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

We show that the mesh mutations are the minimal relations among the g${\bm{g}}$ ‐vectors with respect to any initial seed in any finite‐type cluster algebra. We then use this algebraic result to derive geometric properties of the g${\bm{g}}$ ‐vector fan: we show that the space of all its polytopal realizations is a simplicial cone, and we then observe that this property implies that all its realizations can be described as the intersection of a high‐dimensional positive orthant with well‐chosen affine spaces. This sheds a new light on and extends earlier results of Arkani‐Hamed, Bai, He, and Yan in type A$A$ and of Bazier‐Matte, Chapelier‐Laget, Douville, Mousavand, Thomas, and Yıldırım for acyclic initial seeds. Moreover, we use a similar approach to study the space of polytopal realizations of the g${\bm{g}}$ ‐vector fans of another generalization of the associahedron: nonkissing complexes (also known as support τ$\tau$ ‐tilting complexes) of gentle algebras. We show that the space of realizations of the nonkissing fan is simplicial when the gentle bound quiver is brick and 2‐acyclic, and we describe in this case its facet‐defining inequalities in terms of mesh mutations. Along the way, we prove algebraic results on 2‐Calabi–Yau triangulated categories, and on extriangulated categories that are of independent interest. In particular, we prove, in those two setups, an analogue of a result of Auslander on minimal relations for Grothendieck groups of module categories.


Introduction 2
Structure of the paper: main results and logical dependencies 6 Part I. Type cones of g-vector fans 9 1. Polytopal realizations and type cone of a simplicial fan 9 1. 1

Introduction
The associahedron is a "mythical polytope" [Hai84] whose facial structure encodes Catalan families: its vertices correspond to parenthesizations of a non-associative product, triangulations of a convex polygon, or binary trees; its edges correspond to applications of the associativity rule, diagonal flips, or edge rotations; and in general its faces correspond to partial parenthesizations, diagonal dissections, or Schröder trees. Its combinatorial and topological structure was introduced in early works of D. Tamari [Tam51] and J. Stasheff [Sta63], and a first 3-dimensional polytopal model was realized by J. Milnor for the PhD defense of J. Stasheff. The first systematic polytopal realizations were constructed by M. Haiman [Hai84] and C. Lee [Lee89]. Since then, the associahedron has been largely "demystified" with several polytopal constructions and generalizations (some are discussed below). Its realizations can be classified into three species: the secondary polytopes [GKZ08,BFS90], the g-vectors realizations [Lod04, HL07, LP18, HLT11, HPS18, Pos09, PS12, PS15] and the d-vector realizations [CFZ02,CSZ15]. See [CSZ15] for a discussion of some of these realizations and their connections.
The associahedron appears as a fundamental structure in several mathematical theories, such as moduli spaces and topology [Sta63,Kel01], operads and rewriting theory [Str12,MTTV21], combinatorial Hopf algebras [LR98,CP17,Pil18], diagonal harmonics [BPR12,PRV17], mathematical physics [AHBHY18], etc. Another striking illustration of the ubiquity of the associahedron lies in the theory of cluster algebras introduced by S. Fomin and A. Zelevinsky in [FZ02] with motivation coming from total positivity and canonical bases. Cluster algebras are commutative rings generated by variables obtained from an initial seed by the discrete dynamical process of seed mutations. Finite type cluster algebras, whose mutation graph is finite, were classified in [FZ03a] using the Cartan-Killing classification for crystallographic root systems. Finite type cluster complexes were geometrically realized first by F. Chapoton, S. Fomin and A. Zelevinsky [CFZ02] based on the d-vector fans [FZ03b,FZ03a] of the bipartite initial seed, then by C. Hohlweg, C. Lange and H. Thomas [HLT11] based on the Cambrian fans [Rea06,RS09] of acyclic initial seeds, and finally by C. Hohlweg, V. Pilaud and S. Stella [HPS18] based on the g-vector fans [FZ07] with respect to an arbitrary initial seed. The resulting polytopes are known as generalized associahedra, and the classical associahedra corresponds to the type A cluster algebras. This paper focuses on a surprising construction of the associahedron that recently appeared in mathematical physics. Motivated by the prediction of the behavior of scattering particles, N. Arkani-Hamed, Y. Bai, S. He, and G. Yan recently described in [AHBHY18,Sect. 3.2] the kinematic associahedron. It is a class of polytopal realizations of the classical associahedron obtained as sections of a high-dimensional positive orthant with well-chosen affine subspaces. This construction provides a large degree of freedom in the choice of the parameters defining these affine subspaces, and actually produces all polytopes whose normal fan is affinely equivalent to that of J.-L. Loday's associahedron [Lod04] (see Section 2.1). These realizations were then extended by V. Bazier-Matte, G. Douville, K. Mousavand, H. Thomas and E. Yıldırım [BMDM + 18] in the context of finite type cluster algebras using tools from representation theory of quivers. More precisely, they fix a finite type cluster algebra A and consider the real euclidean space R V indexed by the set V of cluster variables of A. Starting from an acyclic initial seed Σ • , they consider a set M of mesh mutations and compute the corresponding linear dependences between the g-vectors, which can be interpreted as linear spaces in R V . Finally, they perturb these mesh linear spaces by a collection ∈ R M >0 of positive parameters and intersect the resulting perturbed mesh affine spaces with the positive orthant R V ≥0 (see Section 2.2). They show that the resulting polytope is always a generalized associahedron, and that its normal fan is the g-vector fan of A with respect to the initial acyclic seed Σ • . An implicit by-product of their construction is that the space of all polytopal realizations of the g-vector fan of A with respect to the initial acyclic seed Σ • is a simplicial cone.
In this paper, we revisit, extend and explore further this construction using a reversed approach. Given a complete simplicial fan F, we consider the space TC(F) of all its polytopal realizations. This space was called type cone in [McM73] and deformation cone in [Pos09,PRW08], who studied the case when F is the braid arrangement leading to the rich theory of generalized permutahedra (recently extended to all finite Coxeter arrangements in [ACEP20]). The type cone is known to be a polyhedral cone defined by a collection of inequalities corresponding to the linear dependences among the rays of F contained in pairs of adjacent maximal cones of F (see Definition 1.3). Our approach is based on an elementary but powerful observation: for any fan F, all polytopal realizations of F can be described as sections of a high dimensional positive orthant with a collection of affine subspaces parametrized by the type cone TC(F) (see Proposition 1.10); if moreover the type cone TC(F) is a simplicial cone, it leads to a simple parametrization of all polytopal realizations of F by a positive orthant corresponding to the facets of the type cone TC(F) (see Corollary 1.11). To prove that the type cone TC(F) is simplicial, we just need to identify which pairs of adjacent maximal cones of F correspond to the facets of TC(F) and to show that the corresponding linear dependences among their rays positively span the linear dependence among the rays of any pair of adjacent maximal cones of F. When applied to the g-vector fans of cluster algebras (see Section 2.2), the type cone approach yields all polytopal realizations of the g-vector fans and thus efficiently revisits and extends the construction of [BMDM + 18]. This new perspective has several advantages, as our proof uniformly applies to: • any initial seed, regardless of whether it is acyclic or not (see Example 2.18 and Figures 6 to 8 for examples in types A 3 and C 3 cyclic). In contrast, the proof of [BMDM + 18] only treats acyclic initial seeds (although H. Thomas announced in a lecture given at RIMS in June 2019 that this restriction will disappear in a future version of [BMDM + 18]). • all finite type cluster algebras, regardless of whether it is simply-laced or not (see Example 2.18 for an example in type C 3 ). In contrast, even with the acyclicity restriction, the result of [BMDM + 18] is first proved in simply-laced cases (A, D and E) using representation theory of quivers. Extension to the non simply-laced cases is then argued by a folding argument. Although not presented in detail, this argument is subtle and technical. • any positive real-valued parameters, regardless of whether they are rational or not. In contrast, the proof of [BMDM + 18] naturally applies to rational parameters and then requires an approximation argument.
These advantages all follow from one essential feature of the type cone approach. Namely, it enables to completely separate the algebraic aspects from the geometric aspects of the problem: • On the algebraic side, we study the relations between the g-vectors of a finite type cluster algebra and we show that the mesh mutations minimally generate these relations. For this, we use representation theory in the setting of 2-Calabi-Yau triangulated categories, and we adapt M. Auslander's proof that the relations in the Grothendieck group of an Artin algebra are generated by the ones given by almost-split sequences precisely when the algebra is of finite representation type [Aus84] (see Section 3). • On the geometric side, we use elementary manipulations to show that a fan whose type cone is simplicial is affinely equivalent to the normal fan of sections of a high dimensional positive orthant with well-chosen affine subspaces (see Section 1). We then observe that our algebraic result precisely states that all type cones of g-vector fans of cluster algebras are simplicial from which we derive all our geometric consequences (see Section 2).
These two aspects are interesting in their own right and might be useful in different communities: the geometric side enables to simply describe all polytopal realizations of a fan with simplicial type cone (which is sometimes easier to check than to find an explicit realization), while the algebraic side provides fundamental new results on the g-vectors of cluster algebras and on (an extriangulated variant of) the Grothendieck groups of 2-Calabi-Yau triangulated categories with cluster tilting objects. We have therefore deliberately split our presentation into two clearly marked parts that can be read independently. The geometric side, which uses elementary techniques, is presented in Part I together with all its consequences on polytopal realizations. The algebraic side, which is of independent interest, is presented in Part II.
Besides revisiting the construction of [AHBHY18, BMDM + 18] and extending it to any initial seed (acyclic or not) in any finite type cluster algebra (simply-laced or not), our type cone approach is also successful when applied to the g-vector fans of other families of generalizations of the associahedron. In the present paper, we explore specifically gentle associahedra. These complexes where constructed in [PPP21] as polytopal realizations of finite non-kissing complexes of gentle bound quivers. They encompass two families of simplicial complexes studied independently in the literature: on the one hand the grid associahedra introduced by T. K. Petersen, P. Pylyavskyy and D. Speyer in [PPS10] for a staircase shape, studied by F. Santos, C. Stump and V. Welker [SSW17] for rectangular shapes, and extended by T. McConville in [McC17] for arbitrary grid shapes; and on the other hand the Stokes polytopes and accordion associahedra studied by Y. Baryshnikov [Bar01], F. Chapoton [Cha16], A. Garver and T. McConville [GM18] and T. Manneville and V. Pilaud [MP19]. The latter, sometimes called accordiohedra, have also recently appeared in the mathematical physics litterature concerning scattering amplitudes [BLR19,Ram19], with realizations inspired by those in [AHBHY18]. It was shown in [PPP21,BDM + 20] that the nonkissing complex of a gentle bound quiver provides a combinatorial model for the support τ -tilting complex [AIR14] of the associated gentle algebra. Even for finite non-kissing complexes, it turns out that the type cone of the non-kissing fan is not always simplicial (see Remark 2.47). We prove however that the type cone is always simplicial when the gentle bound quiver is brick (at least 2 relations in any cycle) and 2-acyclic (no oriented cycle of length 2). Moreover, we precisely describe the facet-defining inequalities of the type cone in terms of linear dependences of g-vectors involved in certain mesh relations. We thus automatically derive a construction similar to that of [AHBHY18, BMDM + 18] describing all polytopal realizations of the g-vector fans of these brick and 2-acyclic gentle bound quivers as sections of a positive orthant by well-chosen affine spaces (see Section 2.2). Again, this result mainly relies on finding which relations minimally generate the relations between the g-vectors in the non-kissing complex. We present both a purely combinatorial proof (see Section 2.2) and a purely algebraic proof for (non-necessarily gentle) brick algebras making use of extriangulated categories (see Section 4). Our algebraic proof is based on an analogue of a result of M. Auslander [Aus84] relating to minimal generators of Grothendieck groups. Along the way, we need to generalize, for extriangulated categories, several results from cluster-tilting theory on indices [DK08] and on abelian quotients [BMR07,KR07] that were known for triangulated categories. We note, however, that our assumptions are not quite the expected ones. This is related to the fact that the Grothendieck groups of cluster categories are badly behaved (e.g. the Grothendieck group of a cluster category of type A 2 is trivial). So as to overcome this difficulty, we replace the triangulated structure of cluster categories by some relative extriangulated structure, before considering Grothendieck groups. This explains why we consider projective objects rather than cluster-tilting objects for extriangulated categories.
Further families of generalizations of the associahedron certainly deserve to be studied with our type cone approach. We are investigating in particular graph associahedra 1 [CD06,Pos09,FS05,Zel06], brick polytopes [PS12,PS15] and quotientopes [Rea04,Rea06,PS19]. In fact, as already mentioned, it seems sometimes easier to find all polytopal realizations of a fan (by describing its type cone in terms of certain specific pairs of adjacent maximal cones) than to identify one specific explicit realization. This is clearly the case when the type cone is simplicial as illustrated by the results of this paper: we naturally obtained all polytopal realizations of the g-vector fans of cluster algebras from cyclic seeds and of gentle algebras whose polytopality was only established very recently [HPS18,PPP21]. We believe that this approach gives reasonable hope that the polytopality of further fans could in the future be established using our type cone approach. Tempting candidates are quotientopes for hyperplane arrangements beyond type A, which are particularly interesting since the coarsenings of a fan essentially correspond to the faces of its type cone (see Section 1.5).
Last but not least, the type cone approach naturally opens many research questions. First, it raised the fundamental problem of describing the minimal relations among g-vectors in 2-Calabi-Yau and extriangulated categories that we treat in Part II. It also motivates the study of relevant objects revealed by the type cone dictionary between geometry and algebra: • The facet-defining inequalities of the type cone of F correspond to very specific pairs of adjacent maximal cones of F whose corresponding linear dependences minimally generate all such dependences. While they correspond to mesh mutations for cluster algebras and brick and 2-acyclic gentle algebras, they are not understood in general (see for instance Figure 14 (right)). It would be particularly interesting to fully understand these specific mutations for arbitrary (non-kissing finite) gentle algebras. • The rays of the type cone of F correspond to specific polytopes which form a positive Minkowski basis of all realizations of F (see Section 1.6). In particular, when the type cone is simplicial, any polytopal realization of F has a unique representation (up to translation) as a Minkowski sum of positive dilates of these polytopes. For cluster algebras, it follows from [BMDM + 18, Sect. 6] that these polytopes are Newton polytopes of the F -polynomials. It would be interesting to obtain similar algebraic interpretations for the rays of the type cones of the non-kissing fans of gentle algebras. • Once we get a representation of a polytope P as a signed Minkowski sum P = i∈[k] α i Q i , we get a formula for the volume of P as a multivariate polynomial in α 1 , . . . , α k whose coefficients are the mixed volumes of Q 1 , . . . , Q k [ABD10,McM77]. These encode rich combinatorial information when representing generalized permutahedra as positive Minkowski sums of coordinate simplices or hypersimplices [Pos09], and in particular for matroid polytopes [ABD10]. It would be interesting to get such volume formulas for polytopal realizations of g-vector fans of cluster algebras and gentle algebras, and combinatorial interpretations for their coefficients.
Another question related to this paper is to understand the complete realization space of the combinatorics of the associahedra. The associahedra constructed in [AHBHY18, BMDM + 18] and the present paper describe all polytopal realizations of the g-vector fans, but there are other polytopal realizations of the combinatorial type of the associahedron with different normal fans. Is it possible to similarly understand the complete realization space of the cluster complex, including all realizations not having the g-vector fan as their normal fan?
To conclude, it is our hope that the present paper will participate to the interactions between combinatorial geometry, representation theory, and mathematical physics; in particular through further developments of the geometric approach to scattering amplitudes that has flourished during the last years [AHT14, AHBC + 16]. The kinematic associahedron from [AHBHY18] is one of the most recent"positive geometries" [AHBL17] that has emerged in the study of scattering amplitudes.

Structure of the paper: main results and logical dependencies
We now give a more detailed overview of our main results and the structure of the article. In particular, some of the geometric realizations in Part I depend on algebraic results whose proofs are delayed until Part II. The goal of this section is to clarify their logical dependencies, summarized in the schematic diagram depicted at the end.
Type cone approach. (Section 1) Let F be an essential complete polyhedral fan in R n (all definitions are recalled in Section 1.1). The type cone TC(F) of F (Section 1.2), introduced by P. McMullen [McM73], is a polyhedral cone that parametrizes the set of all possible polytopal realizations of F. These realizations can be described explicitly as affine sections of the non-negative orthant R N ≥0 , where N is the number of rays of F (Proposition 1.10). If F is a simplicial fan and has the unique exchange relation property (Definition 1.8), then its type cone is easier to describe and study. And if moreover the type cone TC(F) is simplicial (Remark 1.6), then all possible polytopal realizations of F can be described explicitely in terms of N − n positive real parameters as follows Corollary 1.11. Assume that the type cone TC(F) is simplicial and let K be the (N −n)×N -matrix whose rows are the inner normal vectors of the facets of TC(F). Then the polytope R := z ∈ R N Kz = and z ≥ 0 is a realization of the fan F for any positive vector ∈ R N −n >0 . Moreover, the polytopes R for ∈ R N −n >0 describe all polytopal realizations of F.
This gives a new point of view on [AHBHY18, BMDM + 18], illustrated in Section 2.1 for classical associahedra, that we then apply to two families of g-vector fans generalizing the g-vector fans of classical associahedra: the g-vector fans of cluster algebras of finite type (generalized associahedra) in Section 2.2, and the g-vector fans of gentle algebras (non-kissing associahedra, non-crossing associahedra, generalized accordiohedra) in Section 2.3.
Cluster algebras of finite type and generalized associahedra. (Section 2.2) We first consider the g-vector fan F(B • ) of a skew-symmetrizable cluster algebra of finite type, with respect to any initial exchange matrix B • (acyclic or not). We provide the following polytopal realizations of F(B • ). , the polytope is a generalized associahedron, whose normal fan is the cluster fan F(B • ). Moreover, the poly- Here, M(B • ) denotes the set of all pairs {x, x } related by non-initial mesh mutations (Definition 2.19), and V(B • ) the set of cluster variables. For {x, x } ∈ M(B • ) and y ∈ V(B • ), the coefficient α x,x (y) is defined as follows: α x,x (y) is set to be |b xy | if y ∈ X ∩ X and 0 otherwise, where X, X are any two clusters such that X {x} = X {x }. By the unique exchange relation property, the coefficients α x,x (y) do not depend on the specific choice of clusters X, X .
Some crucial pieces of the proof of Theorem 2.26 use cluster categories (2-Calabi-Yau triangulated categories) and are deferred to Section 3. Namely, a reformulation of [BMR + 06, Thm. 7.5], proved in Section 3, gives the first step towards applying our type cone strategy. And the second step follows from our main result in Section 3, which makes use of representation theory of finite dimensional algebras and additive categorification of cluster algebras.
Gentle algebras, non-kissing complexes, and generalized accordiohedra. (Section 2.3) The same strategy is applied to a class of finite dimensional algebras, called gentle algebras, whose representations are combinatorially well understood. We note that the τ -tilting theory of gentle algebras is the algebraic counterpart of the combinatorics of accordions on surfaces [PPP21, BDM + 20, PPP19]. We identify two conditions, called brick and 2-acyclicity (Definition 2.40), that enable us to make use of the type cone approach. LetQ = (Q, I) be a gentle bound quiver, and let F(Q) be the fan of its g-vectors (Section 2.3.1), also called non-kissing fan. We provide the following polytopal realizations of F(Q).
Theorem 2.44. For any brick and 2-acyclic gentle quiverQ and any ∈ R S(Q) >0 , the polytope R (Q) := z ∈ R W(Q) z ≥ 0, z ω = 0 for any straight or self-kissing walk ω, is a realization of the non-kissing fan F(Q). Moreover, the polytopes R (Q) for ∈ R S(Q) >0 describe all polytopal realizations of F(Q).
Here, S(Q) is the set of all strings ofQ, and W(Q) the set of all walks onQ (i.e. of all strings of the blossoming bound quiver ofQ that join two blossom vertices). If σ is a string ofQ, we make use of the evocative notations σ , σ , σ , σ for the walks obtained by adding, in the blossoming bound quiver, hooks or cohooks at each end of σ.
The two main ingredients are similar to those for cluster algebras. As a first step, we obtain the following consequence of Proposition 2.34 (ii), which is proved in Section 4.6 by representationtheoretic methods.
Proposition 2.34 (iv). The non-kissing fan F(Q) has the unique exchange relation property.
The second step is the following statement, for which we provide a purely combinatorial proof in Section 2.3.3 and an algebraic proof in Section 4.6, using the main result of Section 4. Corollary 2.43. If the gentle bound quiverQ is brick and 2-acyclic, then the type cone TC F(Q) is simplicial.

Cluster categories. (Section 3)
We prove an analogue for cluster categories of a result of M. Auslander for module categories of Artin algebras [Aus84]. We are mainly motivated by the fact that it implies that the type cones of the cluster fans of finite type are simplicial. However, this result is of independent interest and sheds new lights on the Grothendieck groups of cluster categories.
Let C be a cluster category with finitely many isomorphism classes of indecomposable objects (see Section 3.1 for the precise and more general setting in which the theorem is proven). Let K sp 0 (C) be the split Grothendieck group of C. Fix a cluster-tilting object T ∈ C, and let , where E is the middle term of an almost split triangle starting at X. For any objects X, Y ∈ C, we write X, Y for dim K Hom Λ (F X, F Y ), where Λ is the cluster-tilted algebra End C (T ) and F is the equivalence of categories C(T, −) : C/(ΣT ) → mod Λ.
Theorem 3.8. The set L C := { X | X ∈ ind(C) add(ΣT )} is a basis of the kernel of g and, for any x ∈ ker(g), we have Extriangulated categories. (Section 4) We prove a more general version of Theorem 3.8 that applies not only to cluster fans, but also to non-kissing fans. Once more, our main motivation is to prove that the type cone of the non-kissing fan of a brick and 2-acyclic gentle algebra is simplicial, but the results are of independent interest, and might be useful for studying the type cones of other families of simplicial fans.
Let C be an extriangulated category (Section 4.4) with a fixed projective object T such that the morphism T → 0 is an inflation and, for each X ∈ C, there is an extriangle T X Let ΣT be the cone of the inflation T 0.
Our assumptions allow for a well-defined notion of index: . Proposition 4.11. The index induces an isomorphism of abelian groups from K 0 (C) to K 0 (add T ).
The previous two statements also hold when T is an additive subcategory rather than a mere object. Assume moreover that C is K-linear, Ext-finite, Krull-Schmidt and has Auslander-Reiten-Serre duality. Then, with notations analogous to those used for cluster categories above, we obtain the following statement.
Theorem 4.13. The extriangulated category C has only finitely many isomorphism classes of indecomposable objects if and only if the set L C generates the kernel of the canonical projection g : K sp 0 (C) → K 0 (C). In that case L C is a basis of the kernel of g and any x ∈ ker(g) decomposes as Schematic diagram of logical dependencies.
We conclude with a schematic diagram representing the logical dependencies between the different sections of the paper. We briefly recall basic definitions and properties of polyhedral fans and polytopes, and refer to [Zie98] for a classical textbook on this topic. A hyperplane H ⊂ R n is a supporting hyperplane of a set X ⊂ R n if H ∩ X = ∅ and X is contained in one of the two closed half-spaces of R n defined by H.
We denote by R ≥0 R := r∈R λ r r λ r ∈ R ≥0 the positive span of a set R of vectors of R n . A polyhedral cone is a subset of R n defined equivalently as the positive span of finitely many vectors or as the intersection of finitely many closed linear halfspaces. The faces of a cone C are the intersections of C with the supporting hyperplanes of C. The 1-dimensional (resp. codimension 1) faces of C are called rays (resp. facets) of C. A cone is simplicial if it is generated by a set of linearly independent vectors.
A polyhedral fan is a collection F of polyhedral cones such that • if C ∈ F and F is a face of C, then F ∈ F, • the intersection of any two cones of F is a face of both.
A fan is simplicial if all its cones are simplicial, complete if the union of its cones covers the ambient space R n , and essential if it contains the cone {0}. Note that every complete fan is the product of an essential fan with its lineality space (the largest linear subspace contained in all the cones). For two fans F, G in R n , we say that F refines G (and that G coarsens F) if every cone of F is contained in a cone of G.
A polytope is a subset P of R n defined equivalently as the convex hull of finitely many points or as a bounded intersection of finitely many closed affine halfspaces. The dimension dim(P ) is the dimension of the affine hull of P . The faces of P are the intersections of P with its supporting hyperplanes. The dimension 0 (resp. dimension 1, resp. codimension 1) faces are called vertices (resp. edges, resp. facets) of P . A polytope is simple if each vertex is incident to dim(P ) facets (or equivalently to dim(P ) edges).
The (outer) normal cone of a face F of P is the cone generated by the outer normal vectors of the facets of P containing F . In other words, it is the cone of vectors c such that the linear form x → c | x on P is maximized by all points of the face F . The (outer) normal fan of P is the collection of the (outer) normal cones of all its faces. We say that a complete polyhedral fan F in R n is polytopal when it is the normal fan of a polytope P of R n , and that P is a polytopal realization of F.

Type cone.
Fix an essential complete simplicial fan F in R n , with an (arbitrary) indexing of its rays by [N ] := {1, . . . , N }. Let G be the N × n-matrix whose rows are representative vectors of the rays of F. Let K be a (N − n) × N -matrix that spans the left kernel of G (i.e. KG = 0 and rank(K) = N − n). For any height vector h ∈ R N , we define the polytope where Gx ≤ h is the standard shorthand for the corresponding system of inequalities.
We say that h is F-admissible if P h is a polytopal realization of F. The following classical statement characterizes the F-admissible height vectors. It is a reformulation of regularity of triangulations of vector configurations, introduced in the theory of secondary polytopes [GKZ08], see also [DRS10]. We present here a convenient formulation from [CFZ02, Lem. 2.1]. Proposition 1.1. Let F be an essential complete simplicial fan in R n . Then the following are equivalent for any height vector h ∈ R N : (1) The fan F is the normal fan of the polytope P h := {x ∈ R n | Gx ≤ h}.
(2) For any two adjacent maximal cones R ≥0 R and R ≥0 R of F with R {r} = R {r }, is the unique (up to rescaling) linear dependence with α, α > 0 between the rays of R ∪ R .
Notation 1.2. For any adjacent maximal cones R ≥0 R and R ≥0 R of F with R {r} = R {r }, we denote by α R,R (s) the coefficient of s in the unique linear dependence between the rays of R ∪ R , i.e. such that s∈R∪R α R,R (s) s = 0.
These coefficients are a priori defined up to rescaling, but we additionally fix the rescaling so that α R,R (r) + α R,R (r ) = 2 (this convention is arbitrary, but will be convenient in Section 2).
In this paper, we are interested in the set of all possible realizations of F as the normal fan of a polytope P h . This was studied by P. McMullen in [McM73] (see [DRS10,Sect. 9.5] for a formulation in terms of chambers of triangulations of vector configurations).
for any adjacent maximal cones R ≥0 R and R ≥0 R of F .
Note that the type cone is an open cone and contains a lineality subspace of dimension n (it is invariant by translation in GR n ). It is sometimes useful to get rid of the lineality space by considering the projection KTC(F).
We denote by TC(F) the closure of TC(F), and call it the closed type cone of F. It is the closed polyhedral cone defined by the inequalities s∈R∪R α R,R (s) h s ≥ 0 for any adjacent maximal cones R ≥0 R and R ≥0 R . If F is the normal fan of the polytope P , then TC(F) is called the deformation cone of P in [Pos09], see also [PRW08].  Figure 1 (left) (we will encounter this fan again in Examples 1.14, 1.19 and 2.5). It has five rays labeled 1, . . . , 5 and five maximal cones labeled a, . . . , e. For the matrices G and K, we consider Figure 1. A 2-dimensional fan F with five rays 1, . . . , 5 and five maximal cones a, . . . , e (left), its polytopal realization corresponding to the height vector ( 1 /2, 3 /4, 1, 1, 5 /4) (middle), and the intersection of KTC(F) with a hyperplane to get a 2-dimensional slice (right).
The type cone TC(F) lies in R 5 , but has a 2-dimensional lineality space. The five pairs of adjacent maximal cones of F give rise to following five defining inequalities for TC(F): ij denotes the halfspace defined by the inequality corresponding to the two adjacent maximal cones i and j. Note that the inequalities H > ae and H > de are redundant. For example, the height vector ( 1 /2, 3 /4, 5 /4, 1, 5 /4) belongs to the type cone TC(F), and the corresponding polytope is represented in Figure 1 (middle). To represent this type cone, it suffices to represent KTC(F) which is an essential simplicial cone in R 3 , given as the intersection of the following five open halfspaces: One can further reduce the dimension by intersecting with a transversal hyperplane to get a 2-dimensional representation. This is the red triangle depicted in Figure 1 (right). In other words, extremal adjacent pairs define the extremal rays of the polar of the closed type cone TC(F). Understanding the extremal adjacent pairs of F enables to describe its type cone TC(F) and thus all its polytopal realizations. For instance, for the 2-dimensional fan of Example 1.4, the extremal adjacent pairs are {a, b}, {b, c} and {c, d}. Remark 1.6. Since the type cone is an N -dimensional cone with a lineality subspace of dimension n, it has at least N − n facets (thus N − n extremal adjacent pairs). The type cone is simplicial when it has precisely N − n facets.
Although we will only deal with simplicial fans, note for completeness that Proposition 1.1 and Definition 1.3 can be easily adapted to non-simplicial fans. To describe the type cone of an arbitrary complete fan, it suffices to consider any simplicial refinement, and to set some of the strict inequalities of the definition of the type cone to equality (see [PS19,Prop. 3] for a proof). Proposition 1.7. Let F be a complete fan that coarsens the essential complete simplicial fan G. The type cone TC(F) of all F-admissible height vectors h ∈ R N is for any adjacent maximal cones R ≥0 R and R ≥0 R of G belonging to the same maximal cone of F, We say that the fan F has the unique exchange relation property if any two exchangeable rays r, r of F admit a unique exchange relation.
When F has the unique exchange relation property, we change the notation α R,R (s) to α r,r (s) and we obtain that the type cone of F is expressed as TC(F) = h ∈ R N s α r,r (s) h s > 0 for any exchangeable rays r and r of F . Definition 1.9. In a fan F with the unique exchange relation property, an extremal exchangeable pair is a pair of exchangeable rays {r, r } such that the corresponding inequality s α r,r (s) h s ≥ 0 defines a facet of the closed type cone TC(F).
In this paper, we will only consider fans with the unique exchange relation property, and our objective will be to describe their extremal exchangeable pairs. 1.4. Alternative polytopal realizations. In this section, we provide alternative polytopal realizations of the fan F. We also discuss the behavior of these realizations in the situation when the type cone TC(F) is simplicial.
We still consider an essential complete simplicial fan F in R n , the N × n-matrix G whose rows are the rays of F, and an (N − n) × N -matrix K that spans the left kernel of G (i.e. KG = 0 and rank(K) = N − n).
Proof. This result is standard and is proved for instance in [DRS10, Coro. 9.5.7]. We include a short argument here for the convenience of the reader. For x in P h , we have Ψ(x) ≥ 0 by definition and KΨ(x) = Kh−KGx = Kh since K is the left kernel of G. Therefore Ψ(x) ∈ Q h . Moreover, the map Ψ : P h → Q h is: • injective: Indeed, Ψ(x) = Ψ(x ) implies G(x − x ) = 0 and G has full rank since F is essential and complete. • surjective: Indeed, for z ∈ Q h , we have K(h − z) = 0 so that h − z belongs to the right kernel of K, which is the image of G (because G is of full rank). Hence, there exists x ∈ R n such that h − z = Gx. We have z = Ψ(x) and x ∈ P h since h − Gx = z ≥ 0.
Note that K is only defined up to linear transformation, as it depends on the choice of a basis of the left kernel of G. Note also that, by construction, the coefficients of the normal vectors of the facets of TC(F) arise from a linear dependence among the rays of F, and hence belong to this kernel. When TC(F) is simplicial, its (N − n) facets are necessarily linearly independent, and hence form a basis. They provide a particularly interesting choice of K. Corollary 1.11. Assume that the type cone TC(F) is simplicial and let K be the (N −n)×N -matrix whose rows are the inner normal vectors of the facets of TC(F). Then the polytope is a realization of the fan F for any positive vector ∈ R N −n >0 . Moreover, the polytopes R for ∈ R N −n >0 describe all polytopal realizations of F.
Proof. Let ∈ R N −n >0 . Since K has full rank there exists h ∈ R N such that Kh = . Since Kh ≥ 0 and the rows of K are precisely all inner normal vectors of the facets of the type cone TC(F), we obtain that h belongs to TC(F). Since R = Q h ∼ P h by Proposition 1.10, we conclude that R is a polytopal realization of F. Since TC(F) is simplicial, we have KTC(F) = R N −n >0 , so that we obtain all polytopal realizations of F in this way.
1.5. Faces of the type cone. We now relate the faces of the closed type cone TC(F) with the coarsenings of the fan F. Proof. Let G be a simplicial refinement of F. By Proposition 1.7, the type cone of any coarsening of F is a relatively open cell of the hyperplane arrangement defined by adjacent pairs of G. The type cone of the coarsening is not empty whenever it is polytopal. When non-empty, distinct coarsenings give rise to distinct cells (one will have an equality that is a strict inequality in the other), and all lie in the closure of TC(F) (since they are all in the closure of all the defining half-spaces). Thus every coarsening describes a relatively open face of TC(F).
For the converse, observe that for h ∈ TC(F), the normal fan of P h is the polytopal complete fan obtained by merging all pairs of adjacent maximal cones R ≥0 R and R ≥0 R of F such that Remark 1.13. Some observations are in place: • Note that any extremal adjacent pair {R ≥0 R, R ≥0 R } of F gives rise to a facet of TC(F), but that this pair might not be unique for this facet. There might be several extremal adjacent pairs giving rise to the same facet. This happens often when we have the unique exchange relation property and there are many pairs of maximal cones separating the same exchangeable rays. • Mergings induced by extremal adjacent pairs are not the only ones that can take place in the boundary of TC(F). Indeed, if G is the coarsening defining a lower-dimensional face TC(G) of TC(F), then to get F we have to perform all the coarsenings given by facets of TC(F) containing TC(G). But we might have to perform more coarsenings arising from exchangeable pairs {R ≥0 R, R ≥0 R } whose inequality is redundant for TC(F) but tight for all points in TC(G). This situation arises when, in order to have a polytopal polyhedral fan, more pairs of adjacent cones must be merged. • Even if F is essential, not all its polytopal coarsenings need to be. Indeed, it can happen that there is a certain polytopal coarsening G in which all the cells have a common lineality space. In this case, G is the normal fan of a lower dimensional polytope (whose codimension is the dimension of the lineality space of G). For this to happen, it is necessary (but not sufficient) that there is a positive linear dependence between some rays of F. Example 1.14. Consider the 2-dimensional fan F of Figure 1 (left) whose type cone has been described in Example 1.4 and represented in Figure 1 (right). The relative interior of the three facets of TC(F) are the type cones of three coarsenings of F, obtained by merging certain adjacent maximal cones (those that are extremal). One of the rays generating KTC(F) corresponds to a triangle, and its normal fan is obtained by merging two pairs of extremal adjacent maximal cones. The two remaining rays lie not only in hyperplanes H ij defining facets of the type cone, but also in one extra such hyperplane. Their normal fans have only two cones, one arising as a merging of three cones and one arising as a merging of two cones. These fans are not essential, they have a one-dimensional lineality space. Hence, the corresponding polytopes are not full-dimensional, they are of codimension 1.
1.6. Minkowski sums. We now discuss the connection between convex combinations in the type cone and Minkowski sums of polytopes. Recall that the Minkowski sum of two polytopes P and Q is the polytope P + Q := {p + q | p ∈ P and q ∈ Q}.
Lemma 1.15. For h, h ∈ R N , the polytope P h+h is the Minkowski sum of the polytopes P h and P h .
Proof. This follows from the definition of P h and of Minkowski sums: We say that P is a Minkowski summand of R if there is a polytope Q such that P +Q = R, and a weak Minkowski summand if there is a λ ≥ 0 and a polytope Q such that P + Q = λR. Lemma 1.15 ensures that convex combinations in the type cone correspond to Minkowski combinations of polytopes. Actually, the set of weak Minkowski summands of a polytope P with normal fan F is isomorphic to the closed type cone TC(F) modulo lineality (the cone of weak Minkowski summands was studied by W. Meyer in [Mey74], to see that it is equivalent to the type cone and other formulations, see for example the appendix of [PRW08]).
This provides natural Minkowski summands for all polytopal realizations of F: the rays of the closed type cone. These correspond to indecomposable polytopes, see [Mey74] or [McM73]. A polytope P ⊆ R n is called indecomposable if all its weak Minkowski summands are of the form λP + t for some λ ≥ 0 and t ∈ R n . Corollary 1.16. Any polytope in TC(F) is the Minkowski sum of at most N − n indecomposable polytopes corresponding to some rays of KTC(F).
If P = Q + R, then we say that R is the Minkowski difference of P and Q, denoted R = P − Q. Note that this is only defined when Q is a Minkowski summand of P . For instance, we have P h − P h = P h−h when P h is a Minkowski summand of P h . There are two natural ways this construction generalizes to arbitrary pairs of polyhedra. On the one hand, one can consider that On the other hand, considering arbitrary differences of support functions gives rise to virtual polyhedra, which is the Grothendieck group of the semigroup of polytopes with Minkowski sums [PK92]. Proof. It suffises to express the projection of h onto the left kernel of G in terms of h (1) , . . . , h (N −n) to obtain the coefficients of P h as a signed Minkowski sum of dilates of P h (1) , . . . , P h (N −n) . This is used in [ABD10] to show that every generalized permutahedron can be written uniquely as a signed Minkowski sum of simplices. In the particular case when the type cone is simplicial, then we can get rid of the Minkowski differences. Corollary 1.18. If the type cone TC(F) of F is simplicial, then every P h with h ∈ TC(F) (resp. with h ∈ TC(F)) has a unique representation, up to translation, as a Minkowski sum of positive (resp. non-negative) dilates of the N − n indecomposable polytopes P h (1) , . . . , P h (N −n) arising from the rays Note that such a family of polytopes giving rise to unique positive/non-negative representations only exists when the type cone is simplicial. We get unicity with any basis, but in general we cannot guarantee non-negativity unless we are in the simplicial cone they span. If we take all rays of the type cone, we get non-negativity, but the representation is not unique unless the cone is simplicial. Figure 1 (left) whose type cone has been described in Example 1.4 and represented in Figure 1 (right). As illustrated in Figure 2, the rays of the closed type cone TC(F) are directed by the height vectors (1, 2, 1, 2, 1) (corresponding to a triangle), (0, 1, 1, 0, 1) (corresponding to a vertical segment) and (1, 0, 1, 1, 0) (corresponding to a horizontal segment). Therefore, any polytope in TC(F) is (up to translation) a positive Minkowski combination of a triangle and two edges. Figure 3 illustrates this property for the polytope of Figure 1 (middle) corresponding to the height vector ( 1 /2, 3 /4, 5 /4, 1, 5 /4). Note that F is the normal fan of J.-L. Loday's 2-dimensional associahedron, whose realization as a Minkowski sum of faces of a simplex was given by A. Postnikov in [Pos09].

Applications to two generalizations of the associahedron
In this section, we study the type cones of complete simplicial fans arising as normal fans of two families of generalizations of the associahedron: the generalized associahedra of finite type cluster algebras [FZ02, FZ03a, FZ07, HLT11, HPS18] and the gentle associahedra [PPP21]. Both families contain the classical associahedra Asso(n) constructed in [SS93, Lod04].
2.1. Classical associahedra. We first describe the associahedra of [SS93,Lod04] and their type cones as they are the prototypes of our constructions.
2.1.1. Associahedra and their normal fans. We quickly recall the combinatorics and the geometric construction of [SS93,Lod04] for the associahedron Asso(n). The face lattice of Asso(n) is the reverse inclusion lattice of dissections (i.e. pairwise non-crossing subsets of diagonals) of a convex (n + 3)-gon. In particular, its vertices correspond to triangulations of the (n + 3)-gon and its facets correspond to internal diagonals of the (n + 3)-gon. Equivalently, its vertices correspond to rooted binary trees with (n + 1) internal nodes, and its facets correspond to proper intervals of [n + 1] := {1, . . . , n + 1} (i.e. intervals distinct from ∅ and [n + 1]). These bijections become clear when the convex (n + 3)-gon is drawn with its vertices on a concave curve labeled by 0, . . . , n + 2.
The following statement provides three equivalent geometric constructions of Asso(n).
Theorem 2.1. The associahedron Asso(n) can be described equivalently as: • the convex hull of the points L(T ) ∈ R n+1 for all rooted binary trees T with n + 1 internal nodes, where the ith coordinate of L(T ) is the product of the number of leaves in the left subtree by the number of leaves in the right subtree of the ith node of T in inorder [Lod04], with the halfspaces  We now focus on the normal fan of Asso(n). Since Asso(n) lives in a hyperplane of R n+1 , its normal fan has a one-dimensional lineality. Let H := x ∈ R n+1 ∈[n+1] x = 0 and π : R n+1 → H denote the orthogonal projection. We denote by F(n) the intersection of the normal fan of Asso(n) with H, which is an essential complete simplicial fan.
• the normal vector of the facet corresponding to an internal diagonal (a, b) of the (n+3)-gon is the vector g(a, b) := π a< <b e = (n + 1) a< <b e − (b − a − 1) 1≤ ≤n+1 e . • the normal cone of the vertex corresponding to a rooted binary tree T is the incidence cone x ∈ H x i ≤ x j for all edges i → j in T . Figure 4 and Example 2.5 in dimension 2 and 3. Note that in Figure 4 and Example 2.5 we express the g-vectors in the basis given by g(0, i + 1) for i ∈ [n] (to be coherent with the upcoming Definition 2.10).

Theorem 2.2 is illustrated in
Let us also recall the linear dependencies in this fan and observe that it has the unique exchange relation property discussed in Section 1.3. From now on, we use the convention that g(a, b) = 0 when (a, b) is a boundary edge of the (n + 3)-gon. Proposition 2.3. Let (a, b) and (a , b ) be two crossing diagonals with 0 ≤ a < a < b < b ≤ n + 2, and let T and T be any two triangulations such that Then both triangulations T and T contain the square aa bb , and the linear dependence between the g-vectors of T ∪ T is given by In particular, the fan F(n) has the unique exchange relation property.

2.1.2.
Type cones of associahedra. From the linear dependencies of Proposition 2.3, we obtain a redundant description of the type cone of the fan F(n). To simplify the presentation and write the wall-crossing inequalities in a uniform way, we embed the type cone in a larger space, adding dummy variables for the boundary edges of the polygon.
Corollary 2.4. Let n ∈ N and X(n) := {(a, b) | 0 ≤ a < b ≤ n + 2}. Then the type cone of the normal fan F(n) of Asso(n) is given by Example 2.5. Consider the fan F(2) and F(3) illustrated in Figure 4. The type cone of F(2) has dimension 5 and a lineality space of dimension 2 (this is the type cone studied in Examples 1.4, 1.14 and 1.19). It has 3 facet-defining inequalities (given below), which correspond to the flips described in Proposition 2.6 and illustrated in Figure 5 (left). diagonals g-vectors 1 0 In all our tables, we just record the coefficients of the inequalities, which are then easily reconstructed. For instance, the inequality A above is given by h + h > h . The type cone of F(3) has dimension 9 and a lineality space of dimension 3. It has 6 facetdefining inequalities (given below), which correspond to the flips described in Proposition 2.6 and illustrated in Figure 5 (right). diagonals g-vectors Motivated by Example 2.5, we now describe the facets of this type cone TC F(n) .
Proposition 2.6. Two internal diagonals (a, b) and (a , b ) of the (n + 3)-gon form an extremal exchangeable pair for the fan F(n) if and only if a = a + 1 and b = b + 1, or the opposite.
Proof. Let (f (a,b) ) 0≤a<b≤n+2 be the canonical basis of R ( n+3 2 ) . Consider two crossing internal diagonals (a, b) and (a , b ) with 0 ≤ a < a < b < b ≤ n + 2. By Proposition 2.3, the linear dependence between the corresponding g-vectors is given by Therefore, the inner normal vector of the corresponding inequality of the type cone TC F(n) is Indeed, on the right hand side, the basis vector f (c,d) appears with a positive sign in m(c, d This shows that any exchange relation is a positive linear combination of the exchange relations corresponding to all pairs of diagonals (a, b) and (a , b ) of the (n + 3)-gon such that a = a + 1 and b = b + 1 or the opposite. Conversely, since F(n) has dimension n and n(n + 3)/2 rays (corresponding to the internal diagonals of the (n + 3)-gon), we know from Remark 1.6 that there are at least n(n + 1)/2 extremal exchangeable pairs. We thus conclude that all exchangeable pairs of diagonals {(a, b−1), (a+1, b)} for 1 ≤ a < b − 2 ≤ n are extremal.
Our next statement follows from the end of the previous proof.
Corollary 2.7. The type cone TC F(n) is simplicial.
Combining Corollaries 1.11 and 2.7 and Proposition 2.6, we derive the following description of all polytopal realizations of the fan F(n), recovering all associahedra of [AHBHY18,Sect. 3.2]. Note that the arguments of [AHBHY18, Sect. 3.2] were quite different from the present approach.
is an n-dimensional associahedron, whose normal fan is F(n). Moreover, the polytopes describe all polytopal realizations of the fan F(n).

2.2.
Finite type cluster complexes and generalized associahedra. Cluster algebras were introduced by S. Fomin and A. Zelevinsky [FZ02] with motivation coming from total positivity and canonical bases. Here, we will focus on finite type cluster algebras [FZ03a] and more specifically on properties of their g-vectors [FZ07]. These g-vectors support a complete simplicial fan, which is called gvector fan or cluster fan. It is known to be the normal fan of a polytope called generalized associahedron. These polytopal realizations were first constructed for bipartite initial seeds by F. Chapoton, S. Fomin and A. Zelevinsky [CFZ02] using the d-vector fans of [FZ03b], then for acyclic initial seeds by C. Hohlweg, C. Lange and H. Thomas [HLT11] using Cambrian lattices and fans of N. Reading and D. Speyer [Rea06,RS09], then revisited by S. Stella [Ste13], and by V. Pilaud  ] to any finite type (simply-laced or not) cluster algebra with respect to any seed (acyclic or not) by providing a direct proof that the type cone of the cluster fan of any finite type cluster algebra with respect to any initial seed is simplicial.
2.2.1. Cluster algebras and cluster fans. We present some definitions and properties of finite type cluster algebras and their cluster fans, following the presentation of [HPS18].
Cluster algebras. Let Q(x 1 , . . . , x n , p 1 , . . . , p m ) be the field of rational expressions in n + m variables with rational coefficients, and P m denote its abelian multiplicative subgroup generated by • the exchange matrix B is an integer n × n skew-symmetrizable matrix, i.e. such that there exists a diagonal matrix D with −BD = (BD) T , • the coefficient tuple P is any subset of n elements of P m , • the cluster X is a set of n cluster variables in Q(x 1 , . . . , x n , p 1 , . . . , p m ) algebraically independent over Q(p 1 , . . . , p m ).
To simplify our notations, we use the convention to label B = (b xy ) x,y∈X and P = {p x } x∈X by the cluster variables of X. For a seed Σ = (B, P, X) and a cluster variable x ∈ X, the mutation in direction x creates a new seed µ x (Σ) = Σ = (B , P , X ) where: • the new cluster X is obtained from X by replacing x with the cluster variable x defined by the following exchange relation: y −bxy and leaving the remaining cluster variables unchanged so that X {x} = X {x }. • the row (resp. column) of B indexed by x is the negative of the row (resp. column) of B indexed by x, while all other entries satisfy b yz = b yz + 1 2 |b yx |b xz + b yx |b xz | , • the elements of the new coefficient tuple P are An important point is that mutations are involutions: µ x µ x (Σ) = Σ. We say that two seeds are adjacent (resp. mutationally equivalent) when they can be obtained from each other by a mutation (resp. a sequence of mutations). We also use the same terminology for exchange matrices. We denote by V(B • , P • ) the collection of all cluster variables in the seeds mutationally equivalent to an initial seed Σ is the simplicial complex whose vertices are the cluster variables of A(B • , P • ) and whose facets are the clusters of A(B • , P • ).
Finite type. In this paper, we only consider finite type cluster algebras, i.e. those with a finite number N of cluster variables. It turns out that these finite type cluster algebras were classified by S. Fomin and A. Zelevinsky [FZ03a] using the Cartan-killing classification for crystallographic root systems. Define the Cartan companion of an exchange matrix B as the matrix A(B) given by a xy = 2 if x = y and a xy = −|b xy | otherwise.
is already a Cartan matrix, and cyclic otherwise. An acyclic exchange matrix B • is bipartite if each row of B • consists either of nonpositive or non-negative entries. We use the same terminology for the seed Σ • .
From now on, we fix a finite type cluster algebra A(B • , P • ) and we consider the root system of type A(B • ) (again, this root system is finite only when the initial seed B • is acyclic). We use the following classical bases of the underlying vector spaces: • The next definition gives another family of integer vectors, introduced implicitly in [FZ07], that are relevant in the structure of A (B • ).
These two families of vectors are connected via the isomorphism x → x ∨ between the cluster complexes of A (B • ) and A (B ∨ • ) described above.
Cluster fan and generalized associahedron. The following statement is well known and admits several possible proofs as discussed in [HPS18,Sect. 4]. Examples are illustrated in Figure 6.
Theorem 2.13. For any finite type exchange matrix B • , the collection of cones , together with all their faces, forms a complete simplicial fan F(B • ), called the g-vector fan or cluster fan of B • . Figure 6. Two cluster fans F(B • ) for the type A 3 (left) and type C 3 (right) cyclic initial exchange matrices. As the fans are 3-dimensional, we intersect them with the sphere and stereographically project them from the direction (−1, −1, −1). Illustration from [HPS18].
Moreover, this fan is known to be polytopal. More precisely, consider a vector h ∈  Note that Lemma 2.15 implies Theorem 2.14 since it ensures that the type cone TC F(B • ) of the cluster fan contains the cone of height vectors h ∈ R V(B•) such that h x +h x > y∈X∩X , bxy<0 −b xy and h x + h x > y∈X∩X , bxy>0 b xy h y for any seeds (B, X) and (B , X ) with X {x} = X {x }. Unfortunately, Lemma 2.15 is less precise than Proposition 2.3. Indeed, which of the two possible linear dependences is satisfied by the g-vectors of X ∪ X depends on the initial exchange matrix B • . However, for cluster algebras of finite type, this linear dependence is independent of the choice of the adjacent seeds containing x and x . As explained in Corollary 3.30, this statement follows from [ . We obtain the following statement.
Corollary 2.17. For any finite type exchange matrix B • , the type cone of the cluster fan F(B • ) is given by Example 2.18. Consider the cluster fans illustrated in Figure 6. The type cone of the left fan of Figure 6 lives in R 9 and has a lineality space of dimension 3. It has 6 facet-defining inequalities (given below), which correspond to the mesh mutations of Theorem 2.23 as illustrated in Figure 7.
variables The type cone of the right fan of Figure 6 lives in R 12 and has a lineality space of dimension 3. It has 9 facet-defining inequalities (given below), which correspond to the mesh mutations of Theorem 2.23 as illustrated in Figure 8.
x 3 x 2 x 3 x 3 x 1 x 3 x 1 x 2 x 3 x 1 x 2 x 3 g-vectors Figure 8. The facet-defining inequalities of the type cone TC F(B • ) of the cluster fan correspond to the mesh mutations described in Theorem 2.23. See Section 3 for a representation theoretic interpretation.
In order to describe the facets of this type cone, we need the following special mutations.
Definition 2.19. The mutation of a seed Σ = (B, X) in the direction of a cluster variable x ∈ X is a mesh mutation that starts (resp. ends) at x if the entries b xy for y ∈ X are all non-negative (resp. all non-positive). A mesh mutation is initial if it ends at a cluster variable of an initial seed.  For {x, x } ∈ M(B • ) and y ∈ V(B • ), we denote by α x,x (y) the coefficient of g(B • , y) in the linear dependence between the g-vectors g(B • , x) and g(B • , x ). In other words, according to Lemma 2.21, if (B, X) and (B , X ) are two adjacent seeds with X {x} = X {x }, we have α x,x (y) = |b xy | for y ∈ X ∩ X and α x,x (y) = 0 otherwise. The following statement will be shown in Corollary 3.29.
Proposition 2.22. For any finite type exchange matrix B • (acyclic or not, simply-laced or not), the linear dependence between the g-vectors of any mutation can be decomposed into positive combinations of linear dependences between g-vectors of non-initial mesh mutations.
We derive from Proposition 2.22 the following description of the facets of the type cone of the cluster fan.  As a follow-up to Theorem 2.23, we conjecture the following surprising property. , the polytope is a generalized associahedron, whose normal fan is the cluster fan F(B • ). Moreover, the poly-  describe all polytopal realizations of F(T ).

2.3.
Non-kissing complexes and gentle associahedra. Gentle associahedra were constructed by Y. Palu, V. Pilaud and P.-G. Plamondon [PPP21] in the context of support τ -tilting for gentle algebras. For a given τ -tilting finite gentle quiverQ (defined in the next section), theQassociahedron Asso(Q) is a simple polytope which encodes certain representations ofQ and their τ -tilting relations. Combinatorially, theQ-associahedron is a polytopal realization of the nonkissing complex ofQ, defined as the simplicial complex of all collections of walks on the blossoming quiverQ ; which are pairwise non-kissing. The non-kissing complex encompasses two families of simplicial complexes studied independently in the literature: on the one hand the grid associahedra introduced by T. K. Petersen, P. Pylyavskyy and D. Speyer in [PPS10] for a staircase shape, studied by F. Santos, C. Stump and V. Welker [SSW17] for rectangular shapes, and extended by T. McConville in [McC17] for arbitrary grid shapes; and on the other hand the Stokes polytopes and accordion associahedra studied by Y. Baryshnikov [Bar01], F. Chapoton [Cha16], A. Garver and T. McConville [GM18] and T. Manneville and V. Pilaud [MP19]. These two families naturally extend the classical associahedron, obtained from a line quiver. Non-kissing complexes are geometrically realized by polytopes called gentle associahedra, whose normal fan is called the non-kissing fan: its rays correspond to walks in the quiver and its cones are generated by the non-kissing walks. In this section, we describe the type cone of the non-kissing fan of a quiverQ with no self-kissing walks.

2.3.1.
Non-kissing complex and non-kissing fan of a gentle quiver. We present the definitions and properties of the non-kissing complex of a gentle quiver, following the presentation of [PPP21]. See also [BDM + 20] for an alternative presentation.
Gentle quivers. Consider a bound quiverQ = (Q, I), formed by a finite quiver Q = (Q 0 , Q 1 , s, t) and an ideal I of the path algebra kQ (the k-vector space generated by all paths in Q, including vertices as paths of length zero, with multiplication induced by concatenation of paths) such that I is generated by linear combinations of paths of length at least two, and I contains all sufficiently large paths. See [ASS06] for background. Following M. Butler and C. Ringel [BR87], we say thatQ is a gentle bound quiver when (i) each vertex a ∈ Q 0 has at most two incoming and two outgoing arrows, (ii) the ideal I is generated by paths of length exactly two, (iii) for any arrow β ∈ Q 1 , there is at most one arrow α ∈ Q 1 such that t(α) = s(β) and αβ / ∈ I (resp. αβ ∈ I) and at most one arrow γ ∈ Q 1 such that t(β) = s(γ) and βγ / ∈ I (resp. βγ ∈ I). The algebra kQ/I is called a gentle algebra.
The blossoming quiverQ ; of a gentle quiverQ is the gentle quiver obtained by completing all vertices ofQ with additional incoming or outgoing blossoms such that all vertices ofQ become 4-valent. For instance, Figure 9,(left) shows a blossoming quiver: the vertices ofQ appear in black, while the blossom vertices ofQ ; appear in white. We now always assume thatQ is a gentle quiver with blossoming quiverQ ; .

Strings and walks. A string inQ = (Q, I) is a word of the form
there is no path π ∈ I such that π or π −1 appears as a factor of ρ, and (iv) ρ is reduced, in the sense that no factor αα −1 or α −1 α appears in ρ, for α ∈ Q 1 . The integer is called the length of the string ρ. For each vertex a ∈ Q 0 , there is also a string of length zero, denoted by ε a , that starts and ends at a. We implicitly identify the two inverse strings ρ and ρ −1 , and call it an undirected string ofQ. Let S(Q) denote the set of strings ofQ.
A walk ofQ is a maximal string of its blossoming quiverQ ; (meaning that each endpoint is a blossom). As for strings, we implicitly identify the two inverse walks ω and ω −1 , and call it an undirected walk ofQ. Let W(Q) denote the set of walks ofQ.
• the position of σ as a factor of ω matters (the same string at a different position is considered a different substring). • the string ε a is a substring of ω for each occurence of a as a vertex of ω (take j = i + 1).
A peak (resp. deep) of a walk ω is a substring of ω of length zero which is on top (resp. at the bottom) of ω. A walk ω is straight if it has no peak or deep (i.e. if ω or ω −1 is a path inQ ; ), and bending otherwise. We denote by peaks(ω) (resp. deeps(ω)) the multisets of vertices of peaks (resp. deeps) of ω.
Non-kissing complex. Let ω and ω be two undirected walks onQ. We say that ω kisses ω if Σ top (ω) ∩ Σ bot (ω ) = ∅. In other words, ω and ω share a common substring σ, and both arrows of ω (resp. of ω ) incident to the endpoints of σ but not in σ are outgoing (resp. incoming) at the endpoints of σ. See Figure 10 for a schematic representation and Figure 9 (middle) where the three walks are pairwise kissing. We say that ω and ω are kissing if ω kisses ω or ω kisses ω (or both). Note that we authorize the situation where the common finite substring is reduced to a vertex a, that ω can kiss ω several times, that ω and ω can mutually kiss, and that ω can kiss itself. For example, the orange walk in Figure 9 (middle) is self-kissing (at its self-intersection). We say that a walk is proper if it is not straight nor self-kissing, and improper otherwise. We denote by W prop (Q) the set of all proper walks ofQ. Figure 10. A schematic representation of kissing and non-kissing walks. The walks ω and ω kiss (left) while the walks µ and ν share a common substring but do not kiss (right). Illustration from [PPP21].
The (reduced) non-kissing complex ofQ is the simplicial complex N K(Q) whose faces are the collections of pairwise non-kissing proper walks ofQ. For example, Figure 9 (right) represents a non-kissing facet. As shown in [PPP21, Thm. 2.46], the non-kissing complex is a combinatorial model for the support τ -tilting complex on τ -rigid modules over kQ/I. The quiverQ is called τ -tilting finite or non-kissing finite when this complex is finite (in other words,Q has finitely may non-kissing walks).
gand c-vectors. We now define two families of vectors associated to walks in the non-kissing complex. Let (e v ) v∈Q0 denote the canonical basis of R Q0 . For a multiset V = {{v 1 , . . . , v k }} of Q 0 , we denote by m V := i∈[k] e vi .
To define the other family of vectors, we need to recall the notion of distinguished substring of a walk defined in [PPP21, Def. 2.25]. Consider an arrow α ∈ Q 1 contained in two distinct walks ω, ω of a non-kissing facet F ∈ N K(Q), and let σ denote the common substring of ω and ω . We write ω ≺ α ω if ω enters and/or exits σ with arrows in the direction pointed by α, while ω enters and/or exits σ with arrows in the direction opposite to α. For example, in Figure 10, we have µ ≺ γ ν but ν ≺ δ µ. The distinguished walk of the non-kissing facet F ∈ N K(Q) at the arrow α is the maximum walk of F for ≺ α . The distinguished arrows of the walk ω in the non-kissing facet F ∈ N K(Q) are the arrows where ω is the distinguished walk. It is shown in [PPP21, Prop. 2.28] that each bending walk in a non-kissing facet has precisely two distinguished arrows, pointing in opposite directions. For example, we have marked in Figure 9 (right) the two distinguished arrows on each walk by triple arrows. The distinguished substring ds(ω, F ) of the walk ω in the non-kissing facet F ∈ N K(Q) is the substring located between its two distinguished arrows.
The following result will also be essential in our discussion. We say that a string σ is distinguishable if it is the distinguished substring of a walk in a non-kissing facet, and we denote by S dist (Q) the set of distinguishable strings ofQ.
Non-kissing fan and gentle associahedron. The g-vectors support a complete simplicial fan realization of the non-kissing complex N K(Q). Examples are illustrated in Figure 11. is a complete simplicial fan of R Q0 , called non-kissing fan ofQ, which realizes the non-kissing complex N K(Q).
It is proved in [PPP21,Thm. 4.27] that the non-kissing fan comes from a polytope. For a walk ω, denote by kn(ω) the sum over all other walks ω of the number of kisses between ω and ω . (i) the convex hull of the points ω∈F kn(ω) c(ω ∈ F ) for all facets F ∈ N K(Q), or (ii) the intersection of the halfspaces x ∈ R Q0 g(ω) | x ≤ kn(ω) for all walks ω onQ. Figure 11. Two non-kissing fans. As the fans are 3-dimensional, we intersect them with the sphere and stereographically project them from the direction (−1, −1, −1). Illustration from [PPP21].  Remark 2.35. In view of Proposition 2.34, it is tempting to look for a characterization of the exchangeable pairs ω, ω using the kisses between ω and ω . This question was discussed in [BDM + 20, Sect. 9]. However, as illustrated for instance in the non-kissing complex of Figure 11 (right), • two exchangeable walks may kiss along more than one string (only one is distinguished), • two non-exchangeable walks can kiss along more than one distinguishable string, • two walks that kiss along a single distinguishable string are not always exchangeable, • not all strings are distinguishable. In Section 2.3.3, we will restrict to a situation that avoids all these patologies. (1) Grid quivers: Consider the infinite grid quiverQ Z 2 , whose vertices are all integer points of Z 2 , whose arrows are (i, j) −→ (i, j + 1) and (i, j) −→ (i + 1, j) for any (i, j) ∈ Z 2 , and whose relations are (i − 1, j) −→ (i, j) −→ (i, j + 1) and (i, j − 1) −→ (i, j) −→ (i + 1, j) for any (i, j) ∈ Z 2 . A gentle grid quiver is any subquiverQ A ofQ Z 2 induced by a finite subset A ⊂ Z 2 of the integer grid. See Figure 13. The non-kissing complex of gentle grid quivers were introduced in [McC17] with motivation coming from [PPS10] and [SSW17].
(2) Dissection quivers: Consider a dissection D of a convex polygon P (that is a crossing-free set of diagonals of P ) and its gentle dissection quiverQ D , whose vertices are the internal diagonals of D, whose arrows connect pairs of consecutive internal diagonals along the boundary of a face of D, and whose relations correspond to triples of consecutive internal diagonals along the boundary of a face of D. See Figure 13. The non-kissing complex ofQ D then corresponds to non-crossing sets of accordions of D, where an accordion is a segment connecting the middles of two boundary edges of P and crossing a connected set of diagonals of D. This accordion complex was studied in [GM18] and [MP19] with motivation coming from [Bar01], and [Cha16].  Figure 13. A gentle grid quiver (left), a gentle dissection quiver (middle left), and a path quiver which is both the gentle grid quiver of a ribbon (middle right) and the dissection quiver of a triangulation (right). Illustration from [PPP21].
These two families of non-kissing complexes are well-behaved as they avoid all pathologies of Remark 2.35. However, they still provide good examples of non-kissing complexes. In particular, both families contain the classical associahedron. Namely, the gentle associahedron Asso(Q) is the classical associahedron of [SS93,Lod04] presented in Theorem 2.1 whenQ =Q A for the path A = {(0, j) | j ∈ [n]} or equivalentlyQ =Q D for the fan triangulation D (where all internal diagonals are incident to the same point). More generally, Asso(Q) is an associahedron of [HL07] whenQ =Q A for a ribbon A or equivalentlyQ =Q D for a triangulation D. See Figure 13. Note that, it was shown in [PPP19] that the accordion complexes can be extended to dissections of arbitrary orientable surfaces with marked points and then provide a geometric model for all non-kissing complexes of gentle quivers.

2.3.2.
Type cones of non-kissing fans. We now discuss the type cones of the non-kissing fans defined in Theorem 2.32. We obtain from Proposition 2.34 the following redundant description. where the walks µ and ν for two exchangeable walks ω, ω are defined in Proposition 2.34.
Example 2.37. Consider the non-kissing fans illustrated in Figure 11. The type cone of the left fan of Figure 11 lives in R 8 and has a lineality space of dimension 3. It has 5 facet-defining inequalities (given below), which correspond to the flips described in Propositions 2.41 and 2.42 and illustrated in Figure 14 (left).
walks g-vectors The type cone of the right fan of Figure 11 lives in R 11 and has a lineality space of dimension 3. It has 9 facet-defining inequalities (given below), which correspond to the flips illustrated in Figure 14 (right). In particular, it is not simplicial.
walks g-vectors Figure 14. The facet-defining inequalities of the type cone TC F(Q) of the non-kissing fan, represented on the Auslander Reiten quiver of the gentle algebra ofQ. While they correspond to meshes on the left, they are not as clear in the general case as on the right. See also Section 4.
As illustrated in Example 2.37, the type cone of the non-kissing fan is not always simplicial and we do not always understand its extremal exchangeable pairs. In the next section section, we will explore a special family of gentle quivers for which we can completely describe the type cone. The combination of this special family with computer experiments in the general case supports the following conjecture.
Conjecture 2.38. Consider a distinguishable string σ ∈ S dist (Q), and let ω (resp. ω ) be the walk obtained from σ by adding two hooks (resp. two cohooks) at the endpoints of σ. Then (1) The c-vector m σ is the direction of at least one extremal exchangeable pair.
(2) If the walks ω and ω are non-self-kissing and exchangeable, then they form the unique extremal exchangeable pair directed by σ. These extremal exchangeable pairs correspond to meshes of the Auslander-Reiten quiver. In this section, we will restrict our attention to the following family of quivers. Note that the family of brick and 2-acyclic gentle quivers already contains a lot of relevant examples, including the gentle grid and dissection quivers discussed in Section 2.3.1. In particular, the classical associahedron is the gentle associahedron of a brick and 2-acyclic gentle quiver. We will see in Corollary 2.43 that the type cone of the non-kissing fan F(Q) of a brick and 2-acyclic gentle quiverQ happens to be simplicial, and we will derive in Theorem 2.44 a simple description of all polytopal realizations of F(Q).
SinceQ is brick, these walks are not self-kissing by Proposition 2.39 and since they are bending, they are proper walks. We want to show that there are non-kissing facets F, F of the non-kissing complex N K(Q) containing both µ and ν and such that F {ω} = F {ω }. We first show that µ and ν are compatible. Let τ be a maximal common substring of µ and ν. If σ and τ are not disjoint, then σ = τ and µ and ν are not kissing at τ . We can therefore assume that τ appears completely before or completely after σ in both µ and ν. We distinguish two different cases: • If τ appears before σ in both µ and ν, then ατ α forms a cycle inQ. If τ is reduced to a vertex, then we have a 2-cycle. Otherwise, since ατ is a substring of ν and α τ is a substring of ν, the cycle ατ α contains a unique relation α α. This rules out this case under the assumption thatQ is brick and 2-acyclic. The case when τ appears after σ in both µ and ν is symmetric.
• If τ appears before σ in µ and after σ in ν, then ω, µ and ν finish at the same blossom so that µ and ν are not kissing at τ . The case when τ appears after σ in µ and before σ in ν is symmetric. We conclude that µ and ν are non-kissing, which also implies that {µ, ν, ω} and {µ, ν, ω } are non-kissing faces of N K(Q). We can therefore consider a non-kissing facet F containing {µ, ν, ω}.
We claim that ω is distinguished at α and β in F , that µ is distinguished at α in F and that ν is distinguished at β in F . Let us just prove that µ is distinguished at α , the other statements being similar. Consider any walk λ of F containing α , and let τ be the maximal common substring of ω and λ. Note that ω has the outgoing arrow α while λ has the incoming arrow α at one end of τ .
Since ω and λ are compatible, it ensures that either τ ends at a blossom, or λ has an outgoing arrow at the other end of τ . This shows that λ ≺ α µ since λ separates from µ with an arrow in the same direction as α .
Since this precisely coincides with the description of the flip in the non-kissing complex (see Proposition 2.34 (i) or [PPP21, Prop. 2.33] for an alternative presentation), we conclude that the flip of ω in F creates a facet F containing ω . Therefore, ω and ω are exchangeable with distinguished substring σ.
The following statement describes the type cone of the non-kissing fan of a brick and 2-acyclic gentle quiver. We provide here an elementary combinatorial proof, a more general representation theoretic perspective is discussed in Theorem 4.46.
Proposition 2.42. For any brick and 2-acyclic gentle quiverQ, the extremal exchangeable pairs for the non-kissing fan ofQ are precisely the pairs { σ , σ } for all strings σ ∈ S(Q).
Our next statement follows from the end of the previous proof. See also Corollary 4.49.  We also obtain from Proposition 2.42 the following surprising property. Remark 2.46. Although not needed in the proof of Proposition 2.42, we note that the extremal exchangeable pairs { σ , σ } and their linear dependencies g( σ ) + g( σ ) − g( σ ) − g( σ ) precisely correspond to the meshes of the Auslander-Reiten quiver ofQ.
Remark 2.47. Note that Propositions 2.41 and 2.42 and therefore Corollary 2.43 and Theorem 2.44 fail when the quiver is not brick or 2-cyclic. The smallest exemples are the 2-cycles whose nonkissing fans are represented in Figure 16. The left one is not brick: it has a non-distinguishable string and a self-kissing walk. The right one is brick but is 2-cyclic. 1 2 1 2 Figure 16. The non-kissing fans of two 2-cyclic gentle quivers. Illustration from [PPP21].
Part II. Grothendieck group and relations between g-vectors 3. Relations for g-vectors in finite type cluster algebras via 2-Calabi-Yau triangulated categories In [But81] and [Aus84], M. Butler and M. Auslander showed that the relations in the Grothendieck group of an Artin algebra are generated by the ones given by almost-split sequences precisely when the algebra is of finite representation type. This result was generalized to triangulated categories with certain finiteness conditions by J. Xiao and B. Zhu in [XZ02], to Hom-finite, Krull-Schmidt exact categories with enough projectives by H. Enomoto in [Eno19], to (n + 2)-angulated categories by F. Fedele in [Fed21], and to Hom-finite, Krull-Schmidt triangulated categories with a cogenerator by J. Haugland in [Hau19]. Relations in the Grothendieck group of cluster categories have also been studied in [Pal09]. Inspired by these results, we will show in this Section that a similar statement holds (see Theorem 3.8) for the g-vectors (or indices) of objects in a 2-Calabi-Yau triangulated category. We will further generalize these results in Section 4 to the setting of Extfinite, Krull-Schmidt, extriangulated categories with Auslander-Reiten-Serre duality admitting projective objects with certain properties.
3.1. Setting. Let K be a field. Let C be a K-linear triangulated category with suspension functor Σ. We fix a collection ind(C) of representatives of isomorphism classes of indecomposable objects of C. We will assume the following: • C is essentially small (in particular, ind(C) is a set); • C is Hom-finite: for each pair of objects X and Y , the K-vector space C(X, Y ) is finitedimensional; • C is Krull-Schmidt: the endomorphism algebra of any indecomposable object is local; • C is 2-Calabi-Yau: for each pair of objects X and Y , there is an isomorphism of bifunctors where (ΣT ) is the ideal of morphisms factoring through an object of add(ΣT ) and mod Λ is the category of finite-dimensional right Λ-modules. This equivalence induces further equivalences between add(T ) and the category of projective modules, and between add(Σ 2 T ) and that of injective modules.
For categories with a cluster-tilting object, the 2-Calabi-Yau condition implies other duality results which we shall need. Remark 3.3. Although the field K is assumed to be algebraically closed in [Pal08], this assumption is not needed in the proof, and the result is valid over any field.
Finally, we need the existence of almost-split triangles in C. Recall that a triangle in C is almost-split if X and Z are indecomposable, h is non-zero, and any non-section X → X factors through f (or equivalently, any non-retraction Z → Z factors through g). We say that a triangulated category has almost-split triangles if there is an almost-split triangle as above for any indecomposable object X. . Any triangulated category admitting a Serre functor has almost-split triangles. In particular, any 2-Calabi-Yau triangulated category has almost-split triangles.
(1) Let K sp 0 (C) be the split Grothendieck group of C, that is, the free abelian group generated by symbols [X], where [X] denotes the isomorphism class of X in C, modulo the following relations: for any objects Y and Z, we let [ with h ∈ (ΣT ). Denote by g : K sp 0 (C) → K 0 (C; T ) the canonical projection. In particular, K sp 0 (C) is isomorphic to a free abelian group over the set ind(C). Considering the modified group K 0 (C; T ) instead of the usual Grothendieck group is motivated by Section 3.4, where we study relations between g-vectors.
Definition 3.6. For any two objects X and Y of C, define This defines a bilinear form −, − : Notation 3.7.
(1) For any indecomposable object X of C, let be an almost split triangle (unique up to isomorphism). We let be an almost split triangle (unique up to isomorphism). We let . We can finally state the main theorem of this section, which is an analogue of the main result of [Aus84].
Theorem 3.8. Let C be a category satisying the hypotheses of Section 3.1. Then C has only finitely many isomorphism classes of indecomposable objects if and only if the set L := X X ∈ ind(C) add(ΣT ) generates the kernel of g : K sp 0 (C) → K 0 (C; T ). In this case, the set L is a basis of the kernel of g, and for any x ∈ ker(g), we have that Proof. We know that x = A∈ind(C) add(ΣT )

x,[A]
A ,[A] A ; since A , [A] is positive by Lemma 3.16, we only need to show that each x, [A] is non-negative. The functor F = C(T, −) induces an exact sequence F X → F E → F Y → 0, which in turn induces an exact sequence 3.3. Proof of Theorem 3.8. Before we can prove Theorem 3.8, we need to recall the following results and definition from [DK08,Pal08].
with T X 1 and T X 0 in add(T ). Definition 3.11. The index of an object X is the element The notion of index is very close to the definition of the map g : K sp 0 (C) → K 0 (C; T ) of Definition 3.5. The link is given by the following result. Then if and only if h ∈ (ΣT ).
Corollary 3.13. There is an isomorphism φ : K 0 (C; T ) → K sp 0 add(T ) such that ind T = φ • g. In particular, K 0 (C; T ) is a free abelian group generated by the [T i ].
We can now begin the proof of Theorem 3.8 Lemma 3.14. If X or Y lie in add(ΣT ), then X, Y = 0.
Proof. This is because F ΣT = 0. Proof. If X ∈ add(ΣT ), then h cannot be in (ΣT ), otherwise it would be zero since C(T, ΣT ) = 0.
Assume now that X / ∈ add(ΣT ). Let K X be the residue field of the algebra End C (X). By definition of an almost-split triangle, h is in the socle of the right End C (X)-module C(Σ −1 X, ΣX). Moreover, this socle is a one-dimensional K X -vector space; indeed, the 2-Calabi-Yau condition gives an isomorphism C(Σ −1 X, ΣX) ∼ = DC(X, X). Thus the socle of the right module C(Σ −1 X, ΣX) has the same K X dimension as the top of the left module C(X, X). Since X is indecomposable, C(X, X) is local, and its top is one-dimensional over K X . Now, (ΣT )C(Σ −1 X, ΣX) is a sub-module of C(Σ −1 X, ΣX). Therefore, if (ΣT )C(Σ −1 X, ΣX) is non-zero, then it contains the one-dimensional socle of C(Σ −1 X, ΣX), and thus contains h. By [Pal08], (ΣT )C(Σ −1 X, ΣX) ∼ = DC(X, X)/(ΣT ). The identity morphism of X is not in (ΣT ), since X is not in add(ΣT ). Thus the right-hand side is non-zero, and so neither is the left-hand side. By the above, this implies h ∈ (ΣT )C(Σ −1 X, ΣX), which finishes the proof. (1) If A / ∈ add(ΣT ), then (2) If B / ∈ add(ΣT ), then Proof. We only prove the first assertion; the second one is proved dually. Assume that A / ∈ add(ΣT ). Let be an almost-split triangle. By Lemma 3.15, the morphism h is in (ΣT ). Applying the functor F = C(T, −), we get an exact sequence Applying now the functors C(−, B) and Hom Λ (−, F B), we get a commutative diagram whose rows are exact sequences and whose vertical maps are surjective by Proposition 3.1.
If B A, then the definition of an almost-split triangle implies that f * is surjective. Thus coker(f * ) = 0, so that coker(F f * ) = 0, and by additivity of the dimension in exact sequences, we get that A , B = 0.
If B ∼ = A, then coker(f * ) is isomorphic to the residue field K A of C(A, A). Since A is not in add(ΣT ), the ideal of endomorphisms of A factoring through an object of add(ΣT ) is contained in the maximal ideal of C(A, A). Therefore, the rightmost morphism in the commutative diagram is an isomorphism, so coker(F f * ) is isomorphic to K A . Lemma 3.17. Let x ∈ K sp 0 (C), and write Then for any A ∈ ind(C) add(ΣT ), we have that Proof. Let B ∈ ind(C) add(ΣT ). Applying Lemma 3.16, we get that The equality λ A = x,r A A,r A is proved in a similar way. Proof. We will only proove that (1) is equivalent to (2); the proof that (1) is equivalent to (3) is similar.
We only prove (1); the proof of (2) is similar. Assume that  Proof. By Proposition 3.19, the set is free. Let x ∈ ker g. Consider the element Then for any B ∈ ind C, we have that where the second equality is obtained by using Lemma 3.16. By Corollary 3.18, this implies that z ∈ K sp 0 add(ΣT ) . Since z ∈ ker(g) and since g is injective on K sp 0 add(ΣT ) by Corollary 3.13, we get that z = 0. This finishes the proof.
is a basis of K sp 0 (C). Proof. By Proposition 3.19, the set is free. It suffices to prove that it generates K sp 0 (C). Let x ∈ K sp 0 (C). Consider Then for any B ∈ ind(C), we have that z, [B] = 0. By Corollary 3.18, this implies that z ∈ K sp 0 add(ΣT ) , and finishes the proof.
All that remains is to prove the converse in the statement of Theorem 3.8. 3.4. Application to g-vectors of cluster algebras of finite type. In this section, we apply the results of Section 3.2 in order to prove Propositions 2.16 and 2.22 of Section 2.2.
We first recall the results on categorification of cluster algebras that we will use in the proofs. The following definition is a cluster-tilting version of the positive mutation for τ -tilting modules, or for 2-term silting complexes.
Definition 3.24. Let x ∈ X be a cluster variable in a given cluster and let x ∈ X be obtained by mutating X at x. The pair (x, x ) is said to be a positive mutation with respect to some initial cluster X • if there are X, Y ∈ ind(C) such that φ(X) = x, φ(Y ) = x , dim K X C(X, ΣY ) = dim K Y C(X, ΣY ) = 1 and, for any non-split triangle (this triangle is unique up to isomorphism) we have h ∈ (ΣT ), where T ∈ C is any basic clustertilting object such that φ(T ) = X • . The mutation is a mesh mutation if the triangle can be chosen to be an almost-split triangle (with no assumption on h). Proof. Up to isomorphism, there is precisely one almost-split triangle of C starting in X, where X is the object such that φ(X) = x. Thus there is precisely one mesh relation starting at x. By Lemma 3.15, the pair (x, x ) is a positive mutation if and only X / ∈ add(ΣT ), that is, if and only if x is not an initial variable. Proof. If (x, x ) is a positive mesh mutation, then {x, x } is in M(B • ). Indeed, if X and X are the corresponding objects in C, then there is an almost-split triangle so that X = τ −1 X. Moreover, there is a slice of the Auslander-Reiten quiver of C containing X and all indecomposable direct factors of E. The direct sum of indecomposable objects in this slice is a cluster-tilting object in which X is a source, and mutation at X gives X . Thus {x, x } is in M(B • ). This gives an injective map from the set of positive mesh mutations to M(B • ).
Next, assume that {x, x } ∈ M(B • ) starts in x. Then there is a cluster-tilting object T in C having X as a direct factor, and such that X is a source in T and mutation at X yields X . Since X is a source, the mutation triangle starting and ending in X are Therefore X = Σ −1 X = τ −1 X. Thus the first triangle is the almost-split triangle starting in X, since the dimension of C(X , ΣX) is 1 over K X . Since the pair {x, x } is not initial, φ(X ) is not an initial variable; in other words, X is not in add(T ). Therefore, X = τ X = ΣX is not in add ΣT , so the mutation is a positive mesh mutation by Lemma 3.15. This gives an injective map from M(B • ) to the set of positive mutations. Proof. This is a direct consequence of Lemmas 3.25 and 3.26.
Lemma 3.28. Let A be any cluster algebra admitting a categorification by a Hom-finite, 2-Calabi-Yau, Krull-Schmidt, triangulated category with cluster-tilting objects. Let x ∈ X be a cluster variable in a given cluster and let x ∈ X be obtained by mutating X at x. Then precisely one of the two mutations µ x (X) and µ x (X ) is a positive mutation.
Proof. In view of Theorem 3.23 (iii), it follows from [Pal08,Lem. 3.3]. Proof. Let C be a cluster category of Dynkin type A, B, C, D, E, F or G, and let T ∈ C be some basic cluster-tilting object categorifying the given initial exchange matrix B • . By Lemma 3.28, we may assume that the mutation under consideration is a positive mutation. In view of Theorem 3.23 (ii) and (iv) it is enough to show that the relation in K 0 (C; T ) given by any triangle is a positive linear combination of relations coming from Auslander-Reiten triangles with third term not in add(ΣT ). This categorified statement is precisely Corollary 3.9.
It also follows from Theorem 3.23 above that any cluster algebra of finite type has the unique exchange relation property (see Definition 1.8).
Corollary 3.30. (Proposition 2.16) Let B • be any finite type exchange matrix, and let A(B • ) be the associated cluster algebra without coefficients. Then, for any exchangeable cluster variables x and x , the linear dependence y∈X∪X α X,X (g y ) g y = 0 only depends on the pair (x, x ) and not on the specific choice of clusters X and X containing x and x respectively and such that X {x} = X {x }.
Proof. By the results in additive categorification of cluster algebras recalled in Theorem 3.23, the statement follows from Lemma 3.31 below.
Lemma 3.31. Let C be a cluster category of Dynkin type A, B, C, D, E, F or G, and let X, Y ∈ C be such φ(Y ) is obtained by performing a positive mutation at φ(X) in some cluster containing it. Then the linear dependence between the g-vectors given by the positive mutation is Proof. Let T ∈ C be a basic cluster tilting object such that φ(T ) is the initial cluster. According to Lemma 3.28, the morphism h belongs to the ideal (ΣT ). Therefore, the triangle . Theorem 3.23 (ii) gives the equality g φ(X) + g φ(Y ) = i g φ(Ei) , where the g-vectors are computed with respect to the initial seed φ(T ). By Theorem 3.23 (iii), the middle term of the triangle is uniquely defined, up to isomorphism, by X, and Y , and the triangle is therefore an exchange triangle, with respect to any mutation at X: it does not depend on the choice of a cluster tilting object containing X. By Theorem 3.23 (iv) this equality is one of the two equalities in Lemma 2.15. Setting for Section 4.2. In that section, we let C be an extriangulated category with a fixed full additive subcategory T , stable under isomorphisms, under taking direct summands, and satisfying the following three properties:

Relations for g-vectors in brick algebras via extriangulated categories
(1) Every T ∈ T is projective in C; (2) For each T ∈ T , the morphism T → 0 is an inflation for the extriangulated structure of C; (3) For each X ∈ C, there is an extriangle T X Setting for Section 4.3. In that section, we keep the previous setting, but assume moreover that C is Krull-Schmidt, K-linear, Ext-finite, and has Auslander-Reiten-Serre duality, and that the subcategory T is of the form add T , where T = T 1 ⊕ · · · ⊕ T n is a basic object. • for any Artin algebra Λ, the category K [−1,0] (proj Λ) of complexes of finitely generated projective Λ-modules concentrated in degrees −1 and 0, with morphisms considered up to homotopy (see Section 4.6); • more generally, extriangulated categories C constructed as follows: if T is a rigid subcategory of an extriangulated category E (with some assumption ensuring that condition (2) holds) we let C be the full subcategory of E whose objects satisfy condition (3), equipped with the extriangulated structure obtained from that of E by considering those extriangles in E all whose terms belong to C and whose deflation is T -epi (see Section 4.7.2).

4.2.
Statement of preliminary results on extriangulated categories. So as to be able to state the main theorem of Section 4, we need a few results on extriangulated categories. However, all proofs are postponed to Section 4.5.
We let C be as in Section 4.1.

Notation 4.2.
For any object T ∈ T , we fix an extriangle T 0 ΣT . Remark 4.3. As will be proven below (Remark 4.32 and Corollary 4.35), this notation extends to an equivalence of categories from the category T of projective objects in C to the category ΣT of injective objects in C.
Notation 4.4. We let Mod T denote the category of additive functors from T to Abelian groups, and mod T its full subcategory of functors that are finitely presented, i.e. cokernels of morphisms between representable functors. We let F : C → Mod T be the functor defined on objects by sending X ∈ C to C(−, X)| T .
Lemma 4.5. For any X ∈ C, the functor F X is finitely presented. We thus have a functor Proposition 4.6 below extends similar results from [BMR07, KR07, KZ08, IY08] (see also Proposition 3.1) to the setting under consideration. We note that the proof requires minor modifications. Remark 4.7. Because the category C does not have weak kernels in general, the category mod T might not be abelian. However, in all our applications, the subcategory T is of the form add(T ) for some object T . In that case, mod T is equivalent to mod End C (T ) and thus abelian.
Definition 4.8. We let K 0 (C) denote the Grothendieck group of C, that is, the quotient of the free abelian group generated by symbols [X], for each X ∈ C, by the relations Remark 4.9. Since T is extension-closed in C, it inherits an extriangulated structure. Because T is made of projective objects in C, its extriangulated structure splits and we have K 0 (T ) ∼ = K sp 0 (T ). The notion of index from [DK08,Pal08] generalises to our current setting. Remark 4.12. Assume moreover that C is K-linear and Ext-finite. By the preliminary results of Section 4.4, we can apply [INP18,Prop. 4.2] to obtain that mod T is Hom-finite. In particular, if T = add T , then End C (T ) is a finite-dimensional K-algebra. It thus also follows that Definition 3.6 still makes sense in this more general setup: For any X, Y ∈ C, When this makes sense, we make use of Notation 3.7: if is an almost-split sequence (see [INP18] for a definition in this setting), we let . Theorem 4.13. Assume that C is a K-linear, Ext-finite, Krull-Schmidt, extriangulated category with Auslander-Reiten-Serre duality. Assume that T is a projective object of C such that any X ∈ C admits a conflation T X 1 T X 0 X with T X 0 , T X 1 ∈ add(T ), and the morphism T → 0 is an inflation. Fix a conflation T → 0 → ΣT . Then C has only finitely many isomorphism classes of indecomposable objects if and only if the set generates the kernel of the canonical projection g : K sp 0 (C) → K 0 (C). In this case, the set L is a basis of the kernel of g, and for any x ∈ ker(g), we have that  [Jin20]. We also note the generalization, called n-exangulated categories [HLN21,HLN22], to a version suited for higher homological algebra.
An extriangulated category is the data of an additive category C, of an additive bifunctor E : C op × C → Ab modelling the Ext 1 -bifunctor, and of an additive realization s sending each element δ ∈ E(Z, X) to some (equivalence classe of) diagram X → Y → Z modelling the short exact sequences or triangles. Some axioms, inspired from the case of extension-closed subcategories of triangulated categories have to be satisfied.
More specifically: fix an additive category C, and an additive bifunctor E : C op × C → Ab, where Ab is the category of abelian groups. Definition 4.15. For any X, Z ∈ C, an element δ ∈ E(Z, X) is called an E-extension. A split Eextension is a zero element 0 ∈ E(Z, X), for some objects X, Z ∈ C. For any two E-extensions δ ∈ E(Z, X), δ ∈ E(Z , X ), the additivity of C, E permits to define the E-extension δ ⊕ δ ∈ E(Z ⊕ Z , X ⊕ X ).
Definition 4.18. Let X, Z ∈ C be any two objects. Two sequences of morphisms in C are said to be equivalent if there exists an isomorphism g ∈ C(Y, Y ) such that the following diagram commutes.
Notation 4.19. For any X, Y, Z, A, B, C ∈ C, and any [X Definition 4.20. An additive realization s is a correspondence associating, with E-extension δ ∈ E(Z, X), an equivalence class s(δ) = [X x − → Y y − → Z] and satisfying the following condition: ( * ) Let δ ∈ E(Z, X) and δ ∈ E(Z , X ) be any pair of E-extensions, with Then, for any morphism (f, h) : δ → δ , there exists g ∈ C(Y, Y ) such that the following diagram commutes:  (1) For any X, Z ∈ C, the realization of the split E-extension 0 ∈ E(Z, X) is given by s(0) = 0.
We use the following terminology.
In which case the morphism X x − → Y is called an inflation, written X Y , and the morphism Y y − → Z is called a deflation, witten Y Z.
(2) An extriangle is a diagram X Z is a conflation realizing the E-extension δ ∈ E(Z, X).
(3) Similarly, we call morphism of extriangles any diagram The axioms above ensure that any extriangle induce long-ish exact sequences after application of some covariant or contravariant Hom-functor. In particular in any conflation, the inflation is a weak kernel of the deflation, and the deflation is a weak cokernel of the inflation.
Any variant of the axiom (ET4) that would hold in an extension-closed subcategory of a triangulated category by applying the octahedron axiom also holds in any extriangulated category. See [NP19, Sect. 3.2] for more details.
Definition 4.25. An almost-split E-extension δ ∈ E(Z, X) is a non-split E-extension such that: (AS1) f * δ = 0 for any non-section f ∈ C(X, X ). (AS2) g * δ = 0 for any non-retraction g ∈ C(Z , Z). Definition 4.26. A non-zero object X ∈ C is said to be endo-local if C(X, X) is local. . For any non-split E-extension δ ∈ E(Z, X), the following holds.
Definition 4.28. An object P ∈ C is called projective if, for any X ∈ C, we have E(P, X) = 0. Dually, an object I ∈ C is called injective if for any X ∈ C, we have E(X, I) = 0. (proj Λ) be the full subcategory of the homotopy category K b (proj Λ) consisting of complexes concentrated in degrees −1 and 0 (using cohomological conventions). Then any endo-local non-projective object X ∈ C permits an almost-split conflation τ X E X and any endo-local non-injective object Y ∈ C permits an almost-split conflation Y E τ − Y .

4.5.
Proofs of preliminary results and of Theorem 4.13. We let C and T satisfy the assumptions (1), (2) and (3) of Section 4.1. We will make it explicit when more assumptions are needed.
We start by stating some immediate implications of the existence of conflations T 0 ΣT , for T ∈ T .
Lemma 4.30. The following holds: (i) An object belongs to T if and only if it is projective in C.
(v) The full subcategory ΣT is rigid in C.
Proof. (i) If X is projective in C, then the deflation T X 0 X splits and X is a summand of T X 0 . (ii) This follows from the fact that T is projective and 0 ΣT is a deflation.
Proof. The first extriangle is obtained by applying the dual of [NP19, Prop. 3.15] to δ X and δ 1 , and the second one is obtained similarly from δ 0 and (f X ) * δ 1 .
Remark 4.32. For any T ∈ T , fix an extriangle T 0 ΣT ε T . As noticed in Lemma 4.31, for any morphism T f → T in T , there is a unique morphism Σf ∈ C(ΣT, ΣT ) such that f * ε T = (Σf ) * ε T (i.e. such that (f, Σf ) is a morphism from ε T to ε T ), as illustrated below: This gives an equivalence of additive categories where ΣT is the full subcategory of C whose objects are isomorphic to objects of the form ΣT , for some T ∈ T . Proof. Let T t → X be a morphism with T ∈ T . Using the notation of Remark 4.32, there is an there is a commutative diagram of extriangles Since the objects in T are projective, the two conflations with middle term Y split and there are isomorphisms Remark 4.37. We note that Lemma 4.36 only requires the subcategory T to be rigid, and not made of projective objects. However, this stronger assumption is used in the proof of Lemma 4.38 below.
Lemma 4.38. The index descends to K 0 (C): for any conflation where we have used the fact that T Z 0 is projective in C. Moreover, the conflation with middle term B splits, which implies that B is isomorphic to T X 1 ⊕ T Z 1 .
Proof of Proposition 4.11. By Lemma 4.36 and Lemma 4.38, the index induces an well-defined morphism of abelian groups ind T : K 0 (C) → K 0 (T ). It remains to prove that it is an isomorphism. Let π : K 0 (T ) → K 0 (C) send a class [T ] ∈ K 0 (T ) to the class [T ] ∈ K 0 (C). The existence, for any T ∈ T , of a conflation 0 T T shows that π is right inverse to ind T . To see that it is also a left inverse, note that if T 1 T 0 X is a conflation in C, then the equality Proof of Theorem 4.13. The proof is now similar to the proof of Theorem 3.8, by using Proposition 4.6 instead of Proposition 3.1, Proposition 4.11 instead of Corollary 3.13, and the fact that T is made of projective objects instead of Lemma 3.15.
Proof of Corollary 4.14. The proof is similar to that of Corollary 3.9. 4.6. Application to 2-term complexes of projectives and to gentle algebras. Let Λ be an Artin algebra. Then the homotopy category K b (proj Λ) is a triangulated category, and we consider its full subcategory K [−1,0] (proj Λ) of complexes concentrated in degrees −1 and 0 (with cohomological conventions). Proof. The category K [−1,0] (proj Λ) is closed under extensions in K b (proj Λ), so it is an extriangulated category whose conflations are induced by triangles in K b (proj Λ) by [NP19,Rem. 2.18]. Now, the object Λ is projective in K [−1,0] (proj Λ), since if X Y Z is a conflation, then we get a long exact sequence ΣX), and the last term vanishes, since ΣX is a complex concentrated in degrees −2 and −1, while Λ is in degree 0. This proves that Λ is projective.
The extriangulated category K [−1,0] (proj Λ) is the setting in which we can finally prove point (ii) of Proposition 2.34, using results of [DIJ19] on g-vectors. Before doing so, we introduce the notion of mutation conflation for two-term silting objects.   is called a mutation conflation if there are basic, two-term, silting objects X ⊕ R, Y ⊕ R, with X and Y indecomposable, such that the inflation X E is a left (add R)-approximation (i.e. any morphism from X to an object in add R factors through X E).
Remark 4.43. In Definition 4.42, the requirement that X and Y are indecomposable implies the map E Y is a right (add R)-approximation, and that both approximations are minimal.
Proposition 4.44. Let Λ be a finite-dimensional K-algebra, where K is a field. Let X and Y be objects of K [−1,0] (proj Λ).
(1) If there is a mutation conflation (see Definition 4.42) of the form X E Y , then there can be no mutation conflation of the form Y E X.
(2) Assume that X E Y and X E Y are two mutation conflations. Then E and E are isomorphic.
Proof. We first prove (1). Assume that X E Y and Y E X are two mutation conflations. Applying the functor Hom K b (Σ −1 Y, −) to the second sequence, we get an exact sequence of abelian groups By definition of a mutation conflation, we have that Hom K b (Σ −1 Y, E ) = 0. Moreover, since Y is concentrated in homological degrees −1 and 0, we get that Hom K b (Σ −1 Y, ΣY ) = 0. Therefore, Hom K b (Σ −1 Y, X) = 0. However, since the conflation X E Y is not split, we have that Hom K b (Σ −1 Y, X) = 0, a contradiction. This proves (1).
We now prove (2). Let X E Y and X E Y be two mutation conflations. By  Proof. Let ω and ω be in facets F and F as in the statement of Proposition 2.34. For every walk ω , let P (ω ) be the object of K [−1,0] (proj Λ) corresponding to ω . Then the walks µ and ν defined in the same statement yield a mutation conflation P (ω) P (µ) ⊕ P (ν) P (ω ) or P (ω ) P (µ) ⊕ P (ν) P (ω).
In particular, the leftmost morphisms of both mutation conflations are minimal add P (µ) ⊕ P (ν)approximations of P (ω). As such, they are isomorphic as morphisms. They determine common substrings of ω and µ and of ω and ν, which in turn determine σ. Point (ii) of Proposition 2.34 is proved.
We now come to the main result of this section.  Corollary 4.48. Let Λ be a finite dimensional brick algebra of finite representation type. Assume moreover that, for any indecomposable Λ-module M , the space Hom Λ (M, τ 2 M ) vanishes. Then the support τ -tilting fan of Λ has the unique exchange relation property and its type cone is simplicial.
As a consequence of Corollary 4.48, the method of Part I apply and give an explicit description of all realizations of the support τ -tilting fan of Λ. Another consequence is an algebraic proof of Corollary 2.43, that we restate below.
Proof. The almost-split conflation induces an exact sequence Because X is concentrated in non-positive degrees, F ΣX = 0. Let us verify that the map → Z is right almost-split, and Z is indecomposable not in add(ΣΛ), the morphism Σf factors through b. This implies the required surjectivity, and we have a short exact sequence Because mod Λ and K b (proj Λ) are Krull-Schmidt, it is immediate to check that F b (resp. F a) inherits from b (resp. a) the property of being right (resp. left) almost-split.  Note also that Lemma 4.50 implies that F τ X is isomorphic to τ M . By assumption, X is not in add(ΣΛ). The same is true of τ X since objects in add(ΣΛ) are injective in K [−1,0] (proj Λ) and an Auslander-Reiten conflation does not split. It thus follows from [AIR14,Lem. 3.4 ] that X and τ X are rigid in K [−1,0] (proj Λ). We have to prove that both E ⊕ X and E ⊕ τ X are rigid.
The Auslander-Reiten conflation induces an exact sequence As already remarked, X is rigid so that the left-most term vanishes. By assumption Hom Λ (F X, τ F τ X) = Hom Λ (M, τ 2 M ) = 0, so that the right-most term also vanishes by [AIR14,Lem. 3.4 ].
We have an exact sequence The first map is surjective since E → X is right almost-split (note that E cannot contain any summand isomorphic to X). We can conclude by (4). (1) If Hom K b (Y, ΣE) = 0, we have Hom Λ (F X, τ F Y ) = 0.
We fix this issue by considering a relative extriangulated structure. Let E T be the additive sub-bifunctor of C(−, Σ−) formed by those morphisms that factor through add ΣT . Then (C, E T ) becomes extriangulated by either [HLN21,Prop. 3.16 (2) and (3) are extriangles for this relative structure, which shows that those two conditions are still satisfied for the extriangulated structure (C, E T ).
Remark 4.55. The index of an object X in the triangulated category C, coincide with its class in the Grothendieck group of C for the relative extriangulated structure given by E T . Let (E, E, s) be an extriangulated category (for example, a small exact category or a triangulated category), and let T be an essentially small subcategory of E.
For any X, Y ∈ E, let E T (X, Y ) be the subset of E(X, Y ) consisting of those δ that satisfy f * δ = 0, for any T ∈ T and any T f − → X. It is easily seen that E T is an additive subfunctor of E. Consider (E, E T , s T ) endowed with the resctriction of the additive realization s. Since the deflations in (E, E T , s T ) are precisely the deflations in (E, E, s) that are T -epic, they are closed under composition. We can thus apply [HLN21,Prop. 3.16] to obtain that (E, E T , s T ) is an extriangulated category. With that relative structure, the objects in T become projective (alternatively, one can directly apply [HLN21,Prop. 3.19]). Define C to be the full subcategory of E whose objects X admit an s T -conflation T 1 T 0 X.
Lemma 4.57. The full subcategory C is extension-closed in (E, E T , s T ), and thus inherits an extriangulated structure.
Proof. This follows from the proof of Lemma 4.38.
Notation 4.58. Write T ⊥ for the full subcategory of E whose objects are those objects X ∈ E that satisfy E(T, X) = 0, for any T ∈ T .
Proposition 4.59. Assume that, for any T ∈ T , there is an s-conflation T 0 X T in E with X T ∈ T ⊥ . Then the extriangulated category C satisfies the assumptions (1) to (3) of Section 4.1.
Proof. For any T ∈ T the existence of an extriangle 0 T T 0 shows that T is a full subcategory of C. By Lemma 4.57, the category C is extriangulated and, by definition of this extriangulated structure, the objects in T are projective in C. Assumption (3) is satisfied by the definition of C, and assumption (2) is equivalent to that of the proposition.