dictionary between the two definitions:

present paper A. Garver and T. McConville [GM16] reference dissection D • ←→ embedded tree D • diagonal u • v • of P • ←→ path connecting the leaves u • and v • of D • D • -accordion diagonal ←→
arc (path where any two consecutive edges belong to the boundary of a face of the complement of D

• in the unit disk) D • -subaccordion ←→ segment D • -accordion complex ←→ non-crossing complex of D •
The g-, cand d-vectors defined in Section 2.1 could as well be defined in terms of D • . In fact, c-vectors were already implicitly considered in [START_REF] Garver | Oriented flip graphs and noncrossing tree partitions[END_REF], while gand d-vectors are new in this context. For this paper, we find more convenient to work directly with dissections, in particular in Sections 3 and 4.

1.2. Two structural observations. Before studying the accordion complex in details in Section 1.3, we present two simple structural observations. For this, let us recall two classical notions on simplicial complexes. The join of two simplicial complexes ∆, ∆ with disjoint ground sets X, X is the simplicial complex ∆ * ∆ with ground set X X whose faces are disjoint unions of faces of ∆ with faces of ∆ . For a face D in a simplicial complex ∆ on X, the link of D is the simplicial complex on X D whose faces are the subsets D of X D such that D ∪ D is a face of ∆. [START_REF] Garver | Oriented flip graphs and noncrossing tree partitions[END_REF] (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 60.

Recall that we denote by D • the set formed by D • together with all boundary edges of the hollow polygon. An angle u

• v • w • of D • is a pair {u • v • , v • w • } of two consecutive diagonals of D • around a common vertex v • , called the apex. Note that D • has 2|D • | + n = 2|D • | -n angles.
Observe also that an accordion A • of D • can be seen as a sequence of |A • | -1 angles where two consecutive angles are separated by a diagonal of A • . We say that a solid vertex p • belongs to an 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.

Lemma 7. Let D • be a maximal D • -accordion dissection, and let p

• , q • , r • , s • denote four consec- utive vertices of a cell C • of D • (with possibly p • = s • if C • is a triangle).
Then p • and s • belong to the same angle of the accordion of D • which is crossed by q • r • .

Proof. Let A • be the accordion of D • which is crossed by q • r • . Assume that p • and s • belong to distinct angles of A • . Then they are separated by a diagonal ε • of A • . Therefore, there are two boundary edges q • r • and u

• v • of C • with distinct vertices such that the hollow diagonal ε • separates the vertices q • , u • from the vertices r • , v • . Let γ 1 • , . . . , γ i • = ε • , . . . , γ a • (resp. δ 1 • , . . . , δ j • = ε • , . . . , δ b • )
increasing flip denote the diagonals of D • crossed by q

• r • from q • to r • (resp. crossed by u • v • from u • to v • ). Then the hollow diagonals γ 1 • , . . . , γ i • = ε • = δ j • , . . . , δ b • which are crossed by q • v • also form an accordion. It follows that D • is not maximal as we can still include q • v • . Consider now an angle u • v • w • of D • . In any maximal D • -accordion dissection D • , the set X • of diagonals of D • that cross both u • v • and v • w • is non-empty (since it contains the boundary edge (v -1) • (v + 1) • ) and totally ordered (since the diagonals of D • do not cross). Let δ • be the largest diagonal of X • (meaning the farthest from v • ). We say that the diagonal δ • closes the angle u • v • w • . Note that each angle of D • is closed by precisely one diagonal of D • .
The following lemma is stated and proved in [START_REF] Garver | Oriented flip graphs and noncrossing tree partitions[END_REF] Proof. The first sentence is a consequence of Lemma 7: for any four consecutive vertices p • , q • , r • , s • of a cell of D • , the diagonal q • r • closes the unique angle of the accordion of D • crossed by q • r • that contains the vertices p • and s • . Therefore, q • r • closes precisely two angles (resp. one angle) of D • if it is an internal diagonal (resp. a boundary edge of the solid polygon). We finally obtain by double-counting that 2|D

• | + n = |{angles of D • }| = 2|D • | + n and thus |D • | = |D • | for any maximal D • -accordion dissection D • .
We are now ready to prove that the D • -accordion complex is thin, i.e. that each internal diagonal of a maximal D • -accordion dissection can be flipped into a unique other internal diagonal to form a new maximal D • -accordion dissection. Here and throughout the paper, X Y denotes the symmetric difference of two sets X, Y defined by X

Y := (X Y ) ∪ (Y X).
The following notations are illustrated in 1.4. The accordion lattice. We now define a natural orientation on the D • -accordion flip graph. We use the same notations as in Lemma 9 (see also Figure 4), where D In particular, the D • -accordion oriented flip graph is connected and acyclic, and has a unique source D -

• {δ • } = D • {δ • } and δ • , δ • are the two diagonals of the quadrilateral defined by µ • , ν • . Observe that one of the path µ • δ • ν • and µ • δ • ν • forms a Z while
• := {(i -1) • (j -1) • | i • j • ∈ D • } (obtained by slightly rotating D • counterclockwise) and a unique sink D + • := {(i + 1) • (j + 1) • | i • j • ∈ D • } (obtained
(D • ) = AL(D 1 • ) × • • • × AL(D p • ).
In particular, if two consecutive boundary edges γ • , δ • of the hollow polygon belong to the same non-triangular cell of D • , then contracting γ • and δ • to a single boundary edge preserves the D • -accordion lattice. This shows the following statement conjectured for quadrangulations in [START_REF] Chapoton | Stokes posets and serpent nests[END_REF] and proved in [START_REF] Amir-Hossein | A note on quadrangulations and Stokes complexes[END_REF].

Corollary 13. Consider an accordion dissection A • , i.e. a dissection where each cell has at most 2 edges which are internal diagonals of P • . Then the A • -accordion lattice is a Cambrian lattice.

Remark 14. Call cell-sequence of a dissection the sequence whose ith entry is its number of (i + 2)-cells. For example, the dissection of Figure 2 (left) has cell-sequence 3, 1, 0 ∞ and all (p+2)-angulations of a (pm+2)-gon have cell-sequence 0 p-1 , m, 0 ∞ . Observe that the flip preserves the cell-sequence. Thus, all maximal D • -accordion dissections have the same cell-sequence as D • .

We conclude this section with a reciprocity result on accordion dissections.

Proposition 15. Let D • be a hollow dissection and D • be a solid dissection. Then D • is a maximal D • -accordion dissection if and only if D • is a maximal D • -accordion dissection. Proof. Since D - • := {(i -1) • (j -1) • | i • j • ∈ D • } and D + • := {(i + 1) • (j + 1) • | i • j • ∈ D • } are both D • -accordion dissections, we already know that D • is a D - • -accordion dissection. Observe now in Figure 4 that if D • and D • are maximal D • -accordion dissections connected by a flip, then D • is a D • -accordion dissection if and only if it is a D • -accordion dissection. Indeed, if δ • belongs to the zigzag of the D • -accordion A • of a hollow diagonal δ • , then δ • crosses both µ • and ν • , but then δ • also crosses δ • , and thus δ • crosses the D • -accordion A • {δ • , δ • }. Since the D • -accordion flip graph is connected, we obtain that D • is a D • -

The g-vector fan

In this Section, we construct accordiohedra using gand c-vectors. Our construction is in the same spirit as the Cambrian fans of N. Reading and D. Speyer [Rea06, Rea07, RS09] and their polytopal realizations by C. Hohlweg, C. Lange and H. Thomas [START_REF] Hohlweg | Realizations of the associahedron and cyclohedron[END_REF][START_REF] Hohlweg | Permutahedra and generalized associahedra[END_REF], recently extended in [START_REF] Hohlweg | Polytopal realizations of finite type g-vector fans[END_REF] to any initial triangulation, acyclic or not. A different approach to the g-vector fan together with an alternative polytopal realization will be presented in Section 4.

• ∈ D • . When δ • crosses δ • , we let µ • and ν • be the other diagonals of D • crossed by δ • in the two cells of D • containing δ • . We say that δ • slaloms on δ • if µ • δ • ν • forms a path, and we define ε • δ • ∈ D • | δ • to be 1, -1, or 0 depending on whether µ • δ • ν • forms a Z, a Z, or a V I . Similarly we let µ • and ν • be the other diagonals of D • crossed by δ • in the two cells of D • containing δ • , we say that δ • slaloms on δ • if µ • δ • ν • forms a path, and we define ε • δ • | δ • ∈ D • to be 1, -1, or 0 depending on whether µ • δ • ν • forms a Z, a Z, or a V I . Note that the sign convention for ε • δ • ∈ D • | δ • and ε • δ • | δ • ∈ D •
is opposite: the reciprocity already observed in Proposition 15 naturally reverses the orientation. More informally, we exchange the role of hollow and solid dissections by looking at the picture from the opposite side of the blackboard, which of course reverses the orientation. Finally, if δ • and δ • do not cross, then we let ε

• δ • ∈ D • | δ • = ε • δ • | δ • ∈ D • = 0. Let (e δ• ) δ•∈D• denote the canonical basis of R D• .
As in [START_REF] Hohlweg | Polytopal realizations of finite type g-vector fans[END_REF], we define the following vectors:

(i) the g-vector of δ

• with respect to D • is g D • | δ • := δ•∈D• ε • δ • ∈ D • | δ • e δ• . We also define g D • | D • := g D • | δ • δ • ∈ D • . (ii) the c-vector of δ • ∈ D • with respect to D • is c D • | δ • ∈ D • := δ•∈D• ε • δ • | δ • ∈ D • e δ• . We denote by c D • | D • := c D • | δ • ∈ D • δ • ∈ D • the set of c-
= {3 • 7 • , 3 • 13 • , 9 • 13 • } and the rightmost solid dissection D ex • = {2 • 6 • , 2 • 10 • , 10 • 14 • } of Figure 2. Then we have for example ε • 3 • 13 • ∈ D ex • | 2 • 10 • = 1 since the path 1 • -3 • -13 • -9 • forms a Z, ε • 9 • 13 • ∈ D ex • | 2 • 10 • = -1 since the path 3 • -13 • -9 • -11 • forms a Z, and ε • 3 • 13 • ∈ D ex • | 2 • 6 • = 0 since 3 • connects 1 • , 13 • , 7 • as a V I .
Moreover, we have

g D ex • | 2 • 6 • = e 3•7• , c D ex • | 2 • 6 • ∈ D ex • = e 3•7• , g D ex • | 2 • 10 • = e 3•13• -e 9•13• , c D ex • | 2 • 10 • ∈ D ex • = e 3•13• , g D ex • | 10 • 14 • = -e 9•13• , c D ex • | 10 • 14 • ∈ D ex • = -e 3•13• -e 9•13• .
Example 17. For any hollow diagonal i

• j • ∈ D • , we have g D • | (i -1) • (j -1) • = -e i•j• , c D • | (i -1) • (j -1) • ∈ D - • = -e i•j• , g D • | (i + 1) • (j + 1) • = e i•j• , c D • | (i + 1) • (j + 1) • ∈ D + • = e i•j• . Remark 18.
For a hollow triangulation T • , our definitions of gand c-vectors coincide with the shear coordinates of S. Fomin and D. Thurston [START_REF] Fomin | Cluster algebras and triangulated surfaces. part II: Lambda lengths[END_REF], defined in the much more general context of cluster algebras on surfaces [START_REF] Fomin | Cluster algebras and triangulated surfaces I. Cluster complexes[END_REF]. 5 for a schematic illustration. Observe that there is an even (resp. odd) number of hollow diagonals of Z • that slalom on δ • when δ

• := {(i -1) • (j -1) • | i • j • ∈ D • } (resp. not in the sink dissection D + • := {(i + 1) • (j + 1) • | i • j • ∈ D • }). Proof. By Remark 20 (ii), the support of any c-vector is a subaccordion of D • . Reciprocally, let A • be a subaccordion of D • , let C • and C • denote the two cells of D • containing exactly one diagonal of A • , and let p • , q • , r • , s • (resp. p • , q • , r • , s • ) denote the four consecutive vertices in clockwise order around C • (resp. around C • ) such that q • r • (resp. q • r • ) is the diagonal of A • in C • (resp. in C • ). Let δ • := (s -1) • (s -1) • , µ • := (p + 1) • (s -1) • and ν • := (p + 1) • (s -1) • and consider any D • -accordion dissection D • containing {µ • , δ • , ν • }. Then A • is precisely the support of the negative c-vector c(D • | δ • ∈ D • ). Finally, we have associated to the subaccordion A • of D • a D • -diagonal δ • = (s -1) • (s -1) • which cannot be in D - • as otherwise s • s • would cross q • r • . Reciprocally, A • is
• = γ • (resp. when δ • = γ • ).
Moreover, since they are non-crossing, all hollow diagonals of Z • slaloming on δ • do it the same way (either all as a Z or all as a Z). Finally, when γ • = δ • , consider the first hollow diagonal δ • of the zigzag Z • which slaloms on δ • . Then δ • slaloms on δ • in the opposite way as δ • slaloms on δ • . This shows that

g D • | γ • c D • | δ • ∈ D • = δ•∈D• ε • δ • ∈ D • | γ • • ε • δ • | δ • ∈ D • = 1 1 γ=δ ,
g D • | D • = -c D • | D • t and c D • | D • = -g D • | D • t ,
where we consider the sets of g-vectors g(D

• | D • ) and c-vectors c(D • | D • ) as matrices in R D•×D•
, and M t denotes the transpose of a matrix M .

Proof. We immediately derive from the definitions that for any δ

• ∈ D • and δ • ∈ D • , g D • | D • (δ•,δ•) = ε • δ • ∈ D • | δ • = -ε • δ • | δ • ∈ D • = -c D • | D • (δ•,δ•) , which shows g(D • | D • ) = -c(D • | D • ) t .
Zono(A • ) = k∈[|A•|+1] k(|A • | + 1 -k) [-e δ k • , e δ k • ] = 2 Perm(|A • |) -(|A • | + 2) i∈[|A•|+1] f i ,
where

Perm(|A • |) := conv i∈[|A•|+1] σ(i) f i σ ∈ S |A•|+1 is the classical permutahedron.
The vertices of Zono(D • ) correspond to separable subsets of C(D • ), i.e. those which can be strictly separated from their complement by a hyperplane. Although we could work out all facets of Zono(D • ), we will only need the following specific inequalities.

Proposition 26. For any D • -accordion diagonal γ • , the D • -zonotope Zono(D • ) has a facet defined by the inequality

g D • | γ • x ≤ ω D • | γ • ,
where

ω(D • | γ • ) is the D • -height of γ • , i.e. the number of D • -accordion diagonals that cross γ • . Proof. Let ω(D • | γ • ) denote the maximum of g(D • | γ • ) | x over Zono(D • ). As Zono(D • ) is the Minkowski sum of all c-vectors, we have ω D • | γ • = c∈C(D•) g(D• | γ•) | c >0 g D • | γ • c .
• = Z - • Z + • such that g(D • | γ • ) = 1 1 Z + • -1 1 Z - • (where 1 1 X• := δ•∈X• e δ• for X • ⊆ D • ). Let δ • be a D • -accordion diagonal. Let A - • (resp. A + • ) denote the accordion crossed by δ • = u • v • in D • and not including (u + 1) • or (v + 1) • (resp. (u -1) • or (v -1) • ). Let c -(δ • ) := -1 1 A - • and c + (δ • ) := 1 1 A + • . Recall from Lemma 21 that the negative (resp. positive) c-vectors of C(D • ) are given by c -(δ • ) (resp. c + (δ • )) for all D • -accordion diagonal δ • not in D - • (resp. D + • ).
We let the reader check that:

If γ • and δ • do not cross and have no common endpoint, both

|Z • ∩ A - • | and |Z • ∩ A + • | are even. Thus g(D • | γ • ) | c -(δ • ) = g(D • | γ • ) | c + (δ • ) = 0. If γ • and δ • have a common endpoint, and γ • δ • form a counterclockwise angle, then |Z • ∩ A - • | is even while Z • ∩ A +
• is empty or starts and ends in Z +

• . Thus g(D • | γ • ) | c -(δ • ) = 0 while g(D • | γ • ) | c + (δ • ) ≥ 0. The situation is similar if γ • δ • form a clockwise angle. If γ • and δ • cross, Z • ∩ A - • and Z • ∩ A +
• are empty or start and end both in Z - • or both in Z +

• . Thus, either g(D

• | γ • ) | c -(δ • ) < 0 and g(D • | γ • ) | c + (δ • )
≥ 0 or conversely. We conclude from this case analysis that

ω(D • | γ • ) = | {c ∈ C(D • ) | g(D • | γ • ) | c < 0} | = |{D • -accordion diagonals crossing γ • }|. Finally, the inequality g(D • | γ • ) | x ≤ ω(D • | γ • ) defines a priori a face F(γ • ) of the zono- tope Zono(D • ). This face F(γ • ) is the Minkowski sum of the c-vectors of C(D • ) orthogonal to g(D • | γ • ). Proposition 22 ensures that any D • -accordion dissection D • containing γ • already provides |D • |-1 linearly independent such c-vectors c(D • | δ • ∈ D • ) for δ • ∈ D • {γ • }. We obtain that F(γ • ) has dimension |D • |-1 = |D • |-1 and is therefore a facet of the zonotope Zono(D • ).
Define the half-space and the hyperplane corresponding to a solid D • -accordion diagonal γ • by

H ≤ D • | γ • := x ∈ R D• g D • | γ • x ≤ ω D • | γ • ,
and

H = D • | γ • := x ∈ R D• g D • | γ • x = ω D • | γ • .
2.3. g-vector fan and D • -accordiohedron. In this section, we give a geometric realization of the D • -accordion complex. We start by realizing this simplicial complex as a complete simplicial fan in R D• . We denote by R ≥0 R the nonnegative span of a set R of vectors in R D• .

Theorem 27. The collection of cones

F g (D • ) := R ≥0 g D • | D • D • any D • -
• -accordion dissections D • , D • . Let δ • ∈ D • and δ • ∈ D • be such that D • {δ • } = D • {δ • },
and let µ • and ν • be the other diagonals as in Lemma 9 (see also Figure 4). Note that a diagonal of D • crosses none of (resp. one of, resp. both) the diagonals δ • , δ • if and only if it crosses none of (resp. one of, resp. both) the diagonals µ • , ν • . The same holds for a Z or a Z of D • . Therefore, we have the linear dependence g(D

• | δ • ) + g(D • | δ • ) = g(D • | µ • ) + g(D • | µ • )
. This shows that F g (D • ) satisfies the two conditions of Proposition 28, and thus concludes the proof.

Remark 29. The linear dependence g(D Example 31. Following Example 2, we observe that special reference dissections give rise to the following relevant fans:

• | δ • ) + g(D • | δ • ) = g(D • | µ • ) + g(D • | µ • ) relating the g-vectors of two adjacent maximal D • -accordion dissections D • , D • with D • {δ • } = D • {δ • } shows that det g(D • | D • ) = -det g(D • | D • ) . Since the initial cone R ≥0 g(D • | D - • ) is
For an accordion triangulation A • (i.e. with no interior triangle), the g-vector fan F g (A • ) coincides with a type A Cambrian fan of N. Reading and D. Speyer [START_REF] Reading | Cambrian fans[END_REF].

For an arbitrary triangulation T • (with or without interior triangle), the g-vector fan F g (T • ) was recently constructed in [START_REF] Hohlweg | Polytopal realizations of finite type g-vector fans[END_REF].

Example 32. We now provide a first polytopal realization of the g-vector fan F g (D • ) (see also Section 4). This fan has a maximal cone for each maximal D • -accordion dissection and a ray for each D • -accordion diagonal. For a maximal D • -accordion dissection D • , we define a point p D

• | D • ∈ R D• by p D • | D • := δ•∈D• ω D • | δ • • c D • | δ • ∈ D • ,
where ω(D • | δ • ) still denotes the D • -height of δ • defined as the number of D • -accordion diagonals that cross δ • . We will need the following two technical lemmas in the proof of Theorem 35.

Lemma 33. For any maximal D

• -accordion dissection D • , the point p(D • | D • ) is the intersection of all hyperplanes H = (D • | δ • ) with δ • ∈ D • . Proof. Observe first that the hyperplanes H = (D • | δ • ) with δ • ∈ D • have a unique intersection point, since g(D • | D • )
g D • | γ • p D • | D • = δ•∈D• ω D • | δ • • g D • | γ • c D • | δ • ∈ D • = δ•∈D• ω D • | δ • • 1 1 γ•=δ• = ω D • | γ • . Lemma 34. If D • , D • are two adjacent maximal D • -accordion dissections, and δ • ∈ D • and δ • ∈ D • are such that D • {δ • } = D • {δ • }, then c D • | δ • ∈ D • = -c D • | δ • ∈ D • and p D • | D • -p D • | D • ∈ Z <0 • c D • | δ • ∈ D • . Proof. Let D • , D • be two adjacent maximal D • -accordion dissections, let δ • ∈ D • and δ • ∈ D • be such that D • {δ • } = D • {δ • },
and let µ • and ν • be the other diagonals as in Lemma 9 (see also Figure 4). A quick case analysis then shows that

c D • | γ • ∈ D • =      c D • | γ • ∈ D • for all diagonal γ • ∈ D • {δ • , µ • , ν • }, -c D • | δ • ∈ D • if γ • = δ • , c D • | γ • ∈ D • + c D • | δ • ∈ D • if γ • ∈ {µ • , ν • }.
Summing the contribution of all c-vectors with their coefficients ω(D

• | γ • ), we obtain p D • | D • -p D • | D • = ω D • | µ • + ω D • | ν • -ω D • | δ • -ω D • | δ • • c D • | δ • ∈ D • .
Finally, note that any diagonal of P • that crosses one of (resp. both) the diagonals µ • , ν • also crosses one of (resp. both) the diagonals δ • , δ • . Moreover, δ • and δ • cross each other but do not cross µ • and ν For a triangulation T • with an interior triangle, the T • -accordiohedron Acco(T • ) was recently constructed in [START_REF] Hohlweg | Polytopal realizations of finite type g-vector fans[END_REF]. For example, for the triangulation of the hexagon with an interior triangle, this associahedron appeared as a mysterious realization in [START_REF] Ceballos | Many non-equivalent realizations of the associahedron[END_REF].

• . It follows that ω(D • | µ • ) + ω(D • | ν • ) -ω(D • | δ • ) -ω(D • | δ • ) ≤ -2 < 0.
For a quadrangulation Q • , the Q • -accordiohedron Acco(Q • ) is a realization of the Stokes polytope announced by Y. Baryshnikov [Bar01] and discussed by F. Chapoton in [START_REF] Chapoton | Stokes posets and serpent nests[END_REF].

We conclude this section by an immediate consequence of Theorem 35. To our knowledge, this property of accordion complexes was not observed before. However, using the connection between accordion complexes and support τ -tilting complexes [GM16, PPP17, PPS17, BDM + 17], it can also be obtained from [DIJ17, Thm. 1.7].

Corollary 38. For any reference dissection D • , the D • -accordion complex AC(D • ) is shellable.

2.4. Some properties of Acco(D • ). We conclude this section by pointing out some relevant combinatorial and geometric properties and observations on the D • -accordiohedron.

Proposition 39. The graph of the D • -accordiohedron Acco(D • ) linearly oriented in the direction -1

1 := -δ•∈D• e δ• is the Hasse diagram of the accordion lattice AL(D • ). Proof. Consider two adjacent maximal D • -accordion dissections D • , D • such that the flip from D • to D • is increasing. Let δ • ∈ D • and δ • ∈ D • be such that D • {δ • } = D • {δ • }. As observed in Remark 20 (ii), the c-vector c(D • | δ • ∈ D • ) is the characteristic vector 1 1 A• of the set A • of diagonals of D • crossed by both δ • and δ • .
Applying Lemma 34, we therefore obtain that

-1 1 p D • | D • -p D • | D • = -1 1 λ • c D • | δ • ∈ D • = λ • -1 1 | 1 1 A• = -λ • |A • |,
for some λ ∈ Z <0 . Thus, the linear functional -1

1 indeed orients the edge [p(D • | D • ), p(D • | D • )] from p(D • | D • ) to p(D • | D • ).
Remark 40. Since the c-vector fan F c (D • ) refines the g-vector fan F g (D • ), there is a natural projection π from the vertices of the D • -zonotope Zono(D • ) to that of the D • -accordiohedron Acco(D • ). In analogy to the acyclic case, one could hope to obtain the accordion lattice as a lattice quotient through this projection. However, the transitive closure of the graph of the D • -zonotope Zono(D • ) oriented in the direction -1 1 is not a lattice in general (the first counter-example is the dissection with a central square surrounded by 4 triangles). As shown in [START_REF] Garver | Oriented flip graphs and noncrossing tree partitions[END_REF], the right objects are not the separable subsets of c-vectors (i.e. the vertices of Zono(D • )) but the biclosed subsets of c-vectors.

Para(D • ) := x ∈ R D• g(D • | δ • ) | x | ≤ ω(D • | δ • ) for all δ • ∈ D - • ∪ D + •
defined by the inequalities of the D • -zonotope Zono(D • ) corresponding to the positive and negative basis vectors. Our next statement follows from Proposition 41 and is illustrated in Figure 7.

Corollary 42. For any D • , we have matriochka polytopes:

Zono(D • ) ⊆ Acco(D • ) ⊆ Para(D • ).
In fact, each polytope in this chain is obtained by deleting facets from the previous one.

Consider now an isometry σ of the plane that preserves the hollow polygon P • and the solid polygon P • . For any diagonals and dissections δ

• ∈ D • and δ • ∈ D • , we have δ • is a D • -accordion diagonal ⇐⇒ σ(δ • ) is a σ(D • )-accordion diagonal, D • is a D • -accordion dissection ⇐⇒ σ(D • ) is a σ(D • )-accordion dissection, if Σ : R D• → R σ(D•) denotes the isometry defined by Σ(x) σ(δ•) := ε(σ)•x δ• , (where ε(σ) = 1 if σ is direct and -1 if σ is indirect), then we have g σ(D • ) | σ(δ • ) = Σ g(D • | δ • ) , c σ(D • ) | σ(δ • ) ∈ σ(D • ) = Σ c(D • | δ • ∈ D • ) , ω σ(D • ) | σ(δ • ) = ω D • | δ • , and p σ(D • ) | σ(D • ) = Σ p(D • | D • ) .
This immediately implies the following statement.

Proposition 43. Any P • -preserving isometry σ : R 2 → R 2 induces an isometry Σ :

R D• → R σ(D•) with Σ Zono(D • ) = Zono σ(D • ) , Σ Acco(D • ) = Acco σ(D • ) and Σ Para(D • ) = Para σ(D • ) .
We say that a dissection D is σ-invariant when σ(D) = D. Assume now that σ is a rotation and D • is σ-invariant. We call σ-invariant D • -accordion complex the simplicial complex AC σ (D • ) whose vertices are the crossing-free σ-orbits of D • -accordion diagonals, and whose faces are sets of such orbits whose union is crossing-free. In other words, the faces of AC σ (D • ) are σ-invariant D • -accordion dissections, seen as sets of σ-orbits of diagonals. Finally, the general statement follows from this special case. Indeed, if σ is not a central symmetry, let C • denote the cell of D • containing the center of P • , let u • be a vertex of C • , let D • be the set of diagonals of D • whose endpoints are between u • and σ(u • ), and let ρ be the central symmetry around the middle of u ) belongs to F and to Fix(Σ) and thus to e. We conclude that if v is a common vertex of P and Q, then so are all neighbors of v in the graph of Q. Propagating this property, we obtain that all vertices of Q are also vertices of P , so that P = Q. Finally, there is a clear injection from the σ-invariant accordion complex AC σ (D • ) to the boundary complex of P = Q, thus a bijection (since these complexes are two spheres with the same vertex set).

• σ(u • ). Then AC σ (D • ) is isomorphic to AC ρ D • ∪ ρ(D • ) .

The d-vector fan

In this section, we discuss the generalization to the D • -accordion complex of another classical geometric realization of the associahedron coming from the theory of cluster algebras [START_REF] Fomin | Cluster algebras. I. Foundations[END_REF][START_REF] Fomin | Cluster algebras. II. Finite type classification[END_REF][START_REF] Chapoton | Polytopal realizations of generalized associahedra[END_REF][START_REF] Ceballos | Many non-equivalent realizations of the associahedron[END_REF]. Namely, we define compatibility vectors in analogy with the denominator vectors of cluster variables, and we characterize the reference dissections D • for which these vectors support a complete simplicial fan realizing the D • -accordion complex.

3.1. d-vectors. Fix a dissection D • of the hollow n-gon. For a hollow diagonal δ • = i • j • and a solid diagonal δ • , we denote by

(δ • | δ • ) :=      -1 if δ • = (i -1) • (j -1) • , 0 if δ • and (i -1) • (j -1) • do not cross, 1 if δ • and (i -1) • (j -1) • cross.
For any D • -accordion diagonal δ • , the d-vector of δ • with respect to D • is the vector

d D • | δ • = δ•∈D• (δ • | δ • ) e δ• .
In other words, our Proof. Assume that D • contains an even interior cell C • . Denote its vertices by i 1

• , . . . , i 2p

• (in clockwise order) and its edges δ k

• := i k • i k+1 • for k ∈ [2p] (where i 2p+1 = i 1 by convention). Denote by D k • the set of diagonals of D • separated form C • by δ k • (including δ k • itself), and let D k • := (i -1) • (j -1) • i • j • ∈ D k • . Consider the solid diagonals δ k • := (i k + 1) • (i k+1 + 1) • for k ∈ [2p]. Observe that δ k • only crosses diagonals of D k-1
• and D k • , and that δ k • and δ k+1

• cross precisely the same diagonals of D k

• . Since the cell is even, it ensures that the d-vectors of the diagonals δ k

• for k ∈ [2p] satisfy the linear dependence

k∈[2p] k even d D • | δ k • = k∈[2p] k odd d D • | δ k • .
However, as already mentioned in Section 1.4, the diagonals δ k

• for k ∈ [2p] all belong to the D • -accordion dissection D + • := {(i + 1) • (j + 1) • | i • j • ∈ D • }. Therefore, the cone R ≥0 d(D • | D + • ) is degenerate, so that the d-vectors cannot realize the D • -accordion complex.
Example 48. Consider a hollow octagon and the reference dissection D

• := {1 • 5 • , 5 • 9 • , 9 • 13 • , 13 • 1 • } with an interior square cell 1 • 5 • 9 • 13 • . Then we have d D • | 2 • 6 • = e 1•5• + e 5•9• d D • | 6 • 10 • = e 5•9• + e 9•13• d D • | 10 • 14 • = e 9•13• + e 13•1• d D • | 14 • 2 • = e 13•1• + e 1•5•
so that there is already a linear dependence

d D • | 2 • 6 • + d D • | 10 • 14 • = d D • | 6 • 10 • + d D • | 14 • 2 • among the d-vectors of the D • -accordion dissection D + • = {2 • 6 • , 6 • 10 • , 10 • 14 • , 14 • 2 • }.
On the negative side, we have seen that the presence of even interior cells prohibits the d-vectors from forming a complete simplicial fan. The positive side is that the even interior cells are the only obstructions.

Theorem 49. The collection of cones

F d (D • ) := R ≥0 d D • | D • D • any D • -
d(D • | δ • ) + d(D • | δ • ) = d(D • | µ • ) + d(D • | ν • ).
Assume that γ • crosses neither µ • nor ν • . Then γ • is incident to both µ • and ν • , and therefore is either δ

• or δ • , say γ • = δ • . Then d(γ • | δ • ) = -1 while d(γ • | δ • ) = 1 (since δ • crosses δ • = γ • ), so that d(γ • | δ • ) + d(γ • | δ • ) = 0. Moreover, we have d(γ • | δ • ) = 0 for any diagonal ε • ∈ D • ∩ D • since δ • = γ • cannot cross ε • as they both belongs to D • . Therefore, the set d(D • | δ • ) + d(D • | δ • ) ∪ d(D • | D • ∩ D • ) contains |D • | vectors of R D• whose γ • -
coordinate all vanish, so that it admits a linear dependence.

Otherwise, we can assume that γ • crosses µ • but not ν • . Then γ • has a common endpoint with ν • and δ • (or δ • , but we then permute notations). Changing our initial choice of γ • , we can assume that no diagonal of D -

• separates γ • from δ • . We now denote clockwise -by ν • =: λ 0 • , λ 1 • , . . . , λ • := δ • the edges of the cell C • of D • containing ν • and δ • , -by γ • =: γ 0 • , γ 1 • , . . . , γ k • the edges of the cell C - • of D - • containing γ
• and crossed by δ • . These notations are illustrated on Figure 8. We still distinguish two subcases as in Figure 8:

-If γ i • crosses λ i • for all i as in Figure 8 (middle), then = k and we have the linear dependence

2d(D • | δ • ) + d(D • | δ • ) = d(D • | µ • ) + i∈[ -1] (-1) (i-1) d(D • | λ i • ).
It is essential here that = k is even. This is guarantied by the assumption that D • (and thus D - • ) has no even interior cell, since C - • is an interior cell of D - • of size k. -Otherwise, we are in a situation similar to Figure 8 (right). Considering the maximal index m such that γ i • crosses λ i • for all i ≤ m, and we have the linear dependence Remark 51. To prove that the d-vector fan F d (D • ) is polytopal, we would need to find suitable hyperplanes orthogonal to their rays in order to apply Theorem 36. For the g-vector fan, these hyperplanes were defined using the height function ω(D • | δ • ). It would be natural to use the same height function for the d-vector fan as well. Unfortunately, for this choice of height function, we can only prove Condition (i) of Theorem 36 when D • is a triangulation (see also [START_REF] Ceballos | Many non-equivalent realizations of the associahedron[END_REF]). We were not able to find suitable right hand sides for any dissection D • .

d(D • | δ • ) + d(D • | δ • ) = d(D • | µ • ) + i∈[m] (-1) (i-1) d(D • | λ i • ).
Remark 52. Our d-vectors record the compatibility with the dissection D - • . A priori, we could compute compatibility vectors with respect to any other maximal D • -accordion dissection D ini

• . Experiments suggest that the d-vector construction provides a complete simplicial fan as long as neither D • nor D ini

• contain no even interior cell. We checked it for reference quadrangulations with at most 5 diagonals. The linear dependences involved seem however much more complicated than those of the proof of Theorem 49 (in particular, they may involve d-vectors of diagonals not included in the cells containing δ • and δ • ).

Sections and projections

Recall that for a fan F of R d and a linear subspace V of R d , the section of F by V is the fan 4.2. Coordinate sections of the g-vector fan. More relevant are the sections of the g-vector fan. They provide an alternative approach to polytopal realizations of the accordion complex based on projected associahedra. This approach relies on the following crucial observation. Based on this lemma, we obtain in the following statements an alternative realization on the g-vector fan, which is illustrated on Figure 10. The following statement extends Theorem 56 to arbitrary cluster algebras.

F V := {C ∩ V | C ∈ F}.
Theorem 63. The subset C ∈ F g (X • ) C ⊆ R [m] I of the g-vector fan F g (X • ) of X • coincides with the section F g (X • )

R [m] I = C ∩ R [m] I C ∈ F g (X • ) . Proof. The inclusion C ∈ F g (X • ) C ⊆ R [m] I ⊆ F g (X • ) R [m] I is clear.
For the reverse inclusion, we use the sign coherence property of g-vectors in cluster algebras, which was conjectured in [FZ07, Conj. 6.13] and proved in [START_REF] Gross | Canonical bases for cluster algebras[END_REF]Thm. 5.1] in general. This property implies that the coordinate plane R [m] I intersects any cone C of F g (X • ) in a face C . This shows that C ∩ R [m]

I = C belongs to C ∈ F g (X • ) C ⊆ R [m] I .
Corollary 64. The subcomplex ∆ g (X • , I) induced by the variables y such that g(X • | y) i = 0 for all i ∈ I is a pseudomanifold.

Moreover, extending the result of C. Hohlweg, C. Lange and H. Thomas [START_REF] Hohlweg | Permutahedra and generalized associahedra[END_REF] in the acyclic case, C. Hohlweg, V. Pilaud and S. Stella recently constructed a polytope Asso(X • ) realizing the gvector fan F g (X • ) in [START_REF] Hohlweg | Polytopal realizations of finite type g-vector fans[END_REF]. We can use this associahedron to realize the subcomplex ∆ g (X • , I) as a convex polytope, extending Theorem 57.

Corollary 65. The orthogonal projection of Asso(X • ) on R [m] I is a realization of ∆ g (X • , I).

Finally, when oriented in the suitable direction v (the sum of the positive roots, or equivalently the sum of the fundamental weights), the graph of the generalized associahedron Asso(X • ) is the Hasse diagram of a Cambrian lattice [START_REF] Reading | Cambrian lattices[END_REF]. One can similarly orient the graph of the projection of Asso(X • ) on R [m] I in the direction of the projection of v on R [m] I . Is the resulting graph the Hasse diagram of a lattice? Combining the results of [START_REF] Garver | Oriented flip graphs and noncrossing tree partitions[END_REF] with that of the present paper shows that this property holds in type A. We also computationally verified the statement in types B 4 , B 5 , D 4 and D 5 . Following [START_REF] Garver | Oriented flip graphs and noncrossing tree partitions[END_REF] it seems promising to construct first a lattice structure on biclosed sets of c-vectors, and to obtain then the graph of the projection of Asso(X • ) on R [m] I as the Hasse diagram of a lattice quotient.

To conclude, let us mention that the ideas developed in this section have also inspired further investigation of sections of g-vector fans of support τ -tilting complexes of associative algebras, see [START_REF] Pilaud | A τ -tilting approach to dissections of polygons[END_REF] and [PPP17, Sect. 4.2.6].

Figure 1 .

 1 Figure 1. A dissection D (left) and three accordions whose zigzags are bolded (middle and right).

Example 1 .

 1 As a running example, we consider the reference dissection D ex • of Figure 2 (left). Examples of maximal D ex • -accordion dissections are given in Figure 2 (right). The D ex • -accordion complex is illustrated in Figure 3 (left).

Figure 2 .

 2 Figure 2. A hollow dissection D ex • , a solid D ex • -accordion diagonal whose corresponding hollow accordion is bolded, and two maximal solid D ex • -accordion dissections.

Figure 3 .

 3 Figure 3. The D ex • -accordion complex (left) and the D ex • -accordion lattice (right), oriented from bottom to top, for the reference hollow dissection D ex • of Figure 2 (left).

Figure 4 .

 4 Figure 4. Two maximal D • -accordion dissection D • (left) and D • (right) related by the flip of δ • to δ • . The angles of D • closed by δ • and δ • are shaded. The flip is oriented from D • to D • .

  in terms of the dual tree D • of the dissection D • . Lemma 8 ([GM16]). For any maximal D • -accordion dissection D • , each internal diagonal δ • of D • closes two angles of D • (one apex on each side of δ • ) while each boundary edge of the solid polygon closes one angle of D • . Therefore the accordion complex AC(D • ) is pure of dimension |D • |.

Figure 4 .

 4 Let D • be a maximal D • -accordion dissection and δ • be a diagonal of D • . Let u • and v • be the apices of the angles of D • closed by δ • , let µ • and ν • denote the edges of the cells of D • containing δ • , which separate δ • from u • and v • respectively, and let Q • denote the quadrilateral defined by the four vertices of µ • and ν • . Note that δ • is a diagonal of Q • , and let δ • denote the other diagonal. Lemma 9 ([GM16]). With the previous notations, the collection of diagonals D • := D • {δ • , δ • } is a maximal D • -accordion dissection, and D • and D • are the only maximal D • -accordion dissections containing D • {δ • }. In other words, the accordion complex AC(D • ) is thin. Proof. We first observe that δ • is a D • -accordion diagonal, since the edges of D • crossed by δ • are obtained by merging three subaccordions of D • : the subaccordion formed by the diagonals of D • crossed by µ • but not δ • nor ν • , the subaccordion formed by the diagonals of D • crossed by δ • , µ • and ν • , and the subaccordion formed by the diagonals of D • crossed by ν • but not δ • nor µ • . Moreover, δ • and δ • are the only D • -accordion diagonals compatible with D • {δ • }. Indeed, any other such diagonal would cross δ • and δ • (by maximality of D • and D • ), and thus also the subaccordion A • of D • crossed by δ • and δ • (because it cannot cross µ and ν). But it would then improperly intersect the two cells of D • containing precisely one diagonal of A • . The D • -accordion flip graph is the dual graph AFG(D • ) of the D • -accordion complex: its vertices are the maximal D • -accordion dissections, and its edges are the flips between them, i.e. the pairs {D • , D • } of maximal D • -accordion dissections with D • {δ • } = D • {δ • }. See Figure 3 (right).

  the other forms a Z, see Figure4. We then orient the flip from the dissection containing the Z to that containing the Z. See Figure3(right) for an illustration of D • -accordion oriented flip graph (where the graph is oriented from bottom to top). A. Garver and T. McConville introduced a natural closure on sets of D • -subaccordions, and showed that the inclusion poset of biclosed sets of D • -subaccordions is a well-behaved lattice (namely, semidistributive, congruence-uniform and polygonal). Then, they introduced a lattice quotient map from biclosed sets of D • -subaccordions to maximal D • -accordion dissections, which imply the following statement. Theorem 10 ([GM16]). The D • -accordion oriented flip graph is the Hasse diagram of a lattice, that we call the D • -accordion lattice and denote by AL(D • ).

  by slightly rotating D • clockwise). Example 11. Following Example 2, note that special reference hollow dissections D • give rise to special accordion lattices AL(D • ), as it was already observed in [GM16]: For a fan triangulation F • (i.e. where all internal diagonals are incident to a common vertex), the F • -accordion lattice AL(F • ) is the famous Tamari lattice [Tam51, MHPS12] defined equivalently by slope increasing flips on triangulations of a convex polygon, by right rotations on binary trees, or by flips on Dyck paths. In general, accordion lattices of accordion triangulations (i.e. with no interior triangle) precisely correspond to type A Cambrian lattices defined by N. Reading [Rea06]. For an arbitrary triangulation T • (with or without interior triangle), the T • -accordion oriented flip graph AFG(A • ) is a particular instance of the oriented exchange graphs of 2-acyclic quivers defined by T. Brüstle, G. Dupont and M. Pérotin [BDP14]. These oriented exchange graphs are far more general and their transitive closures are in general not lattices. For a quadrangulation Q • , the Q • -accordion lattice AL(Q • ) is the Stokes poset on Q • -compatible quadrangulations studied by F.Chapoton [Cha16]. The following statement is a direct consequence of Proposition 4. Proposition 12. If the reference hollow dissection D • has a cell containing p boundary edges of the hollow polygon P • , then the D • -accordion lattice AL(D • ) is a Cartesian product of p accordion lattices. Proof. Consider the dissections D 1 • , . . . , D p • as in the proof of Proposition 4. Since any increasing flip in AC(D • ) is an increasing flip in one of the AC(D i • ), we obtain that the D • -accordion lattice is the Cartesian product of the D i • -accordion lattices: AL

  accordion dissection for any maximal D • -accordion dissection D • . Finally, maximality follows since all maximal D • -accordion dissections have |D • | diagonals. The equivalence follows by symmetry.

  2.1. g-and c-vectors. Consider a hollow dissection D • and a solid dissection D • that are maximal accordion dissections of each other (see Proposition 15), and let δ • ∈ D • and δ

  vectors of the diagonals of D • and by C(D • ) := D• c D • | D • the set of all c-vectors with respect to D • . Example 16. Consider the hollow dissection D ex •

Remark 19 .

 19 Consider the quiver Q(D • ) of the reference dissection D • , with one node on each internal diagonal of D • and one arrow between two diagonals counter-clockwise consecutive around a cell of D • . Let W(D • ) be the reflection group whose Dynkin diagram is the underlying graph of Q(D • ). Then all g-vectors of the D • -accordion diagonals are weights of W(D • ) and all c-vectors of C(D • ) are roots of W(D • ). Remark 20. Informally, the gand c-vectors can be interpreted as follows: (i) The g-vector g(D • | δ • ) has coordinate 1 and -1 alternating along the zigzag of the accordion crossed by δ • in D • , and coordinate 0 on all other diagonals of D • . (ii) The c-vector c(D • | δ • ∈ D • ) is, up to a sign, the characteristic vector of the diagonals of the subaccordion of D • crossed by both diagonals µ • and ν • of Lemma 9 (see also Figure 4). Thus, any c-vector is either positive (only non-negative coordinates) or negative (only nonpositive coordinates). In fact, the g-vectors are clearly in bijection with the accordions and with the zigzags in D • . In contrast, many pairs (δ • , D • ) produce the same c-vector c(D • | δ • ∈ D • ). For example, if two dissections D • , D • contain δ • and have the same cells incident to δ • , then c(D • | δ • ∈ D • ) = c(D • | δ • ∈ D • ). The set of c-vectors C(D • ) without repetitions can be understood as follows. Lemma 21. There are bijections between: the negative (resp. positive) c-vectors of C(D • ), the subaccordions of D • , the D • -accordion diagonals not in the source dissection D -

Figure 5 .

 5 Figure 5. Illustration of the proof of Proposition 22. The red hollow diagonals form the zigzag of γ • , and the bolded ones are slaloming on δ • . There are an even number of bolded diagonals when γ • = δ • (left) and an odd number when γ • = δ • (right).

  since we sum an even number of alternating ±1 when γ • = δ • , and an odd number of alternating ±1 starting by a 1 when γ • = δ • . In other words, g(D • | D • ) and c(D • | D • ) form dual bases. Proposition 23. Let D • be a hollow dissection and D • be a solid dissection such that D • and D • are maximal accordion dissections of each other (see Proposition 15). Then

  The other equality follows by exchanging D • and D • . Corollary 24. For any maximal D • -accordion dissection D • , we have the following sign coherence: (i) for any δ • ∈ D • , all coordinates of the c-vector c(D • | δ • ∈ D • ) have the same sign, (ii) for any δ • ∈ D • , the δ • -coordinates of all g-vectors g(D • | δ • ) for δ • ∈ D • have the same sign. Proof. Point (i) was already seen in Remark 20 (ii), and Point (ii) follows by Proposition 23. 2.2. c-vector fan and D • -zonotope. Define the c-vector fan of D • to be the complete polyhedral fan F c (D • ) given by the arrangement of the linear hyperplanes orthogonal to the c-vectors of C(D • ). Be careful: in contrast to the gand d-vector fans defined later, the c-vectors are not the rays of F c (D • ) but the normal vectors of the hyperplanes supporting the facets of F c (D • ). We call D • -zonotope the Minkowski sum Zono(D • ) of all c-vectors: Zono(D • ) := c∈C(D•) c. The normal fan of the D • -zonotope Zono(D • ) is the c-vector fan F c (D • ). Note that the c-vector fan is not always simplicial, and thus the D • -zonotope Zono(D • ) is not always simple. See Figure 7. Example 25. Consider an accordion dissection A • (where each cell has at most 2 edges which are internal diagonals of P • ). Label its internal diagonals by δ 1 • , . . . , δ |A•| • such that δ k • and δ k+1 • belong to the same cell of A • for all k. Identifying e δ k • to the simple root f k -f k+1 of type A |A•| , the c-vectors of C(A • ) are all roots ±(f i -f j ) = ± i≤k≤j e δ k • of type A |A•| . Therefore, the c-vector fan is the type A |A•| Coxeter fan and the A • -zonotope is a permutahedron. More precisely,

By

  Remark 20, we have g(D • | γ • ) | c ∈ {-1, 0, 1} for any c ∈ C(D • ). We thus just need to count the distinct c-vectors c such that g(D • | γ • ) | c > 0. It turns out that it is more convenient and equivalent (since C(D • ) = -C(D • )) to count the distinct c-vectors c such that g(D • | γ • ) | c < 0. For that, let Z • denote the zigzag of the accordion crossed by γ • in D • , and decompose Z

  accordion dissection forms a complete simplicial fan, that we call the g-vector fan of D • . The proof uses the following characterization of complete simplicial fans [DRS10, Coro. 4.5.20]. We will provide as well an alternative proof in Remark 60 based on sections of Cambrian fans. Proposition 28. Consider a pseudomanifold ∆ on a finite vertex set X and a set of vectors R := (r x ) x∈X of R d . For D ∈ ∆, define the cone R D := {r x | x ∈ D}. The collection of cones R ≥0 R D D ∈ ∆ forms a complete simplicial fan if and only if (1) there exists a facet D of ∆ such that R D is a basis of R d and such that the open cones R >0 R D and R >0 R D are disjoint for any facet D of ∆ distinct from D; (2) for two adjacent facets D, D of ∆ with D {x} = D {x }, there is a linear dependence α r x + α r x + y∈D∩D β y r y = 0 on R D∪D where the coefficients α and α have the same sign. (When these conditions hold, these coefficients do not vanish and the linear dependence is unique up to rescaling.) Proof of Theorem 27. By Corollary 24, the cone R ≥0 g(D • | D - • ) is the only cone of F g (D • ) intersecting the interior of the positive orthant (R ≥0 ) D• . Consider now two adjacent maximal D

  generated by the coordinate vectors (see Example 17), we obtain that det g(D • | D • ) = ±1 for all D • -accordion dissection D • , so that the g-vector fan F g (D • ) is always smooth. By Proposition 22, any non-maximal cone of F g (D • ) is supported by a hyperplane orthogonal to a c-vector of C(D • ). We thus obtain the following consequence. Corollary 30. The g-vector fan F g (D • ) coarsens the c-vector fan F c (D • ).

  Figure 6 illustrates the g-vector fans F g (D • ) for various reference dissections D • : the fan, the snake, and the cyclic triangulation of the hexagon, and a dissection of the heptagon. More precisely, we have represented the stereographic projection of the fans from the point [ 1, 1, 1 ].

Figure 6 .

 6 Figure 6. Stereographic projections of the g-vector fans F g (D • ) for various reference hollow dissections D • . See Figure 9 for alternative simplicial fan realizations of these accordion complexes.

  is a basis. Moreover, since g(D • | D • ) and c(D • | D • ) form dual bases by Proposition 22, we have for any γ • ∈ D • :

Theorem 35 .

 35 The g-vector fan is the normal fan of the D • -accordiohedron Acco(D • ) defined equivalently as the convex hull of the points p(D • | D • ) for all maximal D • -accordion dissection D • , or the intersection of the half-spaces H ≤ (D • | γ • ) for all D • -accordion diagonals γ • . Thus, the polar dual of Acco(D • ) is a polytopal realization of the D • -accordion complex AC(D • ). The proof of Theorem 35 is based on the following characterization of polytopal realizations of a complete simplicial fan, whose proof can be found e.g. in [HLT11, Thm. 4.1]. Theorem 36. Given a complete simplicial fan F in R d , consider for each ray r of F a halfspace H ≤ r of R d containing the origin and defined by a hyperplane H = r orthogonal to r. For each maximal cone C of F, let a(C) ∈ R d be the intersection of all hyperplanes H = r with r ∈ C. Then the following assertions are equivalent: (i) The vector a(C ) -a(C) points from C to C for any two adjacent maximal cones C, C of F. (ii) The polytopes conv {a(C) | C maximal cone of F} and r ray of F H ≤ r coincide and their normal fan is F. Proof of Theorem 35. The g-vector fan F g (D • ) has a ray g(D • | δ • ) for each D • -accordion diagonal δ • and a maximal cone C(D • ) = R ≥0 g(D • | D • ) for each maximal D • -accordion dissection D • . Consider the half-spaces H ≤ (D • | γ • ) for all D • -accordion diagonals γ • . Lemma 33 ensures that the point a(C(D • )) coincides with p(D • | D • ) for each maximal D • -accordion dissection D • . Finally, Lemma 34 shows that the conditions of application of Theorem 36 are fulfilled. Example 37. Following Example 2, observe that special reference hollow dissections give rise to the following relevant polytopes, illustrated in Figure 7: For a fan triangulation T • , the T • -accordiohedron Acco(T • ) is the classical associahedron constructed by S. Shnider and S. Sternberg [SS93] and J.-L. Loday [Lod04].

Figure 7 .

 7 Figure 7. The zonotope Zono(D • ), D • -accordiohedron Acco(D • ) and parallelepiped Para(D • ) for different reference dissections D • . The first column is J.-L. Loday's associahedron [Lod04],the second column is one of C. Hohlweg and C. Lange's associahedra[START_REF] Hohlweg | Realizations of the associahedron and cyclohedron[END_REF], the third column appeared in a discussion in C. Ceballos, F. Santos and G. Ziegler's survey on associahedra [CSZ15, Fig.3] and was explained in C. Hohlweg, V. Pilaud and S. Stella's recent paper[START_REF] Hohlweg | Polytopal realizations of finite type g-vector fans[END_REF], and the last column is a Stokes complex discussed by F. Chapoton in[START_REF] Chapoton | Stokes posets and serpent nests[END_REF] and illustrated in Figure3.

Proposition 41 .

 41 The accordiohedron Acco(D • ) has precisely |D • | pairs of parallel facets. Proof. Two facets of Acco(D • ) are parallel if and only if the corresponding g-vectors are opposite. We therefore want to prove that the pairs of opposite coordinate vectors are the only pairs of opposite g-vectors. Assume by contradiction that there exist two hollow diagonals δ • , δ • ∈ D • and two solid D • -diagonals δ • , δ • such that g(D • | δ • ) and g(D • | δ • ) have non-zero opposite coordinate both on δ • and δ • . Then both δ • and δ • cross both δ • and δ • . But this implies that they both slalom on δ • (and on δ • ) in the same way. Contradiction. Recall from Example 17 that the g-vectors of the diagonals of D - • (resp. D + • ) are the coordinate vectors (resp. negative of the coordinate vectors). Consider the D • -parallelepiped

Lemma 44 .

 44 The σ-invariant D • -accordion complex AC σ (D • ) is a pseudomanifold. Proof. Assume first that σ is the central symmetry. In this case, there are two possible types of orbits: the long D • -accordion diagonals and the centrally symmetric pairs of D • -accordion diagonals. One can check that any facet of AC σ (D • ) has a long diagonal if and only if D • has, and has as many centrally symmetric pairs of diagonals as D • . Finally, any orbit in any facet of AC σ (D • ) can be flipped: long diagonals can already be flipped in AC(D • ), and a centrally symmetric pair of diagonals can be flipped by flipping one after the other its two diagonals in AC(D • ).

  Let Σ : R D• → R D• denote the isometry defined by Σ(x) σ(δ•) := x δ• and Fix(Σ) denote the linear subspace of fixed points of Σ. According to the previous discussion, a maximal D• -accordion dissection D • is σ-invariant if and only if p(D • | D • ) ∈ Fix(Σ). We obtain the following statement. Proposition 45. For a σ-invariant dissection D • , the polytope Acco σ (D • ) defined equivalently as the convex hull of p(D • | D • ) for all σ-invariant maximal D • -accordion dissections D • , the intersection of the D • -accordiohedron Acco(D • ) with the fixed space Fix(Σ), is a polytopal realization of the σ-invariant accordion complex AC σ (D • ). Proof. Denote by P = conv {p(D • | D • ) | σ-invariant maximal D • -accordion dissections D • } and by Q = Acco(D • ) ∩ Fix(Σ). The inclusion P ⊆ Q is clear since D • is σ-invariant if and only if p(D • | D • ) ∈ Fix(Σ). We now prove the reverse inclusion. For that, consider an arbitrary σinvariant maximal D • -accordion dissection D • . Its corresponding point p(D • | D • ) is a common vertex of P and Q. Moreover, any edge e of Q incident to p(D • | D • ) is the intersection of Fix(Σ) with a face F of Acco(D • ) that corresponds to a σ-invariant D • -dissection. Since AC σ (D • ) is a pseudomanifold, this dissection can be refined into another maximal σ-invariant D • -accordion dissection D • . The point p(D • | D •

  d-vector d(D • | δ • ) records the compatibility of the diagonal δ • with the dissection D - • . For a D • -accordion dissection D • , we define d D • | D• := d D • | δ • δ • ∈ D • . Example 46. Consider the hollow dissection D ex • = {3 • 7 • , 3 • 13 • , 9 • 13 • } and the rightmost solid dissection D ex • = {2 • 6 • , 2 • 10 • , 10 • 14 • } of Figure 2. Its d-vectors are given by d D ex • | 2 • 6 • = -e 3•7• , d D ex • | 2 • 10 • = e 9•13• , and d D ex • | 10 • 14 • = e 3•13• + e 9•13• . 3.2. d-vector fan. We now consider the set of cones R ≥0 d D • | D • D • any D • -accordion dissection generated by the d-vectors of the D • -accordion dissections. We want to characterize the reference hollow dissections D • for which these cones form a complete simplicial fan realizing the D • -accordion complex. We start with a negative result. An even interior cell of a dissection D is a cell with an even number of edges which are all internal diagonals of D. Proposition 47. If the reference hollow dissection D • contains an even interior cell, then the d-vectors cannot realize the D • -accordion complex.

  accordion dissection forms a complete simplicial fan, that we call the d-vector fan of D • , if and only if D • contains no even interior cell.

Figure 8 .

 8 Figure 8. Illustration of the notations and of the different cases in the proof of Theorem 49.

Figure 9 .

 9 Figure 9. Stereographic projections of the d-vector fans F d (D • ) for various reference hollow dissections D • . See Figure 6 for alternative simplicial fan realizations of these accordion complexes.

  For a polytope P ⊆ R d and a projection π : R d → V , the normal fan of the projected polytope π(P ) is the section of the normal fan of P by V [Zie95, Lem. 7.11]. We now consider sections of the gand d-vector fans by coordinate subspaces. For two dissections D • ⊂ D • , we naturally identify R D• with the subspace spanned by {e δ• | δ • ∈ D • } in R D • . 4.1. Coordinate sections of the d-vector fan. We start by presenting sections of the d-vector fan which are not very surprising. The following lemma is immediate from the definition of d-vectors. Lemma 53. Consider two dissections D • ⊂ D • , and a D • -accordion diagonal δ • . Then we have d(D • | δ • ) ∈ R D• if and only if δ • does not cross any diagonal of {(i -1) • (j -1) • | i • j • ∈ D • D • }.Corollary 54. For two dissections D • ⊂ D • , the face complex of the section of the d-vector fan F d (D • ) by R D• is isomorphic to the link of the dissection {(i -1) • (j -1) • | i • j • ∈ D • D • } in the D • -accordion complex AC(D • ).

Lemma 55 .

 55 Consider two dissections D • ⊂ D • , and a D • -accordion diagonal δ • . Then we have g(D• | δ • ) ∈ R D• if and only if δ • is a D • -accordion diagonal. Moreover, in this case, the g-vectors g(D • | δ • ) and g(D • | δ • ) coincide. Proof. Let δ • ∈ D • D • . By definition, a D • -accordion diagonal δ • does not slalom on δ • if and only if the δ • -coordinate of g(D • | δ • ) vanishes. Thus, δ • is a D • -accordion diagonal if and only if the δ • -coordinate of g(D • | δ • ) vanishes for all δ • ∈ D • D • .

Theorem 56 .

 56 For two dissections D • ⊂ D • , the g-vector fan F g (D • ) is precisely the set of cones C ∈ F g (D • ) C ⊂ R D• and coincides with the section of the g-vector fanF g (D • ) by R D• . Proof. Lemma 55 immediately implies that F g (D • ) = C ∈ F g (D • ) C ⊂ R D• . A priori, it is a subfan of the section F g (D • ) R D• = C ∩ R D• C ∈ F g (D • ) . However, since F g (D • ) is already a complete simplicial fan of R D• , it coincides with F g (D • ) R D• . Theorem 57. For two dissections D • ⊂ D • , the g-vector fan F g (D •) is realized by the orthogonal projection of the D • -accordiohedron Acco(D • ) on R D• , which is equivalently described by: the convex hull of the points δ•∈D• ω(D • | δ • ) • c(D • | δ • ∈ D • ) for all D • -accordion dissections D • , the intersection of the half-spaces x ∈ R D• g(D • | γ • ) | x ≤ ω(D • | δ • ) for all D • -accordion diagonals γ • .subcomplex ∆ g (X • , I) is isomorphic to the D • -accordion complex, where D • is the dissection obtained by deleting in T • the diagonals labeled by I.Example 62. Example 61 extends to cluster algebras on surfaces[START_REF] Fomin | Cluster algebras and triangulated surfaces I. Cluster complexes[END_REF][START_REF] Fomin | Cluster algebras and triangulated surfaces. part II: Lambda lengths[END_REF], using accordions of dissections of surfaces.

  Proposition 4. If the reference hollow dissection D • has a cell containing p boundary edges of the hollow polygon P • , then the D • -accordion complex AC(D • ) is the join of p accordion complexes. • 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 of our constructions work in general, but are just obtained as products or joins of the non-degenerate situation.

	Proof. Assume that D • has a cell C • containing p boundary edges of the hollow polygon P • . Let C 1 • denote the p (possibly empty) connected components of the hollow polygon mi-• , . . . , C p nus C • . For i ∈ [p] := {1, . . . , p}, let D i • denote the dissection formed by the cell C • together with the cells of D • contained in the closure of C i • . Observe that for i = j, the internal diagonals of D i • . Thus, no D • -accordion can contain internal • are not incident to the internal diagonals of D j diagonals from distinct dissections D i • . Therefore, the set of D • -accordion diagonals is • and D j the union of the sets of D i • -accordion diagonals for i ∈ [p]. Moreover, for i = j, the D i • -accordion diagonals do not cross the D j • -accordion diagonals. It follows that the D • -accordion complex is the join of the D i • -accordion complexes: AC(D • ) = AC(D 1 • ) * • • • * AC(D p • ).
	Remark 5. In view of Proposition 4, we can do the following reductions:
	(i) If a non-triangular cell of D Proposition 6. The links in an accordion complex are joins of accordion complexes.
	Proof. 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 a diagonal δ • of a cell C i • is a D • -accordion diagonal if and only if it is a D i • -accordion diagonal. Moreover, for i = j, the diagonals of C i • do not cross the diagonals of C j • . It follows that the link of D • in AC(D • ) is isomorphic to the join AC(D 1 • ) * • • • * AC(D p • ).
	1.3. Pseudo-manifold. We now prove that the accordion complex AC(D • ) is a pseudomanifold,
	i.e. that it is:
	(i) pure: all maximal D • -accordion dissections have the same number of 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

Acknoledgements

We thank C. Hohlweg and S. Stella for many helpful discussions on realizations of the associahedron [HPS18] which were the starting point of this paper. We are grateful to F. Chapoton for various conversations on quadrangulations and Stokes posets, and to A. Garver and T. McConville for introducing us with the accordion complexes during FPSAC'16. Their works [Cha16,GM16] were highly inspiring and motivating. We also thank N. Thiery for a question which led to the approach of Section 4.2, and to P.-G. Plamondon for discussions on the generalization to cluster algebras presented in Section 4.3. Finally, we are grateful to two anonymous referees for their attentive reading and their suggestions on the content and presentation which largely improved our original draft.

Partially supported by the French ANR grant SC3A (15 CE40 0004 01).

Proof. Since F g (D • ) is the normal fan of Acco(D • ), Theorem 56 implies that F g (D

is the normal fan of the orthogonal projection of Acco(D • ) on R D• [Zie95, Lem. 7.11]. We therefore just need to prove the given vertex and facet descriptions of this projection. First, since Proof. Apply Theorem 57 to any triangulation T • that refines D • .

Remark 60. Approaching accordion complexes as coordinate sections of g-vector fans actually provides more concise (but also less instructive) proofs for Sections 1.3 and 2.3. Namely, consider any dissection D • and let T • be a triangulation that refines D • . The sign coherence property for triangulations (see Corollary 24) shows that the section

Therefore, this gives an alternative concise proof that the collection of cones C ∈ F g (T • ) C ⊂ R D• forms a complete simplicial fan. Moreover, this fan has the same combinatorics as the D • -accordion complex AC(D • ) by Lemma 55. We conclude directly that AC(D • ) is a pseudomanifold realized by the fan

and by the orthogonal projection of the associahedron Asso(T • ) on R D• . 4.3. Cluster algebra analogues. The perspective on accordion complexes developed in this section also opens the door to generalizations on arbitrary cluster algebras (finite type or not). Namely, consider an arbitrary cluster X • = (x 1

• , . . . , x m • ) in an arbitrary cluster algebra A. For any cluster variable y ∈ A, we denote by g(X • | y) ∈ R m and d(X • | y) ∈ R m the gand d-vectors of y computed with respect to X • , see [START_REF] Fomin | Cluster algebras. I. Foundations[END_REF][START_REF] Fomin | Cluster algebras[END_REF]. Fix a non-empty proper subset I of [m]. We consider two natural subcomplexes of the cluster complex of A:

the subcomplex ∆ d (X • , I) induced by the variables y such that d(X • | y) i = 0 for all i ∈ I, the subcomplex ∆ g (X • , I) induced by the variables y such that g(X • | y) i = 0 for all i ∈ I. It is well-known that the subcomplex ∆ d (X • , I) is the cluster complex obtained by freezing all variables x i for i ∈ I. For example in type A, it is a join of simplicial associahedra and it can therefore be realized by a product of smaller associahedra. In contrast, we do not know whether the subcomplex ∆ g (X • , I) has been investigated. The present paper dealt with the type A situation.