Polytopal realizations of finite type g -vector fans
PPOLYTOPAL REALIZATIONS OF FINITE TYPE g-VECTOR FANS
CHRISTOPHE HOHLWEG, VINCENT PILAUD, AND SALVATORE STELLA
Abstract.
This paper shows the polytopality of any finite type g -vector fan, acyclic or not. Infact, for any finite Dynkin type Γ, we construct a universal associahedron Asso un (Γ) with theproperty that any g -vector fan of type Γ is the normal fan of a suitable projection of Asso un (Γ). Introduction A generalized associahedron is a polytope which realizes the cluster complex of a finite typecluster algebra of S. Fomin and A. Zelevinsky [FZ03b, FZ02, FZ03a]. Generalized associahedrawere first constructed by F. Chapoton, S. Fomin and A. Zelevinsky [CFZ02] using the d -vector fansof [FZ03b]. Further realizations were obtained by C. Hohlweg, C. Lange and H. Thomas [HLT11]based on the Cambrian lattices of N. Reading [Rea06] and Cambrian fans of N. Reading andD. Speyer [RS09]. These constructions were later revisited by S. Stella [Ste13] using an approachsimilar to the original one of [CFZ02], and by V. Pilaud and C. Stump [PS15a] via brick polytopes.The realizations of [HLT11] start from an acyclic initial exchange matrix B ◦ , and constructa generalized associahedron Asso (B ◦ ) whose normal fan is the g -vector fan of the cluster alge-bra A pr (B ◦ ) with principal coefficients at B ◦ . They rely on a combinatorial understanding of the g -vector fans as Cambrian fans [RS09]. A major obstruction in dropping the acyclicity assumptionin this approach is that this combinatorial description is only partially available beyond acycliccases [RS15]. Therefore, it remained a challenging open problem, since the appearance of general-ized associahedra, to construct polytopal realizations of all finite type g -vector fans including thecyclic cases. This paper answers this problem. Theorem 1.
For any finite type initial exchange matrix B ◦ , the g -vector fan F g (B ◦ ) with respectto B ◦ is the normal fan of a generalized associahedron Asso (B ◦ ) . When we start from an acyclic initial exchange matrix, our construction precisely recovers theassociahedra of [HLT11, Ste13, PS15a]. These can all be obtained by deleting inequalities fromthe facet description of the permutahedron of the corresponding finite reflection group. The maindifficulty to extend the previous approach to arbitrary initial exchange matrices lies in the factthat this property, intriguing as it might be, is essentially a coincidence. First, the hyperplanearrangement H supporting the g -vector fan is no longer the Coxeter arrangement of a finitereflection group. Even worse, we prove that the generalized associahedron Asso (B ◦ ) usually cannotbe obtained by deleting inequalities in the facet description of any zonotope whose normal fan is H .This behavior already appears in type D .To overcome this situation, we develop an alternative approach based on a uniform understand-ing of the linear dependences among adjacent cones in the g -vector fan. In fact, not only we coveruniformly all finite type initial exchange matrices, but we actually can treat them simultaneouslywith a universal object. Theorem 2.
For any given finite Dynkin type Γ , there exists a universal associahedron Asso un (Γ) such that, for any initial exchange matrix B ◦ of type Γ , the generalized associahedron Asso (B ◦ ) isa suitable projection of the universal associahedron Asso un (Γ) . In particular, all g -vector fans oftype Γ are sections of the normal fan of the universal associahedron Asso un (Γ) . CH was supported by NSERC Discovery grant
Coxeter groups and related structures . SS was a Marie Curie- Cofund Fellow at INdAM and is now supported by the ISF grant 1144/16. VP was partially supported by theFrench ANR grant SC3A (15 CE40 0004 01). a r X i v : . [ m a t h . C O ] F e b CHRISTOPHE HOHLWEG, VINCENT PILAUD, AND SALVATORE STELLA
This universal associahedron provides a tool to study simultaneously geometric properties of allgeneralized associahedra of a given finite Dynkin type. For example, it is known that the vertexbarycenters of all generalized associahedra of [HLT11] lie at the origin. In type A , this propertywas observed by F. Chapoton for J.-L. Loday’s realization of the classical associahedron [Lod04],conjectured for all associahedra of C. Hohlweg and C. Lange in [HL07], proved by C. Hohlweg,J. Lortie and A. Raymond [HLR10] and revisited by C. Lange and V. Pilaud in [LP13]. For arbi-trary acyclic finite types, it was conjectured by C. Hohlweg, C. Lange and H. Thomas in [HLT11]and proved by V. Pilaud and C. Stump using the brick polytope approach [PS15b]. In the presentpaper, we use the universal associahedron to extend this surprising property to all generalizedassociahedra Asso (B ◦ ). Theorem 3.
The origin is the vertex barycenter of the universal associahedron
Asso un (Γ) , andthus of all generalized associahedra Asso (B ◦ ) . The paper is organized as follows. In Section 2, we collect all the definitions and propertiesof finite type cluster algebras needed in this paper. In Section 3, we recall convenient criteria tocheck that a collection of cones forms a polyhedral fan and that a simplicial fan is the normalfan of a polytope. Based on a precise understanding of the linear dependences in g -vectors ofadjacent cones described in Section 4, we prove the polytopality of all finite type g -vector fans inSection 5. Section 6 is devoted to two special cases: that of acyclic initial exchange matrices forwhich our construction yields the same generalized associahedra as [HLT11, Ste13, PS15a], andthat of type A which presents several remarkable features. In particular we prove that the facetdescription of the associahedron Asso (B ◦ ) is contained in the facet description of the correspond-ing zonotope Zono (B ◦ ) for any initial exchange matrix B ◦ of type A . Further properties of ourgeneralized associahedra are explored in Section 7, including their connection to green mutations(Section 7.1), the construction of the universal associahedron (Section 7.2), its vertex barycenter(Section 7.3), and a discussion on the relation between Asso (B ◦ ) and Zono (B ◦ ) (Section 7.4).2. Finite type cluster algebras
We begin by recalling some standard notions on cluster algebras simplifying, whenever possible,our notations to deal with the case at hand. This section can be used as a compendium of theresults concerning finite type that are scattered through the literature. We refer to [FZ07] for ageneral treatment of cluster algebras.2.1.
Cluster algebras.
We will be working in the ambient field Q ( x , . . . , x n , p , . . . , p m ) of ra-tional expressions in n + m variables with coefficients in Q and we denote by P m its abelianmultiplicative subgroup generated by the elements { p i } i ∈ [ m ] . Given p = (cid:81) i ∈ [ m ] p a i i ∈ P m we willwrite { p } + : = (cid:89) i ∈ [ m ] p max( a i , i and { p } − : = (cid:89) i ∈ [ m ] p − min( a i , i so that p = { p } + { p } − − .A seed Σ is a triple (B , P , X) consisting of an exchange matrix, a coefficient tuple, and a cluster: • the exchange matrix B is an integer n × n skew-symmetrizable matrix, i.e., such that thereexist 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 cluster variables , n rational functions in the ambient field that arealgebraically independent over Q ( p , . . . , p m ).To shorten our notation we think of rows and columns of B, as well as elements of P, as beinglabeled by the elements of X: we write B = ( b xy ) x,y ∈ X and P = { p x } x ∈ X . Moreover we say that acluster variable x (resp. a coefficient p ) belongs to Σ = (B , P , X) to mean x ∈ X (resp. p ∈ P).Given a seed Σ = (B , P , X) and a cluster variable x ∈ Σ, we can construct a new seed µ x (Σ) =Σ (cid:48) = (B (cid:48) , P (cid:48) , X (cid:48) ) by mutation in direction x , where: OLYTOPAL REALIZATIONS OF FINITE TYPE g -VECTOR FANS 3 • the new cluster X (cid:48) is obtained from X by replacing x with the cluster variable x (cid:48) definedby the following exchange relation : xx (cid:48) = { p x } + (cid:89) y ∈ X b xy > y b xy + { p x } − (cid:89) y ∈ X b xy < y − b xy and leaving the remaining cluster variables unchanged so that X (cid:114) { x } = X (cid:48) (cid:114) { x (cid:48) } . • the row (resp. column) of B (cid:48) indexed by x (cid:48) is the negative of the row (resp. column) of Bindexed by x , while all the other entries satisfy: b (cid:48) yz = b yz + 12 (cid:0) | b yx | b xz + b yx | b xz | (cid:1) , • the elements of the new coefficient tuple P (cid:48) are p (cid:48) y = p − x if y = x (cid:48) ,p y { p x } b xy − if y (cid:54) = x (cid:48) and b xy ≤ ,p y { p x } b xy + if y (cid:54) = x (cid:48) and b xy > , A straightforward computation shows that mutations are involutions, i.e., µ x (cid:48) ( µ x (Σ)) = Σ so theydefine an equivalence relation on the collection of all seeds.Fix a seed Σ ◦ = (B ◦ , P ◦ , X ◦ ) and call it initial . Up to an automorphism of the ambient field wewill assume that X ◦ = { x , . . . , x n } and drop X ◦ from our notation. Definition 4 ([FZ07, Def. 2.11]) . The (geometric type) cluster algebra A (B ◦ , P ◦ ) is the ZP m -sub-ring of the ambient field generated by all the cluster variables in all the seeds mutationally equiv-alent to the initial seed Σ ◦ . Example 5.
The simplest possible choice of coefficient tuple in the initial seed, namely m = 0 andP ◦ = { } i ∈ [ n ] , gives rise to the cluster algebra without coefficients which we will denote by A fr (B ◦ ).Note that this algebra, up to an automorphism of the ambient field Q ( x , . . . , x n ), depends onlyon the mutation class of B ◦ and not on the exchange matrix itself. The appearance of B ◦ in thenotation A fr (B ◦ ) is just to fix the embedding inside the ambient field.2.2. Finite type.
We will only be dealing with cluster algebras of finite type i.e., cluster algebrashaving only a finite number of cluster variables. As it turns out, being of finite type is a propertythat depends only on the exchange matrix in the initial seed and not on the coefficient tuple.The
Cartan companion of an exchange matrix B is the symmetrizable matrix A(B) given by: a xy = (cid:40) x = y, −| b xy | otherwise. Theorem 6 ([FZ03a, Thm. 1.4]) . The cluster algebra A (B ◦ , P ◦ ) is of finite type if and only if thereexists an exchange matrix B obtained by a sequence of mutations from B ◦ such that its Cartancompanion is a Cartan matrix of finite type. Moreover the type of A(B) is uniquely determinedby B ◦ : if B (cid:48) is any other exchange matrix obtained by mutation from B ◦ and such that A(B (cid:48) ) is afinite type Cartan matrix then A(B (cid:48) ) and A(B) are related by a simultaneous permutation of rowsand columns.
In accordance with the above statement, when talking about the (cluster) type of A (B ◦ , P ◦ ) orB ◦ we will refer to the Cartan type of A(B). We reiterate that the Cartan type of A(B ◦ ) need notbe finite: being of finite type is a property of the mutation class.For a finite type cluster algebra A (B ◦ , P ◦ ), we will consider the root system of A(B ◦ ). To avoidany confusion later on let us state clearly the conventions we use in this paper: for us simpleroots { α x } x ∈ X ◦ and fundamental weights { ω x } x ∈ X ◦ are two basis of the same vector space V ;the matrix relating them is the Cartan matrix A(B ◦ ). Fundamental weights are the dual basis tosimple coroots { α ∨ x } x ∈ X ◦ , while simple roots are the dual basis to fundamental coweights { ω ∨ x } x ∈ X ◦ ;coroots and coweights are two basis of the dual space V ∨ and they are related by the transposeof the Cartan matrix. This set of conventions is the standard one in Lie theory but it is not theone generally used in the setting of Coxeter groups [BB05, Chap. 4]. CHRISTOPHE HOHLWEG, VINCENT PILAUD, AND SALVATORE STELLA
A finite type exchange matrix B ◦ is said to be acyclic if A(B ◦ ) is itself a Cartan matrix of finitetype and cyclic otherwise. An acyclic finite type exchange matrix is said to be bipartite if each ofits rows (or equivalently columns) consists either of non-positive or non-negative entries.2.3. Principal coefficients, g- and c-vectors.
Among all the cluster algebras having a fixedinitial exchange matrix, a central role is played by those with principal coefficients. Indeed, thanksto the results in [FZ07], they encode enough informations to understand all the other possiblechoices of coefficients.
Definition 7 ([FZ07, Def. 3.1]) . A cluster algebra is said to have principal coefficients (at theinitial seed) if its ambient field is Q ( x , . . . , x n , p , . . . , p n ) and the initial coefficient tuple consistsof the generators of P n i.e., P ◦ = { p i } i ∈ [ n ] . In this case we will write A pr (B ◦ ) for A (cid:0) B ◦ , { p i } i ∈ [ n ] (cid:1) ,and we reindex the generators { p i } i ∈ [ n ] of P n by { p x } x ∈ X ◦ .In the above definition, it is important to specify that principal coefficients are with respect to aspecific exchange matrix, even though it is usually omitted. In other words A pr (B ◦ ) and A pr (B (cid:48)◦ )are in general different cluster algebras even when B ◦ and B (cid:48)◦ are related by mutations.A notable property of cluster algebras with principal coefficients is that they are Z n -graded (inthe basis { ω x } x ∈ X ◦ of V ). The degree function deg(B ◦ , · ) on A pr (B ◦ ) is obtained by settingdeg(B ◦ , x ) : = ω x and deg(B ◦ , p x ) : = (cid:88) y ∈ X ◦ − b yx ω y for any x ∈ X ◦ . This assignment makes all exchange relations and all cluster variables in A pr (B ◦ )homogeneous [FZ07] and it justifies the definition of the following family of integer vectors asso-ciated to cluster variables. Definition 8 ([FZ07]) . The g -vector of a cluster variable x ∈ A pr (B ◦ ) is its degree g (B ◦ , x ) : = deg(B ◦ , x ) ∈ V. We denote by g (B ◦ , Σ) : = { g (B ◦ , x ) | x ∈ Σ } the set of g -vectors of the cluster variable in theseed Σ of A pr (B ◦ ).The next definition gives another family of integer vectors, introduced implicitly in [FZ07], thatare relevant in the structure of A pr (B ◦ ). Definition 9.
Given a seed Σ in A pr (B ◦ ), the c -vector of a cluster variable x ∈ Σ is the vector c (B ◦ , x ∈ Σ) : = (cid:88) y ∈ X ◦ c yx α y ∈ V of exponents of p x = (cid:81) y ∈ X ◦ ( p y ) c yx . Let c (B ◦ , Σ) : = { c (B ◦ , x ∈ Σ) | x ∈ Σ } denote the set of c -vectors of a seed Σ. Finally, let C (B ◦ ) : = (cid:83) Σ c (B ◦ , Σ) denote the set of all c -vectors in A pr (B ◦ ).It is worth spending few words here to emphasize the fact that, contrary to what happens for g -vectors, c -vectors are not attached to cluster variables per se but depends on the seed in whichthe given cluster variable sits.An important features of c -vectors is that their entries weekly agree in sign. This is one ofthe various reformulation of the sign-coherence conjecture of [FZ07] recently established in fullgenerality by [GHKK18]. In the setting of finite type cluster algebras, this result can also bededuced in several ways from earlier works: one proof is to combine [DWZ10] with [Dem10];another one is to use surfaces [FST08, FT12] and orbifolds [FST12] to study types A n , B n , C n and D n , and to deal with exceptional types by direct inspection. Here we prefer to observe it asa corollary of the following theorem that will be useful later on to justify our notation. Theorem 10 ([NS14, Thm. 1.3]) . The c -vectors of the finite type cluster algebra A pr (B ◦ ) areroots in the root system whose Cartan matrix is A(B ◦ ) . Again note that, since A(B ◦ ) may be not of finite type, the root system in this statement is,in general, not finite. For example, for the cyclic type A exchange matrix, the Cartan compan-ion A(B ◦ ) is of affine type A (1)2 , see its Coxeter arrangement in Figure 5 (top right). More precisely, OLYTOPAL REALIZATIONS OF FINITE TYPE g -VECTOR FANS 5 it is finite if and only if B ◦ is acyclic. We will discuss in more details the relation of c -vectors withroot systems in Remark 21.Another consequence of [NS14, Thm. 1.3] is that in a cluster algebra of finite type, there are nh distinct c -vectors where h is the Coxeter number of the given finite type. We remind to ourreader that the same algebra has ( h + 2) n/ g -vectors, one for each cluster variable.Our next task in this section is to discuss a duality relation in between c -vectors and g -vectors.A first step is to recall the notion of the cluster complex of A (B ◦ , P ◦ ): it is the abstract simplicialcomplex whose vertices are the cluster variables of A (B ◦ , P ◦ ) and whose facets are its clusters.As it turns out, at least in the finite type cases, this complex is independent of the choice ofcoefficients, see [FZ03a, Thm. 1.13] and [FZ07, Conj. 4.3]. In particular this means that, up toisomorphism, there is only one cluster complex for each finite type: the one associated to A fr (B ◦ ).We will use this remark later on to relate cluster variables of different cluster algebras of thesame finite type. Note also that, again when A (B ◦ , P ◦ ) is of finite type, the cluster complex is a pseudomanifold [FZ03a].For a skew-symmetrizable exchange matrix B ◦ , the matrix B ∨◦ : = − B T ◦ is still skew-symmetrizable.The cluster algebras A pr (B ◦ ) and A pr (B ∨◦ ) can be thought as dual to each other. Indeed theirtypes are Langlands dual of each other. Moreover their cluster complexes are isomorphic: byperforming the same sequence of mutations we can identify any cluster variable x of A pr (B ◦ ) witha cluster variable x ∨ of A pr (B ∨◦ ), and any seed Σ in A pr (B ◦ ) with a seed Σ ∨ in A pr (B ∨◦ ). Moreimportantly the following crucial property holds. Theorem 11 ([NZ12, Thm. 1.2]) . For any seed Σ of A pr (B ◦ ) , let Σ ∨ be its dual in A pr (B ∨◦ ) .Then the g -vectors g (B ◦ , Σ) of the cluster variables in Σ and the c -vectors c (B ∨◦ , Σ ∨ ) of the clustervariables in Σ ∨ are dual bases, i.e., (cid:10) g (B ◦ , x ) (cid:12)(cid:12) c (B ∨◦ , y ∨ ∈ Σ ∨ ) (cid:11) = δ x = y for any two cluster variables x, y ∈ Σ . In view of the above results, and since A(B ∨◦ ) = A(B ◦ ) T , the c -vectors of a finite type clusteralgebra A pr (B ∨◦ ) can be understood as coroots for A(B ◦ ) so that the g -vectors of A pr (B ◦ ) becomeweights. This justify our choice to define g -vectors in the weight basis.2.4. Coefficient specialization and universal cluster algebra.
We now want to relate, withina given finite type, cluster algebras with different choices of coefficients. Pick a finite type exchangematrix B ◦ and let A (B ◦ , P ◦ ) ⊂ Q ( x , . . . , x n , p , . . . , p m ) and A (cid:0) B ◦ , P ◦ (cid:1) ⊂ Q ( x , . . . , x n , p , . . . , p (cid:96) )be any two cluster algebras having B ◦ as exchange matrix in their initial seed. As we said, clustervariables and seeds in these two algebras are in bijection because their cluster complexes areisomorphic. Let us write x ←→ x and Σ ←→ Σfor this bijection. We will say that A (cid:0) B ◦ , P ◦ (cid:1) is obtained from A (B ◦ , P ◦ ) by a coefficient special-ization if there exist a map of abelian groups η : P m → P (cid:96) such that, for any p x in some seed Σof A (B ◦ , P ◦ ) η ( { p x } + ) = { p x } + and η ( { p x } − ) = { p x } − and which extends in a unique way to a map of algebras that satisfy η ( x ) = x. Note that this is not the most general definition (see [FZ07, Def. 12.1 and Prop. 12.2]) but it willsuffice here. Armed with the notion of coefficient specialization we can now introduce the lastkind of cluster algebra of finite type we will need.
Definition 12 ([FZ07, Def. 12.3 and Thm. 12.4]) . Pick a finite type exchange matrix B ◦ . The cluster algebra with universal coefficients A un (B ◦ ) is the unique (up to canonical isomorphism)cluster algebra such that any other cluster algebra of the same type as B ◦ can be obtained fromit by a unique coefficient specialization. CHRISTOPHE HOHLWEG, VINCENT PILAUD, AND SALVATORE STELLA
Let us insist on the fact that, in view of the universal property it satisfies, A un (B ◦ ) dependsonly on the type of B ◦ and not on the exchange matrix B ◦ itself. We again keep B ◦ in the notationonly to fix an embedding into the ambient field.Rather than proving the existence and explaining the details of how such a universal algebra isbuilt, we will recall here one of its remarkable properties that follows directly from the g -vectorrecursion [NZ12, Prop. 4.2 (v)] and that we will need in our proofs later on. Denote by X (B ◦ ) theset of all cluster variables in A un (B ◦ ) and let { p [ x ] } x ∈X (B ◦ ) be the generators of P |X (B ◦ ) | . Theorem 13 ([Rea14, Theo. 10.12]) . The cluster algebra A un (B ◦ ) can be realized over P |X (B ◦ ) | .The coefficient tuple P = { p x } x ∈ X at each seed Σ = (B , P , X) of A un (B ◦ ) is given by the formula p x = (cid:89) y ∈X (B ◦ ) (cid:0) p [ y ] (cid:1) [ g (B T ,y T ); x T ] where we denote by [ v ; x ] the x -th coefficient of a vector v in the weight basis ( ω x ) x ∈ X . The bijectionof the elements of X (B ◦ ) with the cluster variables of A pr (cid:0) B T (cid:1) , appearing in the formula, is givenby an isomorphism of the corresponding cluster complexes similar to the one discussed above. Remark 14.
In view of this result, it is straightforward to produce the coefficient specializa-tion morphism to get any cluster algebra with principal coefficients of type B ◦ from A un (B ◦ ).Namely, for any seed Σ (cid:63) = (B (cid:63) , P (cid:63) , X (cid:63) ) of A un (B ◦ ), we obtain A pr (B (cid:63) ) by evaluating to 1 all thecoefficients p [ y ] corresponding to cluster variables y not in Σ (cid:63) .We conclude this review giving an example: the cluster algebra of type B with universalcoefficients. We will do so in terms of “tall” rectangular matrices to help readers, not familiar withthe language we adopt here, recognize an hopefully more familiar setting. Indeed, to pass fromour seeds to the one consisting of extended exchange matrices and clusters with frozen variables,it suffices to observe that each p ∈ P m can be encoded in a vector. In this way any n -tuple ofelements of P m corresponds to a m × n integer matrix and one gets an extended exchange matrixby gluing it below the exchange matrix of the seed. The frozen variables are the generators of P m and the rules of mutations we discussed become then the usual mutations of extended exchangematrices. We prefer to use the notation we set up here following [FZ03a, FZ07] because it makesmore evident the distinction in between coefficients and cluster variables, and because it is moresuited to deal with coefficient specializations. Example 15.
Consider the exchange matrixB ◦ = (cid:20) − (cid:21) . Any cluster algebra built from this matrix will contain 6 cluster variables. We will call them X (B ◦ ) = { x , x , x , x , x , x } . The corresponding cluster algebra with universal coefficients willthen be a subring of Q ( x , x , p [ x ] , p [ x ] , p [ x ] , p [ x ] , p [ x ] , p [ x ]) . Namely it will be the cluster algebra A un (B ◦ ) = A (cid:18) B ◦ , (cid:26) p [ x ] p [ x ] p [ x ] p [ x ] , p [ x ] p [ x ] p [ x ] p [ x ] (cid:27)(cid:19) Figure 1 shows the exchange graph of this algebra listing all the seeds in terms of extended exchangematrices.One final computational remark: there is a simple algorithm to compute one of the rectangularexchange matrices appearing in a cluster algebra of finite type with universal coefficients. Let Bbe an exchange matrix of the given finite type having only non-negative entries above its diagonal;it is acyclic and, by [YZ08, Eqn. (1.4)], it corresponds to the
Coxeter element c = s · · · s n inthe associated Weyl group (note that the labelling of simple roots may not be the standard one A note to the reader that might be scared of a circular reasoning here: the set of generators { p [ x ] } x ∈X (B ◦ ) isjust a collection of symbols and its cardinality can be precomputed: it depends only on the type of the algebra andnot on its coefficients. OLYTOPAL REALIZATIONS OF FINITE TYPE g -VECTOR FANS 7 x x x − x p [ x ] p [ x ] − p [ x ]0 − p [ x ]1 − p [ x ]2 − p [ x ] x x − x x p [ x ] − p [ x ] − p [ x ] − p [ x ] − p [ x ]0 1 p [ x ] x x x x − p [ x ] − p [ x ] − p [ x ] 1 − p [ x ] 2 − p [ x ] 1 0 p [ x ] 0 1 x x x − x p [ x ] − p [ x ] − p [ x ] − p [ x ] 0 1 p [ x ] 1 0 p [ x ] 0 − x x x x − p [ x ] − p [ x ] − p [ x ] 1 0 p [ x ] 0 1 p [ x ] − p [ x ] 0 − x x − x x − p [ x ] p [ x ]1 0 p [ x ]0 − p [ x ] − p [ x ] − p [ x ] Figure 1.
The exchange graph of type B with attached the rectangular matrices giving universalcoefficients. In bold are highlighted the entries of the coefficient part that give the principalcoefficient cluster algebra at the seed attached to the marked node.here). Let w ◦ ( c ) denote the c -sorting word for w ◦ , that is the lexicographically minimal reducedexpression of w ◦ that appears as a subword of c ∞ . For each s i obtained reading from left to rightthe word cw ◦ ( c ), repeat the following two steps: • add a row to the bottom of B whose only non-zero entry is a 1 in column i • replace B by its mutation in direction i .The matrix obtained at the end of the procedure is the desired one.3. Polyhedral geometry and fans
The second ingredient of this paper is discrete geometry of polytopes and fans. We referto [Zie98] for a textbook on this topic.3.1.
Polyhedral fans. A polyhedral cone is a subset of the vector space V defined equivalentlyas the positive span of finitely many vectors or as the intersection of finitely many closed linearhalfspaces. Throughout the paper, we write R ≥ Λ for the positive span of a set Λ of vectors of V . CHRISTOPHE HOHLWEG, VINCENT PILAUD, AND SALVATORE STELLA
The faces of a cone C are the intersections of C with its supporting hyperplanes. In particular,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 independent vectors.A polyhedral fan is a collection F of polyhedral cones of V 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, and complete if the union of its cones covers the ambientspace V . For a simplicial fan F with rays X , the collection { X ⊆ X | R ≥ X ∈ F} of generating setsof the cones of F defines a pseudomanifold (in other words, a pure and thin simplicial complex, i.e., with a notion of flip). The following statement characterizes which pseudomanifolds arecomplete simplicial fans. A formal proof can be found e.g., in [DRS10, Coro. 4.5.20]. Proposition 16.
Consider a pseudomanifold ∆ with vertex set X and a set of vectors (cid:8) r ( x ) (cid:9) x ∈X of V . For X ∈ ∆ , let r (X) : = (cid:8) r ( x ) (cid:12)(cid:12) x ∈ X (cid:9) . Then the collection of cones (cid:8) R ≥ r (X) (cid:12)(cid:12) X ∈ ∆ (cid:9) forms a complete simplicial fan if and only if(1) there exists a facet X of ∆ such that r (X) is a basis of V and such that the open cones R > r (X) and R > r (X (cid:48) ) are disjoint for any facet X (cid:48) of ∆ distinct from X ;(2) for any two adjacent facets X , X (cid:48) of ∆ with X (cid:114) { x } = X (cid:48) (cid:114) { x (cid:48) } , there is a linear dependence γ r ( x ) + γ (cid:48) r ( x (cid:48) ) + (cid:88) y ∈ X ∩ X (cid:48) δ y r ( y ) = 0 on r (X ∪ X (cid:48) ) where the coefficients γ and γ (cid:48) have the same sign. (When these conditionshold, these coefficients do not vanish and the linear dependence is unique up to rescaling.) Polytopes and normal fans. A polytope is a subset P of V ∨ defined equivalently asthe convex hull of finitely many points or as a bounded intersection of finitely many closed affinehalfspaces. The faces of P are the intersections of P with its supporting hyperplanes. In particular,the dimension 0 (resp. dimension 1, resp. codimension 1) faces of P are called vertices (resp. edges ,resp. facets ) of P . The (outer) normal cone of a face F of P is the cone in V generated by the outernormal vectors of the facets of P containing F . The (outer) normal fan of P is the collection ofthe (outer) normal cones of all its faces. We say that a complete polyhedral fan in V is polytopal when it is the normal fan of a polytope in V ∨ . The following statement provides a characterizationof polytopality of complete simplicial fans. It is a reformulation of regularity of triangulations ofvector configurations, introduced in the theory of secondary polytopes [GKZ08], see also [DRS10].We present here a convenient formulation from [CFZ02, Lem. 2.1]. Proposition 17.
Consider a pseudomanifold ∆ with vertex set X and a set of vectors (cid:8) r ( x ) (cid:9) x ∈X of V such that F : = (cid:8) R ≥ r (X) (cid:12)(cid:12) X ∈ ∆ (cid:9) forms a complete simplicial fan. Then the following areequivalent:(1) F is the normal fan of a simple polytope in V ∨ ;(2) There exists a map h : X → R > such that for any two adjacent facets X , X (cid:48) of ∆ with X (cid:114) { x } = X (cid:48) (cid:114) { x (cid:48) } , we have γ h ( x ) + γ (cid:48) h ( x (cid:48) ) + (cid:88) y ∈ X ∩ X (cid:48) δ y h ( y ) > , where γ r ( x ) + γ (cid:48) r ( x (cid:48) ) + (cid:88) y ∈ X ∩ X (cid:48) δ y r ( y ) = 0 is the unique (up to rescaling) linear dependence with γ, γ (cid:48) > between the rays of r (X ∪ X (cid:48) ) .Under these conditions, F is the normal fan of the polytope defined by (cid:8) v ∈ V ∨ (cid:12)(cid:12) (cid:104) r ( x ) | v (cid:105) ≤ h ( x ) for all x ∈ X (cid:9) . OLYTOPAL REALIZATIONS OF FINITE TYPE g -VECTOR FANS 9 The g -vector fan We first recast a well known fact concerning the cones spanned by the g -vectors of a finite typecluster algebra with principal coefficients. We insist on the fact that the following statement isvalid for any finite type exchange matrix B ◦ , acyclic or not. Theorem 18.
For any finite type exchange matrix B ◦ , the collection of cones F g (B ◦ ) : = (cid:8) R ≥ g (B ◦ , Σ) (cid:12)(cid:12) Σ seed of A pr (B ◦ ) (cid:9) , together with all their faces, forms a complete simplicial fan, called the g -vector fan of B ◦ . There are several ways to find or deduce Theorem 18 from the literature. First, it was es-tablished in the acyclic case in [RS09, YZ08, Ste13] (see Example 22). As already observed byN. Reading in [Rea14, Thm. 10.6], the general case then follows from the initial seed recursion on g -vectors [NZ12, Prop. 4.2 (v)], valid thanks to sign-coherence. A second proof would be to use theunique cluster expansion property of any vector in the weight lattice (following from the fact thatcluster monomials are a basis of A pr (B ◦ ) in finite type), and to use approximation by this latticeto show that any vector is covered exactly once by the interiors of the cones of the g -vector fan.Note that contrarily to what sometimes appears in the literature, this approximation argumentis subtle as it relies on the integrity of the g -vectors . Finally, another possible proof is to useProposition 16: the first point is a simplified version of the unique expansion property, and thesecond point is a consequence of the following description of the linear dependence between the g -vectors of two adjacent clusters, which will be crucial in the next section. Lemma 19.
For any finite type exchange matrix B ◦ and any adjacent seeds (B , P , X) and (B (cid:48) , P (cid:48) , X (cid:48) ) in A pr (B ◦ ) with X (cid:114) { x } = X (cid:48) (cid:114) { x (cid:48) } , the g -vectors of X ∪ X (cid:48) satisfy precisely one of the followingtwo linear dependences g (cid:0) B ◦ , x (cid:1) + g (cid:0) B ◦ , x (cid:48) (cid:1) = (cid:88) y ∈ X ∩ X (cid:48) b xy < − b xy g (cid:0) B ◦ , y (cid:1) or g (cid:0) B ◦ , x (cid:1) + g (cid:0) B ◦ , x (cid:48) (cid:1) = (cid:88) y ∈ X ∩ X (cid:48) b xy > b xy g (cid:0) B ◦ , y (cid:1) . Proof.
This is a straightforward consequence of the definition of g -vectors together with signcoherence. Indeed all exchange relations in A pr (B ◦ ) are homogeneous and xx (cid:48) = { p x } + (cid:89) y ∈ X b xy > y b xy + { p x } − (cid:89) y ∈ X b xy < y − b xy means that deg(B ◦ , x ) + deg(B ◦ , x (cid:48) ) = deg(B ◦ , { p x } + ) + (cid:88) y ∈ X ∩ X (cid:48) b xy > b xy deg(B ◦ , y )= deg(B ◦ , { p x } − ) + (cid:88) y ∈ X ∩ X (cid:48) b xy < − b xy deg(B ◦ , y ) . Now, by sign-coherence, exactly one of { p x } + and { p x } − is 1 so that its degree is 0. (cid:3) Remark 20.
Note that which of the two possible linear dependences is satisfied by the g -vectorsof X ∪ X (cid:48) depends on the initial exchange matrix B ◦ . In particular, the geometry of the g -vectorfan F g (B ◦ ) changes as B ◦ varies within a given mutation class. For an illustration of the subtlety, consider the 8 cones in R defined by the coordinate hyperplanes, rotatethe 4 cones with x ≥ x -axis by π/
4, and finally rotate all the cones around the origin by an irrationalrotation so that each rational direction belongs to the interior of one of the 8 resulting cones. Then any vector in Z belongs to a single cone, but the resulting cones do not form a fan (since the cones with x ≥ x ≤ Figure 2.
The dual c -vector fan F c (B ∨◦ ) (thin red) and g -vector fan F g (B ◦ ) (bold blue) for thetype A (left) and type C (right) cyclic initial exchange matrices. Each 3-dimensional fan is inter-sected with the unit sphere and stereographically projected to the plane from the pole ( − , − , − ◦ , the g -vector fan F g (B ◦ ) can be seen as a coarseningof two other fans naturally associated to A(B ◦ ). Denote by F Cox (B ∨◦ ) the dual Coxeter fan i.e., the fan of regions of the hyperplane arrangement given by the root system of A(B ∨◦ ). Similarlylet F c (B ∨◦ ) be the dual c -vector fan i.e., the fan of regions of the arrangement of hyperplanesorthogonal to all the c -vector of A pr (B ∨◦ ). By Theorem 11, F g (B ◦ ) coarsens F c (B ∨◦ ) which, inturn, coarsens F Cox (B ∨◦ ) by Theorem 10. See Figure 5 for examples of these fans for differentexchange matrices of type A . Remark 21.
By further inspecting [NS14, Thm. 1.3] we can say more about F c (B ∨◦ ). Indeedits supporting hyperplane arrangement contains the hyperplanes associated to small roots in theroot system of A(B ∨◦ ). Therefore, it turns out that the dual c -vector fan F c (B ∨◦ ) intersectedwith the Tits cone contains the Shi arrangement for the root system of A(B ∨◦ ) (see [HNW16,Sect. 3.6 & Def. 3.18] for a review on the topic). In order to see that, it is enough to comparethe description of the possible supports of the c -vectors (in terms of simple roots) given in [NS14,Thm. 1.3] with the description of the possible supports of small roots given in [Bri98]. Example 22.
When the exchange matrix B ◦ is acyclic, the g -vector fan is the Cambrian fanconstructed by N. Reading and D. Speyer [RS09], while the dual c -vector fan is the type A(B ∨◦ )Coxeter fan. Section 6.1 provides a detailed discussion of the acyclic case. Example 23.
Figure 2 illustrates the g -vector fans for the initial exchange matrices − − − and − − − (type A cyclic) (type C cyclic)Note that these matrices are the only examples of 3-dimensional cyclic exchange matrices (up toduality and simultaneous permutations of rows and columns). OLYTOPAL REALIZATIONS OF FINITE TYPE g -VECTOR FANS 11 Polytopality
In this section, we show that the g -vector fan F g (B ◦ ) is polytopal for any finite type exchangematrix B ◦ . As discussed in Section 6.1, this result was previously known for acyclic finite type ex-change matrices [HLT11, Ste13, PS15a]. The proof of this paper, although similar in spirit to thatof [Ste13], actually simplifies the previous approaches.We first consider some convenient functions which will be used later in Theorem 26 to lift the g -vector fan. The existence of such functions will be discussed in Proposition 28. Definition 24.
A positive function h on the cluster variables of A (B ◦ , P ◦ ) is exchange submod-ular if, for any pair of adjacent seeds (B , P , X) and (B (cid:48) , P (cid:48) , X (cid:48) ) with X (cid:114) { x } = X (cid:48) (cid:114) { x (cid:48) } , itsatisfies h ( x ) + h ( x (cid:48) ) > max (cid:16) (cid:88) y ∈ X ∩ X (cid:48) b xy < − b xy h ( y ) , (cid:88) y ∈ X ∩ X (cid:48) b xy > b xy h ( y ) (cid:17) . Definition 25.
Let h be a positive function on the cluster variables of A pr (B ◦ ) we define:(i) a point p h (cid:0) B ◦ , Σ (cid:1) : = (cid:88) x ∈ Σ h ( x ) c (cid:0) B ∨◦ , x ∨ ∈ Σ ∨ (cid:1) ∈ V ∨ for each seed Σ of A pr (B ◦ ),(ii) a halfspace H h ≤ (B ◦ , x ) and a hyperplane H h = (B ◦ , x ) by H h ≤ (cid:0) B ◦ , x (cid:1) : = (cid:8) v ∈ V ∨ (cid:12)(cid:12) (cid:10) g (cid:0) B ◦ , x (cid:1) (cid:12)(cid:12) v (cid:11) ≤ h ( x ) (cid:9) and H h = (cid:0) B ◦ , x (cid:1) : = (cid:8) v ∈ V ∨ (cid:12)(cid:12) (cid:10) g (cid:0) B ◦ , x (cid:1) (cid:12)(cid:12) v (cid:11) = h ( x ) (cid:9) for each cluster variable x of A pr (B ◦ ).The following statement is the central result of this paper. We refer again to Proposition 28 forthe existence of exchange submodular functions. Theorem 26.
For any finite type exchange matrix B ◦ and any exchange submodular function h ,the g -vector fan F g (B ◦ ) is the normal fan of the B ◦ -associahedron Asso h (B ◦ ) ⊆ V ∨ equivalentlydefined as(i) the convex hull of the points p h (B ◦ , Σ) for all seeds Σ of A pr (B ◦ ) , or(ii) the intersection of the halfspaces H h ≤ (B ◦ , x ) for all cluster variables x of A pr (B ◦ ) .Proof. Consider two adjacent seeds Σ = (B , P , X) and Σ (cid:48) = (B (cid:48) , P (cid:48) , X (cid:48) ) with X (cid:114) { x } = X (cid:48) (cid:114) { x (cid:48) } .By Lemma 19, the linear dependence between the g -vectors of X ∪ X (cid:48) is of the form g (cid:0) B ◦ , x (cid:1) + g (cid:0) B ◦ , x (cid:48) (cid:1) = (cid:88) y ∈ X ∩ X (cid:48) ε (B ◦ , Σ , Σ (cid:48) ) b xy > ε (cid:0) B ◦ , Σ , Σ (cid:48) (cid:1) b xy g (cid:0) B ◦ , y (cid:1) for some ε (B ◦ , Σ , Σ (cid:48) ) ∈ {± } (depending on the initial exchange matrix B ◦ ). However, by Defini-tion 24, the function h satisfies h ( x ) + h ( x (cid:48) ) > max (cid:16) (cid:88) y ∈ X ∩ X (cid:48) b xy < − b xy h ( y ) , (cid:88) y ∈ X ∩ X (cid:48) b xy > b xy h ( y ) (cid:17) ≥ (cid:88) y ∈ X ∩ X (cid:48) ε (B ◦ , Σ , Σ (cid:48) ) b xy > ε (cid:0) B ◦ , Σ , Σ (cid:48) (cid:1) b xy h ( y ) . Applying the characterization of Proposition 17, we thus immediately obtain that the g -vectorfan F g (B ◦ ) is the normal fan of the polytope defined as the intersection of the halfspaces H h ≤ (B ◦ , x )for all cluster variables x of A pr (B ◦ ). The term “exchange submodular” is inspired from a particular situation in type A (namely, when B ◦ has 1’s onthe upper subdiagonal and − i.e., functions h : 2 [ n ] → R such that h ( X ) + h ( Y ) > h ( X ∪ Y ) + h ( X ∩ Y ) for any distinctnon-trivial subsets X, Y of [ n ]. Finally, to show the vertex description, we just need to observe that, for any seed Σ of A pr (B ◦ ),the point p h (B ◦ , Σ) is the intersection of the hyperplanes H h = (B ◦ , x ) for all x ∈ Σ. Indeed,since g (B ◦ , Σ) and c (B ∨◦ , Σ ∨ ) form dual bases by Theorem 11, we have for any x ∈ Σ: (cid:10) g (cid:0) B ◦ , x (cid:1) (cid:12)(cid:12) p h (cid:0) B ◦ , Σ (cid:1) (cid:11) = (cid:88) y ∈ Σ h ( y ) (cid:10) g (cid:0) B ◦ , x (cid:1) (cid:12)(cid:12) c (cid:0) B ∨◦ , y ∨ ∈ Σ ∨ (cid:1) (cid:11) = (cid:88) y ∈ Σ h ( y ) δ x = y = h ( x ) . (cid:3) Remark 27.
By definition
Asso h (B ◦ ) fulfills the following properties: • the normal vectors are the g -vectors of all cluster variables of A pr (B ◦ ), • for any two adjacent seeds Σ = (B , P , X) and Σ (cid:48) = (B (cid:48) , P (cid:48) , X (cid:48) ) with X (cid:114) { x } = X (cid:48) (cid:114) { x (cid:48) } ,the edge of Asso h (B ◦ ) joining the vertex p h (B ◦ , Σ) to the vertex p h (B ◦ , Σ (cid:48) ) is a negativemultiple of the dual c -vector c (B ∨◦ , x ∨ ∈ Σ ∨ ) = − c (B ∨◦ , x (cid:48)∨ ∈ Σ (cid:48)∨ ). More precisely, p h (cid:0) B ◦ , Σ (cid:48) (cid:1) − p h (cid:0) B ◦ , Σ (cid:1) = (cid:16) (cid:88) y b xy h ( y ) − h ( x (cid:48) ) − h ( x ) (cid:17) c (cid:0) B ∨◦ , x ∨ ∈ Σ ∨ (cid:1) , where the sum runs over the variables y ∈ Σ ∩ Σ (cid:48) such that b xy has the same sign as the c -vector c (B ∨◦ , x ∨ ∈ Σ ∨ ). Note that (cid:80) y b xy h ( y ) − h ( x (cid:48) ) − h ( x ) < h is exchangesubmodular.Our next step is to show the existence of exchange submodular functions for any finite typecluster algebra with principal coefficients. The important observation here is that the definitionof exchange submodular function does not involve in any way the coefficients of A pr (B ◦ ) so thatit suffices to construct one in the coefficient free cases. Indeed, if h is exchange submodular for A fr (B ◦ ), and η is the coefficient specialization morphism given by η : A pr (B ◦ ) −→ A fr (B ◦ ) p i (cid:55)−→ h ( x ) : = h ( η ( x ))for any cluster variable x of A pr (B ◦ ).Recall that, up to an obvious automorphism of the ambient field, there exists a unique clusteralgebra without coefficients for each given finite type [FZ07]. We can therefore, without loss ofgenerality, assume that B ◦ is bipartite and directly deduce our result from [Ste13, Prop. 8.3]obtained as an easy consequence of [CFZ02, Lem. 2.4] which we recast here in our current setting.When B ◦ is acyclic, the Weyl group of A(B ◦ ) is finite and has a longest element w ◦ . Apoint λ ∨ : = (cid:80) x ∈ X ◦ λ ∨ x ω ∨ x in the interior of the fundamental Weyl chamber of A(B ∨◦ ) (that isto say λ ∨ x > x ∈ X ◦ ) is fairly balanced if w ◦ ( λ ∨ ) = − λ ∨ . Proposition 28.
Let A fr (B ◦ ) be any finite type cluster algebra without coefficients and assumethat B ◦ is bipartite. To each fairly balanced point λ ∨ corresponds an exchange submodular function h λ ∨ on A fr (B ◦ ) .Proof. To define h λ ∨ , recall from the construction in [FZ03b, FZ03a] that the set of cluster variablesin A fr (B ◦ ) is acted upon by a dihedral group generated by the symbols τ + and τ − .Each (cid:104) τ + , τ − (cid:105) -orbit of cluster variables meets the initial seed in either 1 or 2 elements. We willdefine h λ ∨ to be constant on the orbits of this action: on any element in the orbit of the initialcluster variable x the function h λ ∨ evaluates to the x -th coordinate of λ ∨ when written in thebasis of simple coroots. The requirement that λ ∨ is fairly balanced, as explained in the proof of[Ste13, Thm. 6], is tantamount to say that h λ ∨ ( x ) = h λ ∨ ( y ) if x and y are initial cluster variablesin the same (cid:104) τ + , τ − (cid:105) -orbit.The fact that the function h λ ∨ defined in this way is exchange submodular is then the contentof [CFZ02, Lem. 2.4] together with the computation following [Ste13, Prop. 8]. The only minorthing to observe is that, there, this function appears as a piecewise linear function on the ambientspace of the root lattice and thus, instead of writing the cluster expansions of the two monomialsin the right hand side of the exchange relations as we do here, their total denominator vectorsappear. (cid:3) OLYTOPAL REALIZATIONS OF FINITE TYPE g -VECTOR FANS 13 Remark 29.
Reading through [Ste13] one may have the impression, as the author did at thetime, that for a given λ ∨ a different exchange submodular function has been constructed for eachchoice of acyclic initial seed. This is not the case. Indeed, if one unravels the definitions, it is easyto see that the functions defined there only differ because the set of g -vectors they use as domainare different. Taken as function on the cluster variables in the respective cluster algebras withprincipal coefficients, under the specialization maps η , they all correspond to the same functionon A fr (B ◦ ).A particular example of fairly balanced point is the point ρ ∨ : = (cid:88) x ∈ X ◦ ω ∨ x . Note that ρ ∨ is both the sum of the fundamental coweights and the half sum of all positive corootsof the root system of finite type A(B ◦ ). In particular h ρ ∨ is the half compatibility sum of x , i.e., the half sum of the compatibility degrees h ρ ∨ ( x ) : = 12 (cid:88) y (cid:54) = x ( y (cid:107) x )over all cluster variables distinct from x . (See [FZ03b, CP15] for the definition and discussionof the relevant properties of compatibility degrees.) The point ρ ∨ is particularly relevant inrepresentation theory and its role in this context has already been observed in [CFZ02, Rem. 1.6].We call balanced B ◦ -associahedron and denote by Asso (B ◦ ) the B ◦ -associahedron Asso h ρ ∨ (B ◦ ) forthe exchange submodular function h ρ ∨ . Example 30.
When B ◦ is acyclic, the B ◦ -associahedron Asso (B ◦ ) was already constructed in[HLT11, Ste13, PS15a]. It is then obtained by deleting inequalities from the facet description ofthe permutahedron of the Coxeter group of type A(B ◦ ). Section 6.1 provides a detailed discussionof the acyclic case. Example 31.
Following Example 23, we have represented in Figure 3 the B ◦ -associahedra Asso (B ◦ )for the same two initial exchange matrices − − − and − − − (type A cyclic) (type C cyclic)Note that the leftmost associahedron of Figure 3 appeared as a mysterious realization of theassociahedron in [CSZ15].We sum up by stating the main result of this paper. Corollary 32.
For any finite type exchange matrix B ◦ , the g -vector fan F g (B ◦ ) is polytopal. Two families of examples
Before investigating more combinatorial and geometric properties of the B ◦ -associahedron Asso (B ◦ ), we take a moment to study two specific families of examples, corresponding to initialexchange matrices that are either acyclic (Section 6.1), or of type A (Section 6.2).6.1. Acyclic case.
We first consider an acyclic initial seed, i.e., with exchange matrix B ◦ whoseCartan companion A(B ◦ ) is itself a Cartan matrix of finite type. We denote by W (B ◦ ) the Weylgroup of type A(B ◦ ) and by w ◦ its longest element. Note that the choice of an acyclic seed isequivalent to the choice of a Coxeter element c of W (B ◦ ).We gather in the following list some relevant facts from the literature (some of which werealready observed earlier in the text). We refer to [Hoh12] for a detailed survey on these properties. Roots and weights: • the cluster variables of A pr (B ◦ ) are in bijection with the almost positive roots of A(B ◦ ), • the dual c -vectors of A pr (B ◦ ) are all the coroots of A(B ◦ ), Figure 3.
The associahedra
Asso (B ◦ ) for the type A (left) and type C (right) cyclic initialexchange matrices. • the g -vectors of A pr (B ◦ ) are some weights of A(B ◦ ). Cambrian fan and Coxeter fan: • the dual c -vector fan F c (B ∨◦ ) coincides with the dual Coxeter fan F Cox (B ∨◦ ), • the g -vector fan F g (B ◦ ) coincides with the c -Cambrian fan of N. Reading and D. Speyer [RS09]. HLT associahedron and permutahedron:
Consider any fairly balanced point λ ∨ : = (cid:80) x ∈ X ◦ λ ∨ x ω ∨ x in the fundamental chamber of A(B ∨◦ ). Then the B ◦ -associahedron Asso h λ ∨ (B ◦ ) coincides with the c -associahedron Asso λ ∨ ( c ) constructed by C. Hohlweg, C. Lange and H. Thomas [HLT11] and laterrevisited by S. Stella [Ste13] and by V. Pilaud and C. Stump [PS15a] in the context of brick poly-topes. In particular, Asso λ ∨ ( c ) is defined by the inequalities normal to the g -vectors of A pr (B ◦ ) inthe facet description of the B ◦ -permutahedron Perm λ ∨ (B ◦ ) : = conv { w ( λ ∨ ) | w ∈ W (B ◦ ) } = (cid:8) v ∈ V ∨ (cid:12)(cid:12) (cid:104) w ( ω x ) | v (cid:105) ≤ λ ∨ x for all x ∈ X ◦ , w ∈ W (B ◦ ) (cid:9) . See [Hoh12] for more details on the relation between
Perm λ ∨ (B ◦ ) and Asso λ ∨ ( c ). Cambrian lattice and weak order:
When oriented in the direction − (cid:80) x ∈ X ◦ ω x , • the graph of the B ◦ -permutahedron is the Hasse diagram of the weak order on W (B ◦ ), • the graph of the B ◦ -associahedron is the Hasse diagram of the c -Cambrian lattice ofN. Reading [Rea06], which is a lattice quotient and a sublattice of the weak order on W (B ◦ ). Vertex barycenter:
For any fairly balanced point λ ∨ , the origin is the vertex barycenter of boththe B ◦ -permutahedron Perm λ ∨ (B ◦ ) and the B ◦ -associahedron Asso λ ∨ (B ◦ ): (cid:88) w ∈ W (B ◦ ) w ( λ ∨ ) = (cid:88) Σ p λ ∨ (cid:0) B ◦ , Σ (cid:1) = . In type A , this property was observed by F. Chapoton for J.-L. Loday’s realization of the clas-sical associahedron [Lod04] and conjectured for arbitrary Coxeter element by C. Hohlweg andC. Lange in [HL07] in the balanced case. It was later proved by C. Hohlweg, J. Lortie and A. Ray-mond [HLR10] and revisited by C. Lange and V. Pilaud in [LP13]. Both proofs use an orbitrefinement of this property. For arbitrary finite types, it was conjectured by C. Hohlweg, C. Langeand H. Thomas in [HLT11] and proved by V. Pilaud and C. Stump using the brick polytopeapproach [PS15b].We will see in Section 7 how these properties of the B ◦ -associahedron Asso h λ ∨ (B ◦ ) for finitetype acyclic initial exchange matrices B ◦ extend to arbitrary initial exchange matrices. OLYTOPAL REALIZATIONS OF FINITE TYPE g -VECTOR FANS 15 Type A . We now consider an initial exchange matrix of type A . It is well known that thevertices of the type A associahedron are counted by the Catalan numbers and are therefore inbijection with all Catalan families. We use in this paper the classical model by triangulations ofa convex polygon [FZ03b], that we briefly recall now. Triangulation model for type A cluster algebras. Consider a convex ( n + 3)-gon Ω n . A dissection is a set of pairwise non-crossing internal diagonals of Ω n , and a triangulation is a maximaldissection (thus decomposing Ω n into triangles). A triangulation T defines a matrix B(T) = ( b γδ )whose rows and columns are indexed by the diagonals of T and where b γδ = γ follows δ in counter-clockwise order around a triangle of T, − γ precedes δ in counter-clockwise order around a triangle of T,0 otherwise.A flip in a triangulation T consists in exchanging an internal diagonal γ by the other diagonal γ (cid:48) of the quadrilateral formed by the two triangles of T containing it. The reader can observe thatthe exchange matrix B(T (cid:48) ) of the resulting triangulation T (cid:48) = T (cid:52) { γ, γ (cid:48) } is obtained from theexchange matrix B(T) by a mutation in direction γ (as defined in Section 2).Moreover, note that if the triangulation T has no internal triangle, then we can order linearlyits diagonals γ , . . . , γ n such that b γ i ,γ i +1 = ± b γ i ,γ j = 0 if | i − j | (cid:54) = 1, so that the Cartancompanion A(B(T)) is precisely the type A Cartan matrix. The reciprocal statement clearlyholds as well: any exchange matrix whose Cartan companion is the type A Cartan matrix is theexchange matrix B(T) of a triangulation T with no internal triangle. Therefore, the flip graph ontriangulations of Ω n completely encodes the combinatorics of mutations in type A .The choice of a type A initial exchange matrix B ◦ is thus equivalent to the choice of an initialtriangulation T ◦ of Ω n . The cluster algebra A fr (B(T ◦ )) has a cluster variable x δ for each (internal)diagonal of the polygon Ω n . Recall that if δ, δ (cid:48) are the two diagonals of a quadrilateral withedges κ, λ, µ, ν in cyclic order, then the corresponding cluster variables are related by the exchangerelation x δ x δ (cid:48) = x κ x µ + x λ x ν . (Here and elsewhere it is understood that if κ is a boundary edge,then x κ = 1 and similarly for λ , µ , and ν .)The compatibility degree in type A is very simple: for γ (cid:54) = δ , we have ( x γ (cid:107) x δ ) = 1 if γ and δ cross, and 0 otherwise. In particular, the function h ρ is given by h ρ ( x γ ) = i ( n − − i ) / γ with i vertices of Ω n on one side and n − − i on the other side.Finally, note that since the exchange matrix B(T ◦ ) is skew-symmetric, the algebra A pr (B(T ◦ ))and its dual A pr (B(T ◦ ) ∨ ) coincide. We therefore take the freedom to omit mentioning duals in allthis type A discussion. Shear coordinates for g- and c-vectors.
We now provide a combinatorial interpretation for the g - and c -vectors in a type A cluster algebra in terms of the triangulation model. Our presentationis a light version (for the special case of triangulations of a disk) of the shear coordinates of [FST08,FT12] developed to provide combinatorial models for cluster algebras from surfaces.We consider 2 n +6 points on the unit circle labeled clockwise by 1 ◦ , 2 • , . . . , (2 n + 5) ◦ , (2 n + 6) • .We say that 1 ◦ , ◦ , . . . , (2 n + 5) ◦ are the hollow vertices and that 2 • , • , . . . , (2 n + 6) • are the solidvertices . We simultaneously consider hollow triangulations (based on hollow vertices) and solidtriangulations (based on solid vertices), but never mix hollow and solid vertices in our triangula-tions. To help distinguishing them, hollow vertices and diagonals appear red while solid verticesand diagonals appear blue in all pictures. See e.g., Figure 4.Let T be a hollow (resp. solid) triangulation, let δ ∈ T, and let γ be a solid (resp. hollow)diagonal. We denote by Q( δ ∈ T) the quadrilateral formed by the two triangles of T incident to δ .When γ crosses δ , we define ε ( δ ∈ T , γ ) to be 1, −
1, or 0 depending on whether γ crosses Q( δ ∈ T)as a Z , as a Z , or in a corner. If γ and δ do not cross, then we set ε ( δ ∈ T , γ ) = 0.Fix once and for all a reference triangulation T ◦ of the hollow polygon and let ( ω δ ◦ ) δ ◦ ∈ T ◦ and ( α δ ◦ ) δ ◦ ∈ T ◦ denote dual bases of R T ◦ . The reference triangulation T ◦ of the hollow poly-gon defines an initial triangulation T −• : = { ( i − • ( j − • | ( i ◦ , j ◦ ) ∈ T ◦ } of the solid polygon,with B(T ◦ ) = B(T −• ). The cluster algebra A pr (B(T ◦ )) has one cluster variable x δ • for each solidinternal diagonal δ • . For a diagonal δ • and a triangulation T • with δ • ∈ T • , we write g (T ◦ , δ • ) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • • • • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • • • • • • • Figure 4.
Examples of hollow (red) and solid (blue) triangulations.and c (T ◦ , δ • ∈ T • ) for the g - and c -vectors of the variable x δ • computed from the initial seedtriangulation T −• . We denote by C (T ◦ ) the set of all c -vectors of A pr (B(T ◦ )).The following statement provides a combinatorial interpretation of the g - and c -vectors. Proposition 33.
For any diagonal δ • and a triangulation T • with δ • ∈ T • , we have g (cid:0) T ◦ , δ • (cid:1) : = (cid:88) δ ◦ ∈ T ◦ ε (cid:0) δ ◦ ∈ T ◦ , δ • (cid:1) ω δ ◦ and c (cid:0) T ◦ , δ • ∈ T • (cid:1) : = (cid:88) δ ◦ ∈ T ◦ − ε (cid:0) δ • ∈ T • , δ ◦ (cid:1) α δ ◦ . Intuitively, the g -vector of δ • is given by alternating ± zigzag of δ • in T ◦ (thediagonals of T ◦ that cross opposite edges of Q( δ ◦ ∈ T ◦ )) and the c -vector of δ • in T • is up to asign the characteristic vector of the diagonals of T ◦ that cross opposite edges of Q( δ • ∈ T • ).For example, the solid diagonal 2 • • in the triangulation T (cid:96) • of Figure 4 (left) has g -vector g (T ◦ , • • ) = ω ◦ ◦ − ω ◦ ◦ and c -vector c (T ◦ , • • ∈ T (cid:96) • ) = − α ◦ ◦ − α ◦ ◦ , while the bluediagonal 4 • • in the triangulation T r • of Figure 4 (right) has g -vector g (T ◦ , • • ) = ω ◦ ◦ and c -vector c (T ◦ , • • ∈ T r • ) = α ◦ ◦ + α ◦ ◦ .Note that there is one g -vector g (T ◦ , δ • ) for each internal diagonal δ • . In contrast, many δ • ∈ T • give the same c -vector c (T ◦ , δ • ∈ T • ). For a diagonal δ • = u • v • , let A −◦ (resp. A + ◦ ) denote theedges of T ◦ crossed by δ • and not incident to the vertices u ◦ + 1 or v ◦ + 1 (resp. u ◦ − v ◦ − c − ( δ • ) : = − (cid:80) δ ◦ ∈ A −◦ ω δ ◦ and c + ( δ • ) : = (cid:80) δ ◦ ∈ A + ◦ ω δ ◦ . The negative (resp. positive) c -vectorsof C (T ◦ ) are then precisely given by the vectors c − ( δ • ) (resp. c + ( δ • )) for all diagonals δ • notin T −• : = { ( i − • ( j − • | ( i ◦ , j ◦ ) ∈ T ◦ } (resp. T + • : = { ( i + 1) • ( j + 1) • | ( i ◦ , j ◦ ) ∈ T ◦ } ).Specializing Theorem 18, the simplicial complex of dissections of Ω n is realized by the g -vectorfan F g (T ◦ ) : = { g (T ◦ , D • ) | D • dissection of Ω n } , which coarsens the c -vector fan F c (T ◦ ) (definedby the arrangement of hyperplanes orthogonal to the c -vectors of C (T ◦ )), which in turn coarsensthe Coxeter fan F Cox (T ◦ ) (defined by the Coxeter arrangement for the Cartan matrix A(B(T ◦ ))).These fans are illustrated in Figure 5 for various initial hollow triangulations T ◦ .T ◦ -zonotope, T ◦ -associahedron and T ◦ -parallelepiped. Using these g - and c -vectors, wenow consider three polytopes associated to T ◦ :(1) The T ◦ -zonotope Zono (T ◦ ) is the Minkowski sum of all c -vectors: Zono (T ◦ ) : = (cid:88) c ∈ C (T ◦ ) c . Its normal fan is the fan given by the arrangement of the hyperplanes normal to the c -vectors of C (T ◦ ).(2) The T ◦ -associahedron Asso (T ◦ ) is the polytope defined equivalently as(i) the convex hull of the points p (cid:0) T ◦ , T • (cid:1) : = (cid:80) δ • ∈ T • h ρ ( δ • ) c (T ◦ , δ • ∈ T • ) for all solidtriangulations T • ,(ii) the intersection of the half-spaces H ≤ (cid:0) T ◦ , δ • (cid:1) : = (cid:8) v ∈ R T ◦ (cid:12)(cid:12) (cid:104) g (T ◦ , δ • ) | v (cid:105) ≤ h ρ ( δ • ) (cid:9) for all solid diagonal δ • .The normal fan of Asso (T ◦ ) is the g -vector fan F g (T ◦ ). OLYTOPAL REALIZATIONS OF FINITE TYPE g -VECTOR FANS 17 Figure 5.
Stereographic projections of the Coxeter fan F Cox (T ◦ ), the c -vector fan F c (T ◦ ),and the g -vector fans F g (T ◦ ) for various reference triangulations T ◦ . The 3-dimensional fan isintersected with the unit sphere and stereographically projected to the plane from the pole indirection ( − , − , − (3) The T ◦ -parallelepiped Para (T ◦ ) is the parallelepiped defined as Para (T ◦ ) : = (cid:8) v ∈ R T ◦ (cid:12)(cid:12) | (cid:104) ω δ ◦ | v (cid:105) | ≤ h ρ (cid:0) ( i − • ( j − • (cid:1) for all δ ◦ = i ◦ j ◦ ∈ T ◦ (cid:9) . Its normal fan is the coordinate fan, defined by the coordinate hyperplanes.These polytopes are illustrated in Figure 6 for various initial hollow triangulations T ◦ . Note thatthe T ◦ -zonotope Zono (T ◦ ) is not simple in general. To simplify the presentation, we restrictedthe definition to the balanced exchange submodular function h ρ , but similar definitions would ofcourse hold with any exchange submodular function.These three polytopes are strongly related: not only their normal fans, but even their inequalitydescriptions, refine each other, as stated in our next proposition. This property is quite specificto type A as shown in Section 7.4. Proposition 34.
All facet defining inequalities of
Para (T ◦ ) are facet defining inequalities of Asso (T ◦ ) ,and all facet defining inequalities of Asso (T ◦ ) are facet defining inequalities of Zono (T ◦ ) .Proof. The first part of the sentence is immediate since for any initial diagonal δ ◦ = i ◦ j ◦ , we have g (cid:0) T ◦ , ( i − • ( j − • (cid:1) = ω δ ◦ , and g (cid:0) T ◦ , ( i + 1) • ( j + 1) • (cid:1) = − ω δ ◦ . For the second part of the statement, let k ( γ • ) denote the maximum of (cid:104) g (T ◦ , γ • ) | v (cid:105) over Zono (T ◦ ).As Zono (T ◦ ) is the Minkowski sum of all c -vectors, we have k ( γ • ) = (cid:88) c ∈ C (T ◦ ) (cid:104) g (T ◦ ,γ • ) | c (cid:105) > (cid:10) g (cid:0) T ◦ , γ • (cid:1) (cid:12)(cid:12) c (cid:11) . To compute this sum, recall that the g -vector g (T ◦ , γ • ) has alternating ± ◦ of γ • in T ◦ . Choose a c -vector c ∈ C (T ◦ ) and let δ • ∈ T • be such that c = c (T ◦ , δ • ∈ T • ).Since all diagonals of Z ◦ that traverse Q( δ • ∈ T • ) cross it in the same way (either all as Z orall as Z ), we have (cid:104) g (T ◦ , γ • ) | c (cid:105) ∈ {− , , } . We thus want to count the c -vectors c ∈ C (T ◦ )for which (cid:104) g (T ◦ , γ • ) | c (cid:105) >
0. It actually turns out that it is more convenient and equivalent(since C (T ◦ ) = − C (T ◦ )) to count the c -vectors c ∈ C (T ◦ ) for which (cid:104) g (T ◦ , γ • ) | c (cid:105) < ◦ into Z ◦ = Z + ◦ (cid:116) Z −◦ such that g (T ◦ , γ • ) = (cid:80) δ ◦ ∈ Z + ◦ ω δ ◦ − (cid:80) δ ◦ ∈ Z −◦ ω δ ◦ .For a diagonal δ • = u • v • , let A −◦ (resp. A + ◦ ) denote the edges of T ◦ crossed by δ • and not incidentto u ◦ + 1 or v ◦ + 1 (resp. u ◦ − v ◦ − c − ( δ • ) : = − (cid:80) δ ◦ ∈ A −◦ ω δ ◦ and c + ( δ • ) : = (cid:80) δ ◦ ∈ A + ◦ ω δ ◦ .Recall that the negative (resp. positive) c -vectors of C (T ◦ ) are then precisely given by thevectors c − ( δ • ) (resp. c + ( δ • )) for all diagonals δ • not in T −• : = { ( i − • ( j − • | ( i ◦ , j ◦ ) ∈ T ◦ } (resp. T + • : = { ( i + 1) • ( j + 1) • | ( i ◦ , j ◦ ) ∈ T ◦ } ). We leave it to the reader to check that:(i) If γ • and δ • do not cross and have no common endpoint, both | Z ◦ ∩ A −◦ | and | Z ◦ ∩ A + ◦ | areeven. Thus (cid:104) g (T ◦ , γ • ) | c − ( δ • ) (cid:105) = (cid:104) g (T ◦ , γ • ) | c + ( δ • ) (cid:105) = 0.(ii) If γ • and δ • have a common endpoint, and γ • δ • form a counterclockwise angle, then | Z ◦ ∩ A −◦ | is even while Z ◦ ∩ A + ◦ is empty or starts and ends in Z + ◦ . Thus (cid:104) g (T ◦ , γ • ) | c − ( δ • ) (cid:105) = 0 while (cid:104) g (T ◦ , γ • ) | c + ( δ • ) (cid:105) ≥
0. The situation is similar if γ • δ • form a clockwise angle.(iii) If γ • and δ • cross, Z ◦ ∩ A −◦ and Z ◦ ∩ A + ◦ are empty or start and end both in Z −◦ or bothin Z + ◦ . Thus, either (cid:104) g (T ◦ , γ • ) | c − ( δ • ) (cid:105) < (cid:104) g (T ◦ , γ • ) | c + ( δ • ) (cid:105) ≥ c -vectors c ∈ C (T ◦ ) for which (cid:104) g (T ◦ , γ • ) | c (cid:105) < δ • crossing γ • . In other words, k ( γ • ) = h ρ ( γ • ).Finally, we obtained that the inequality (cid:104) g (T ◦ , γ • ) | v (cid:105) ≤ k ( γ • ) defines a face F ( γ • ) of thezonotope Zono (T ◦ ). This face F ( γ • ) is the Minkowski sum of the c -vectors of C (T ◦ ) orthogonalto g (T ◦ , γ • ). Theorem 11 ensures that any triangulation T • containing γ • already provides n − c -vectors c (T ◦ , δ • ∈ T • ) for δ • ∈ T • (cid:114) { γ • } . We obtain that F ( γ • ) hasdimension n − Zono (T ◦ ). (cid:3) Vertex barycenter.
We now use this (type A ) interpretation of the T ◦ -associahedron Asso (T ◦ )to show that its vertex barycenter is also at the origin. Our approach is independent, and somewhatsimpler than the previous proofs of this property for type A acyclic associahedra [HLR10, LP13]. OLYTOPAL REALIZATIONS OF FINITE TYPE g -VECTOR FANS 19 Figure 6.
The zonotope
Zono (T ◦ ), associahedron Asso (T ◦ ) and parallelepiped Para (T ◦ ) for dif-ferent reference triangulations T ◦ . The first column is J.-L. Loday’s associahedron [Lod04], thesecond column is one of C. Hohlweg and C. Lange’s associahedra [HL07], the third column appearedin a discussion in C. Ceballos, F. Santos and G. Ziegler’s survey on associahedra [CSZ15, Fig. 3]. Proposition 35.
For any initial triangulation T ◦ , the origin is the vertex barycenter of the T ◦ -zonotope Zono (T ◦ ) , the T ◦ -associahedron Asso (T ◦ ) and the T ◦ -parallelepiped Para (T ◦ ) .Proof. Assume that we started from a regular (2 n + 6)-gon with alternative hollow and solidvertices. Consider a diagonal δ ◦ of T ◦ and let Ψ denote the reflexion of the plane which stabilizes δ ◦ .Note that Ψ sends solid diagonals (resp. triangulations) onto solid diagonals (resp. triangulations).Moreover, a diagonal δ • crosses δ ◦ if and only if its image Ψ( δ • ) crosses δ ◦ . Since Ψ reverses theorientation, we therefore obtain that ε (cid:0) δ • ∈ T • , δ ◦ (cid:1) = − ε (cid:0) Ψ( δ • ) ∈ Ψ(T • ) , δ ◦ (cid:1) , for any δ • ∈ T • . Finally, Ψ preserves the length of a diagonal δ • , so that h ρ (Ψ( δ • )) = h ρ ( δ • ).Summing over all diagonals in all solid triangulations, we obtain that the δ ◦ -coordinate of thevertex barycenter of Asso (T) is given by (cid:16) (cid:88) T • p (cid:0) T ◦ , T • (cid:1)(cid:17) δ ◦ = (cid:88) T • (cid:88) δ • ∈ T • (cid:0) h ρ ( δ • ) c (cid:0) T ◦ , δ • ∈ T • (cid:1)(cid:1) δ ◦ = − (cid:88) T • (cid:88) δ • ∈ T • h ρ ( δ • ) ε (cid:0) δ • ∈ T • , δ ◦ (cid:1) = 0since the contribution of δ • ∈ T • is balanced by that of Ψ( δ • ) ∈ Ψ(T • ). Since this holds forany δ ◦ ∈ T ◦ , we conclude that the vertex barycenter of Asso (T ◦ ) is the origin. It is immediate forthe other two polytopes Zono (T ◦ ) and Para (T ◦ ) since they are centrally symmetric. (cid:3) Proposition 35 will be generalized in Section 7.3 for arbitrary seeds in arbitrary finite types andfor arbitrary fairly balanced point λ .7. Further properties of
Asso (B ◦ )In this section, we discuss further geometric properties of the B ◦ -associahedron Asso h (B ◦ ),motivated by the specific families of examples presented in Section 6. We also introduce the uni-versal associahedron mentioned in Theorem 2, a high dimensional polytope which simultaneouslycontains the associahedra Asso h (B ◦ ) for all exchange matrices B ◦ of a given finite type.7.1. Green mutations.
Motivated by Section 6.1, we consider a natural orientation of muta-tions introduced by B. Keller in the context of quantum dilogarithm identities [Kel11]. For twoadjacent seeds Σ = (B , P , X) and Σ (cid:48) = (B (cid:48) , P (cid:48) , X (cid:48) ) of A pr (B ◦ ) with X (cid:114) { x } = X (cid:48) (cid:114) { x (cid:48) } , the mu-tation Σ → Σ (cid:48) is a green mutation when the dual c -vector c (B ∨◦ , x ∨ ∈ Σ ∨ ) = − c (B ∨◦ , x (cid:48)∨ ∈ Σ (cid:48)∨ ) ispositive. The directed graph G (B ◦ ) of green mutations is known to be acyclic, and even the Hassediagram of a lattice when the type of B ◦ is simply laced, see [GM17, Coro. 4.7] and the referencestherein. Further lattice theoretic properties of G (B ◦ ) are discussed in [GM17]. Example 36.
For an acyclic initial exchange matrix B ◦ , the lattice of green mutations is the c -Cambrian lattice of N. Reading [Rea06] (where c is the Coxeter element corresponding to B ◦ ).It turns out that this green mutation digraph G (B ◦ ) is apparent in the B ◦ -associahedron. Indeed,the following statement is a direct consequence of Remark 27. Proposition 37.
For any finite type exchange matrix B ◦ , the graph of the associahedron Asso h (B ◦ ) ,oriented in the linear direction − (cid:80) x ∈ X ◦ ω x , is the graph G (B ◦ ) of green mutations in A pr (B ◦ ) .Proof. Consider two adjacent seeds Σ = (B , P , X) and Σ (cid:48) = (B (cid:48) , P (cid:48) , X (cid:48) ) of A pr (B ◦ ) with X (cid:114) { x } =X (cid:48) (cid:114) { x (cid:48) } . By Remark 27, we have p h (B ◦ , Σ (cid:48) ) − p h (B ◦ , Σ) = − γ c (B ∨◦ , x ∨ ∈ Σ ∨ )for some positive γ ∈ R > . Therefore, (cid:10) − (cid:88) x ∈ X ◦ ω x (cid:12)(cid:12) p h (cid:0) B ◦ , Σ (cid:48) (cid:1) − p h (cid:0) B ◦ , Σ (cid:1) (cid:11) = γ (cid:10) (cid:88) x ∈ X ◦ ω x (cid:12)(cid:12) c (B ∨◦ , x ∨ ∈ Σ ∨ ) (cid:11) is positive if and only if c (B ∨◦ , x ∨ ∈ Σ ∨ ) is a positive c -vector. (cid:3) OLYTOPAL REALIZATIONS OF FINITE TYPE g -VECTOR FANS 21 Universal associahedron.
For each initial exchange matrix B ◦ of a given type, we con-structed in Section 5 a generalized associahedron Asso h (B ◦ ) by lifting the g -vector fan using anexchange submodular function h on the cluster variables of A pr (B ◦ ). As already observed though,the function h is independent of the coefficients of A pr (B ◦ ), so that all g -vector fans can be liftedwith the same function h . This motivates the definition of a universal associahedron.For this, consider the finite type cluster algebra A un (B ◦ ) with universal coefficients, and let X (B ◦ )denote its set of cluster variables. Consider a |X (B ◦ ) | -dimensional vector space U with ba-sis { β x } x ∈X (B ◦ ) and its dual space U ∨ with basis { β ∨ x ∨ } x ∨ ∈X (B ∨◦ ) . As before, the cluster variablesof A un (B ◦ ) and A un (B ∨◦ ) are related by x ↔ x ∨ . For X ⊆ X (B ◦ ), we denote by H X the coordinatesubspace of U spanned by { β x } x ∈ X .Given a seed Σ in A un (B ◦ ), the u -vector of a cluster variable x ∈ Σ is the vector u (B ◦ , x ∈ Σ) : = (cid:88) y ∈X (B ◦ ) u yx β y ∈ U of exponents of p x = (cid:81) y ∈X (B ◦ ) ( p [ y ]) u yx . Remark 14 then reformulates geometrically in terms of u - and c -vectors as follows. Choose a seed Σ (cid:63) = (B (cid:63) , P (cid:63) , X (cid:63) ) in A un (B ◦ ) that you want to makeinitial. Then, for any cluster variable x in a seed Σ, the c -vector c (B (cid:63) , x ∈ Σ) is the orthogonalprojection of the u -vector u (B ◦ , x ∈ Σ) on the coordinate subspace H X (cid:63) . (Here and elsewhere weidentify H X (cid:63) with V and H X ∨ (cid:63) with V ∨ in the obvious way.)We are now ready to define the universal associahedron. Definition 38.
For any finite type exchange matrix B ◦ and any exchange submodular function h ,the universal B ◦ -associahedron is the polytope Asso h un (B ◦ ) in U ∨ defined as the convex hull of thepoints p h un (cid:0) B ◦ , Σ (cid:1) : = (cid:88) x ∈ Σ h ( x ) u (cid:0) B ∨◦ , x ∨ ∈ Σ ∨ (cid:1) ∈ U ∨ for each seed Σ of A un (B ◦ ).Note that Asso h un (B ◦ ) does not depend on B ◦ but only on its cluster type. We keep B ◦ in thenotation since it fixes the indexing of the spaces U and U ∨ . Example 39.
We illustrate Definition 38 on the type C exchange matrix:B ◦ = (cid:20) − (cid:21) . The cluster algebra A un (B ◦ ) has 6 cluster variables that we denote by X (B ◦ ) = { x , x , x , x , x , x } .It is straightforward to verify that to the point 2 ρ ∨ corresponds the function h ρ ∨ with value 3on x , x , x and 4 on x , x , x . The u -vectors we need to compute our polytope are those as-sociated to the algebra A un (B ∨◦ ) that appears in Example 15. (Notationally one should think ofcluster variables in Example 15 as { x ∨ i } .) Using Figure 1 we then get that Asso h ρ ∨ un (B ◦ ) is theconvex hull of the 6 points(3 , , − , − , − , , (3 , − , − , − , , , (1 , , , − , − , − , ( − , − , − , , , , ( − , , , , − , − , ( − , − , , , , − . These are in general position so
Asso h ρ ∨ un (B ◦ ) is a 5-dimensional simplex embedded in R .Our interest in Asso h un (B ◦ ) comes from the following property. Theorem 40.
Fix a finite type exchange matrix B ◦ and an exchange submodular function h . Forany seed (B (cid:63) , P (cid:63) , X (cid:63) ) of A un (B ◦ ) , the orthogonal projection of the universal associahedron Asso h un (B ◦ ) on the coordinate subspace H X ∨ (cid:63) of U ∨ spanned by { β ∨ x ∨ } x ∨ ∈ X ∨ (cid:63) is the B (cid:63) -associahedron Asso h (B (cid:63) ) .Proof. Denote by π X ∨ (cid:63) the orthogonal projection on H X ∨ (cid:63) . We already observed that c (B ∨ (cid:63) , x ∨ ∈ Σ ∨ ) = π X ∨ (cid:63) (cid:0) u (B ∨◦ , x ∨ ∈ Σ ∨ ) (cid:1) n dimension ofambient space dimension ≤ · ≤
10 30 ≤ · ≤
424 14 13 42 8960 14 ≤ · ≤
28 3463 ≤ · ≤
Table 1.
Some statistics for the universal associahedron of type A n for n ∈ [4].for any cluster variable x in any seed Σ. It follows that p h (cid:0) B (cid:63) , Σ (cid:1) = π X ∨ (cid:63) (cid:0) p h un (cid:0) B ◦ , Σ (cid:1)(cid:1) for any seed Σ. We conclude that Asso h (B ◦ ) : = conv (cid:8) p h (cid:0) B (cid:63) , Σ (cid:1) (cid:12)(cid:12) Σ seed in A pr (B ◦ ) (cid:9) = conv (cid:8) π X ∨ (cid:63) (cid:0) p h un (cid:0) B (cid:63) , Σ (cid:1)(cid:1) (cid:12)(cid:12) Σ seed in A un (B ◦ ) (cid:9) = π X ∨ (cid:63) (cid:0) conv (cid:8) p h un (cid:0) B (cid:63) , Σ (cid:1) (cid:12)(cid:12) Σ seed in A un (B ◦ ) (cid:9) (cid:1) = : π X ∨ (cid:63) (cid:0) Asso h un (B ◦ ) (cid:1) . (cid:3) Remark 41.
Consider the normal fan F of the universal B ◦ -associahedron Asso h un (B ◦ ). Then forany seed Σ (cid:63) = (B (cid:63) , P (cid:63) , X (cid:63) ) in A un (B ◦ ), the section of F by the coordinate subspace H X (cid:63) of U spanned by { β x } x ∈ X (cid:63) is the g -vector fan F g (B (cid:63) ). We therefore call universal g -vector fan thenormal fan F g un (B ◦ ) of the universal B ◦ -associahedron Asso un (B ◦ ).We now gather some observations on the universal B ◦ -associahedron Asso h un (B ◦ ): • The universal associahedron
Asso h un (B ◦ ) is a priori defined in U ∨ . However, computerexperiments indicate that it has codimension 1. The linear space containing Asso h un (B ◦ )seems to be expressed naturally in terms of the Cartan matrix of type B ◦ and h . • As an immediate consequence of Theorem 40, we obtain that the vertices of the universalassociahedron
Asso h un (B ◦ ) are precisely the points p h un (B ◦ , Σ) for all seeds Σ of A un (B ◦ ),and that the mutation graph of the cluster algebra A un (B ◦ ) is a subgraph of the graphof Asso h un (B ◦ ). However, this inclusion is strict in general. • In general
Asso h un (B ◦ ) is neither simple nor simplicial. Table 1 presents some statistics forthe number of vertices per facet and facets per vertex in the universal associahedron oftype A n for n ∈ [4]. • Computer experiments indicate that the face lattice (and thus the f -vector) of the universalB ◦ -associahedron Asso h un (B ◦ ) is independent of h .To conclude, let us insist on the fact that Theorem 40 describes the projection of the universalassociahedron Asso h un (B ◦ ) on coordinate subspaces corresponding to clusters of A un (B ◦ ). It turnsout that the projections on coordinate subspaces corresponding to all faces (not necessarily facets)of the cluster complex create relevant simplicial complexes, fans and polytopes [Cha16, GM16,MP17]. This naturally raises the question to understand all coordinate projections of the universalassociahedron Asso h un (B ◦ ).7.3. Vertex barycenter.
As mentioned in Section 6.1, the vertex barycenters of all associahedraconstructed by C. Hohlweg, C. Lange and H. Thomas in [HLT11] coincide with the origin. Thisintriguing property observed in [HLT11, Conj. 5.1] was proved by V. Pilaud and C. Stump [PS15b].We show in this section that it also extends to all associahedra
Asso h (B ◦ ) for any initial exchangematrix B ◦ and any exchange submodular function h . In fact, it is a consequence of the followingstronger statement. Theorem 42.
For any finite type exchange matrix B ◦ and any exchange submodular function h ,the origin is the vertex barycenter of the universal B ◦ -associahedron Asso h un (B ◦ ) . OLYTOPAL REALIZATIONS OF FINITE TYPE g -VECTOR FANS 23 Our proof of this theorem relies on its validity in the acyclic case [PS15b], and on the followingobservation.
Lemma 43.
Fix a finite type exchange matrix B ◦ and an exchange submodular function h . For anyseed (B (cid:63) , P (cid:63) , X (cid:63) ) of A un (B ◦ ) , the vertex barycenter of the B (cid:63) -associahedron Asso h (B (cid:63) ) is the imageof the vertex barycenter of the universal associahedron Asso h un (B ◦ ) by the orthogonal projection onthe coordinate subspace H X ∨ (cid:63) of U ∨ spanned by { β ∨ x ∨ } x ∨ ∈ X ∨ (cid:63) .Proof. Denote by π X ∨ (cid:63) the orthogonal projection on H X ∨ (cid:63) . Since π X ∨ (cid:63) is linear, we have (cid:88) Σ p h (cid:0) B (cid:63) , Σ (cid:1) = (cid:88) Σ π X ∨ (cid:63) (cid:0) p h un (cid:0) B ◦ , Σ (cid:1)(cid:1) = π X ∨ (cid:63) (cid:16) (cid:88) Σ p h un (cid:0) B ◦ , Σ (cid:1)(cid:17) , where Σ runs over all seeds of A un (B ◦ ). The result follows since these seeds index the vertices ofboth Asso h (B (cid:63) ) and Asso h un (B ◦ ). (cid:3) Proof of Theorem 42.
Consider a cluster variable x of A un (B ◦ ). Let (B (cid:63) , P (cid:63) , X (cid:63) ) be an acyclicseed of A un (B ◦ ) containing x (such a seed exists, we could even require that it is bipartite). Sinceby [PS15b] the vertex barycenter of Asso h (B (cid:63) ) is at the origin, we obtain by Lemma 43 that the x -coordinate of the vertex barycenter of the universal associahedron Asso h un (B ◦ ) vanishes. Applyingthe same argument independently for all cluster variables x of A un (B ◦ ) concludes the proof. (cid:3) Corollary 44.
For any finite type exchange matrix B ◦ and any exchange submodular function h ,the origin is the vertex barycenter of the B ◦ -associahedron Asso h (B ◦ ) .Proof. This is an immediate consequence of Theorem 42 and Lemma 43. (cid:3)
Zonotope.
Motivated by the specific families presented in Section 6, it is natural to investi-gate whether there exists a zonotope
Zono h (B ◦ ) whose facet description contains all inequalitiesof the associahedron Asso h (B ◦ ). In this section, we show that such a zonotope does not alwaysexist in general. A first naive option.
Mimicking Section 6, the natural choice is to consider the zonotope
Zono (B ◦ ) : = (cid:80) c ∈ C (B ∨◦ ) c . Indeed, we have seen in Section 6 that all inequalities of
Asso (B ◦ ) areinequalities of Zono (B ◦ ) when B ◦ is either acyclic or of type A .However, this property already fails for the type C cyclic initial exchange matrixB ◦ = − − − . Figure 7.
The zonotope counter-example in type C . The B ◦ -associahedron Asso (B ◦ ) (left),the B ◦ -zonotope Zono (B ◦ ) (middle), and their superposition (right) for the type C cyclic initialexchange matrix B ◦ . The hyperplanes supporting the shaded facets of Asso (B ◦ ) are parallel tobut do not coincide with the hyperplanes supporting the shaded facets of Zono (B ◦ ). Figure 8.
The polytope defined by the inequalities of
Zono (B ◦ ) defining facets whose normal vec-tors are g -vectors of A pr (B ◦ ) (left), and a zonotope whose facet description contains all inequalitiesof Asso (B ◦ ) (right).One indeed checks that in the direction of the g -vectors ( − , ,
0) and (1 , − , Asso (B ◦ ) and 3 in Zono (B ◦ ). This is visible in Figure 7 where thefacets of Asso (B ◦ ) and Zono (B ◦ ) orthogonal to the g -vectors ( − , ,
0) and (1 , − ,
0) are shaded.Figure 8 (left) shows that the polytope defined by the inequalities of
Zono (B ◦ ) defining facetswhose normal vectors are g -vectors of A pr (B ◦ ) is not an associahedron of type C (it is not evena simple polytope).Note by the way that the two g -vectors ( − , ,
0) and (1 , − ,
0) are opposite (thus correspondto parallel facets of
Asso (B ◦ )). This should sound unusual as the only pairs of opposite g -vectorsin both situations of Section 6 are the pairs of opposite coordinate vectors { ω x , − ω x } for x ∈ X ◦ . General approach.
For any tuple γ : = ( γ c ) c ∈ C (B ∨◦ ) of positive coefficients, we consider the zono-tope Zono γ (B ◦ ) : = (cid:88) c ∈ C (B ∨◦ ) γ c c By definition, its normal fan is F c (B ∨◦ ). For any ray g of this fan, the inequality defining the facetnormal to g is given by (cid:104) g | v (cid:105) ≤ k ( g ) where k ( g ) : = (cid:88) c ∈ C (B ∨◦ ) (cid:104) g | c (cid:105) > γ c (cid:104) g | c (cid:105) . The facet description of the zonotope
Zono γ (B ◦ ) thus contains all inequalities of the associahe-dron Asso h (B ◦ ) if and only if γ = γ ( h ) is a positive solution to the system of linear equations (cid:88) c ∈ C (B ∨◦ ) (cid:104) g (B ◦ ,x ) | c (cid:105) > γ c (cid:10) g (cid:0) B ◦ , x (cid:1) (cid:12)(cid:12) c (cid:11) = h ( x ) for all cluster variable x of A pr (B ◦ ).For example, such a solution exists for the type C cyclic initial exchange matrix, as illus-trated in Figure 8 (right). Note that we had to pick different coefficients for different elements of C (B ∨◦ ) (leading to unnaturally narrow faces) since we already observed in Figure 7 that constantcoefficients are not suitable.In contrast, a quick computer experiment shows that this system has no solution for the type D cyclic initial exchange matrix B ◦ = − − − − − OLYTOPAL REALIZATIONS OF FINITE TYPE g -VECTOR FANS 25 ◦ ◦ ◦ ◦ ◦ • • • • • ◦ ◦ ◦ ◦ ◦ • • • • • Figure 9.
An obstruction to the existence of a centrally symmetric zonotope whose facet de-scription would contain that of the associahedron
Asso (B ◦ ) for a cyclic initial seed B ◦ in type D .The two arcs γ (cid:96) (left) and γ r (right) have opposite g -vectors but different compatibility sums.and the balanced exchange submodular function h ρ ∨ . Rather than showing the detailed linearsystem, we prefer to convince the reader that there is no solution under the natural assumptionthat γ c = γ − c for any c ∈ C (B ∨◦ ), i.e., that the zonotope Zono γ (B ◦ ) is centrally symmetric. Tosee that no such solution exists, consider the classical punctured pentagon model for the type D cluster algebra [FST08]. Recall that(i) cluster variables correspond to internal arcs up to isotopy (with the subtlety that an arcincident to the puncture can be tagged or not),(ii) g -vectors can be read by shear coordinates [FT12] in a similar way as in Section 6.2, and(iii) the compatibility degree between two arcs is the (minimal) number of crossings between them.We therefore obtain that the arcs γ (cid:96) connecting 1 • to 3 • in Figure 9 (left) and the arc γ r connect-ing 4 • to 5 • in Figure 9 (right) satisfy: g (cid:0) B ◦ , x γ (cid:96) (cid:1) = ( − , , , ,
0) and h ρ ∨ ( x γ (cid:96) ) = 7 , g (cid:0) B ◦ , x γ r (cid:1) = (1 , , − , ,
0) and h ρ ∨ ( x γ (cid:96) ) = 9 . In other words, we exhibited two cluster variables with opposite g -vectors while belonging todistinct (cid:104) τ + , τ − (cid:105) -orbits. Since a centrally symmetric zonotope would have the same right-hand-sides on opposite normal vectors, this shows that no such polytope exists. Aknowledgements
We thank N. Reading for helpful discussions on this topic and comments on a preliminary versionof this paper. We are also grateful to A. Garver and T. McConville for bibliographic inputs. Thiswork answers one of the questions A. Zelevinsky posed to the third author as possible Ph.D.problems. He would therefore like to express once again gratitude to his advisor for the guidancereceived.
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E-mail address : [email protected] URL ::