Minkowski decompositions for generalized associahedra of acyclic type

We give an explicit subword complex description of the generators of the type cone of the g-vector fan of a finite type cluster algebra with acyclic initial seed. This yields in particular a description of the Newton polytopes of the F-polynomials in terms of subword complexes as conjectured by S. Brodsky and the third author. We then show that the cluster complex is combinatorially isomorphic to the totally positive part of the tropicalization of the cluster variety as conjectured by D. Speyer and L. Williams.


Introduction and main results
A generalized associahedron for a cluster algebra of finite type is a simple polytope whose face lattice is dual to the cluster complex. Constructing such generalized associahedra has been a fruitful area of mathematical research since the introduction of cluster algebras by S. Fomin and A. Zelevinsky in the early 2000s. We refer to [FZ02,CFZ02,HLT11,PS15,HPS18] in this chronological order for some of the milestones and history. This paper is a continuation of [BS18] and builds on recent results from [BMDM + 18, AHL20] and from [PPPP19].
The paper has three major results, two of which resolve conjectures by S. Brodsky and the third author and, respectively, by D. Speyer and L. Williams. Theorem 1.1 gives a self-contained combinatorial construction of the rays of the type cone of the g-vector fan of a finite type cluster algebra with acyclic initial seed via subword complexes and brick polytopes. Using this construction together with recent results from [BMDM + 18, AHL20] and [PPPP19], Theorem 1.3 yields that this construction also describes the Newton polytopes of the F -polynomials of the cluster algebra. This description was conjectured in [BS18, Conjecture 2.12]. The appearance of the F -polynomials is then as well used to derive Theorem 1.4 showing that the totally positive part of the tropical cluster variety is, modulo its lineality space, linearly isomorphic to the g-vector fan. As the g-vector fan is combinatorially isomorphic to the cluster complex, this affirmatively answers [SW05, Conjecture 8.1] for finite type cluster algebras with principal coefficients and acyclic initial seed.
In order to precisely state the results, let ∆ ⊆ Φ + ⊆ Φ ≥−1 ⊆ Φ denote a finite crystallographic root system and let M denote an initial mutation matrix with principal coefficients for a cluster algebra A(M) of type Φ with cluster complex S(M). Its F -polynomials are denoted by F β β ∈ Φ + and its g-vector fan F g (M) is given by the cones over compatible sets of g-vectors.
It is well-known that the g-vector fan is combinatorially isomorphic to the cluster complex S(M). Let (W, S) denote the Coxeter system generated by S = {s α | α ∈ ∆} and let c ∈ W be a standard Coxeter element given by the product of the reflections in S in some order. It is well-known how to associate to this data an acyclic initial mutation matrix M c with principal coefficients, and as well a brick polytope Asso (M c ) with normal fan given by the g-vector fan F g (M c ) of A(M c ). In particular, Asso (M c ) is a generalized associahedron for A(M c ). Brick polytopes for subword complexes come with natural Minkowski decompositions which in the present context may be written in the form The type cone TC(F g (M c )) of the g-vector fan is the space of all its polytopal realizations. We thus have While motivated by beautiful constructions in [BMDM + 18] and [PPPP19], the following result is entirely self-contained and only uses properties of brick polytopes developed in [PS15] and [BS18].
Theorem 1.1. For an acyclic initial mutation matrix M c with principal coefficients, the type cone of the g-vector fan F g (M c ) is the open simplicial cone generated by the natural Minkowski summands of the brick polytope Asso (M c ), Remark 1.2. The definition of a generalized associahedron Asso (M c ) in [HLT11,PS15] extends verbatim to the noncrystallographic finite types I 2 (m) for m / ∈ {3, 4, 6} and H 3 , H 4 . This theorem also holds for noncrystallographic types when replacing the left-hand side by the type cone of weak Minkowski summands of Asso (M c ) even though mutations of cluster variables, g-vectors and F -polynomials in these types do not behave combinatorially nicely [Lam18].
Combining [BMDM + 18, Theorem 3] (simply-laced types) and [AHL20, Theorem 6.1] (multiplylaced types) with [PPPP19, Theorem 2.26], one obtains that the rays of the type cone of the g-vector fan are also equal to the Newton polytopes of the F -polynomials, and in particular that is a generalized associahedron for A(M c ). According to [PPPP19], H. Thomas announced that a future version of [BMDM + 18] will generalize (⋆) also to cyclic finite types. In this case, (⋆⋆) was conjectured by S. Brodsky  . Let M c be an acyclic initial mutation matrix with principal coefficients. For any positive root β ∈ Φ + , we have In [SW05] the authors associate to the cluster algebra A(M) a polyhedral fan Trop + Spec A(M) by tropicalizing the positive part of the affine variety Spec A(M). Using (⋆⋆), we finally derive the following theorem.
Theorem 1.4. For acyclic initial mutation matrix M c with principal coefficients, the totally positive part of the tropical variety associated to the cluster algebra A(M c ) is, modulo its lineality space L, linearly isomorphic to the g-vector fan, As the g-vector fan is combinatorially isomorphic to the cluster complex, this affirmatively answers a conjecture by D. Speyer and L. Williams for cluster algebras of acyclic finite type with principal coefficients. 1.1. Acknowledgements. The third author would like to thank Arnau Padrol, Markus Reineke, Raman Sanyal and Hugh Thomas for valuable discussions concerning various parts of this paper.

A natural Minkowski decomposition of generalized associahedra
We follow the notions from [BS18] and refer to Section 2 therein for details.
2.1. Generalized associahedra for acyclic type. Let (W, S) be a finite type Coxeter system of rank n and let ∆ ⊆ Φ + ⊆ Φ ≥−1 ⊆ Φ ⊆ V be a finite root system for (W, S) inside an Euclidean vector space V , with simple roots ∆ = {α s | s ∈ S}, positive roots Φ + and almost positive roots Φ ≥−1 = Φ + ⊔ −∆. Let ∇ = {ω s | s ∈ S} ⊆ V be the set of fundamental weights. Fix a Coxeter element c ∈ W and a reduced word c = s 1 · · · s n for c. To avoid double indices we write α i for α si and ω i = ω si . Furthermore we denote by N = |Φ + | the number of positive roots and n + N = |Φ ≥−1 |. In all below considerations we consider V ∼ = R ∆ to have fixed basis ∆, though in the examples we simultaneously consider the vector space with standard basis and standard inner product.
Let w • ∈ W be the unique longest element in weak order. For a given word Q = q 1 · · · q m in the simple system S define the (spherical) subword complex SC(Q) as the simplicial complex of (positions of) letters in Q whose complement contains a reduced word of w • . A more general version of these complexes were introduced by A. Knutson and E. Miller in [KM04]. By definition, the facets of SC(Q) are subwords of Q whose complements are reduced words for w • . We consider facets as sorted lists of indices, written in set notation. Moreover define I g and I ag to be the lexicographically first and last facets, respectively, and call them greedy facet and antigreedy facet. The following notions were introduced and studied for general subword complexes in [CLS14,PS15]. For Q = q 1 · · · q m and any facet I ∈ SC(Q) associate a root function where ΠQ X denotes the product of the simple reflections q x ∈ Q, for x ∈ X ⊆ [m], in the order given by Q. It is well known, see [KM04,Theorem 3.7], that SC(Q) is a simplicial sphere, thus for a given facet I and index i ∈ I there exists a unique adjacent facet J with I \ i = J \ j. We call the transition from I to J the flip of i in I and if i < j such a flip is called increasing, in which case we write I ≺ J. This yields a poset structure on the set of facets of SC(Q) with I g as unique minimal element and I ag as unique maximal element.
Following [CLS14], the (abstract) cluster complex S(M c ) can be seen as a subword complex as follows. Denote by w • (c) the Coxeter-sorting word of w • , i.e., the lexicographically first subword of c N that is a reduced word for w • . The notion of Coxeter-sorting words was introduced by N. Reading in [Rea07] and is an essential ingredient in the combinatorial descriptions of finite type cluster algebras and, in particular, in the description of cluster complexes in terms of subword complexes. In this setting we get the cluster complex as Furthermore the initial mutation matrix with principal coefficients M c is given by Furthermore the initial mutation matrix with principal coefficients M c is given by It was developed in [PS15] how one may obtain a generalized associahedron using subword complexes and brick polytopes. Define the brick vector of the facet I of SC cw • (c) as b(I) = N k=1 w(I, n + k) − w(I ag , n + k) ∈ V, (2.2) and the brick polytope Asso (M c ) in V as the convex hull of all brick vectors of SC cw • (c) , that is, . It was shown in [PS15, Corollary 6.10] that Asso (M c ) is a generalized associahedron. Furthermore, for example by combining [PS15, Proposition 6.6] and [BS18, Corollary 2.10], the normal fan of Asso (M c ) is the g-vector fan of the cluster algebra A(M c ). This definition of the brick polytope differs from the definition given in [PS15] by a translation and is chosen so that the brick vector b(I ag ) of the antigreedy facet is the origin. This translation corresponds to the shifted weight function as used in [BS18, Conjecture 2.12]. Furthermore, we have for any facet I of SC cw • (c) that w(I, k) = w(I ag , k) for all 1 ≤ k ≤ n. This clarifies why we do not consider the first n weight vectors in the summation in (2.2).
The root function of the greedy facet provides a bijection between the set of positive roots and the positions n + 1, . . . , n + N . That is, Lemma 3.7], we moreover have r(I g , n + k) = w(I g , n + k) − w(I ag , n + k) for all 1 ≤ k ≤ N . For β = r(I g , n + k) ∈ Φ + and a facet I, we sometimes write w(I, β) := w(I, n + k) for simplicity, and define Remark 2.5. This identification of the positions n + 1, . . . , n + N and Φ + is the same as the isomorphism in (2.1) in the following sense. As known since [FZ03], sending a cluster variable to its d-vector-this is the exponent vector of its denominator monomial-is a bijection between cluster variables and almost positive roots Φ ≥−1 written as vectors in the basis ∆. Identifying the positions 1, . . . , n with the simple negative roots −α 1 , . . . , −α n in this order and the above identification between positions n + 1, . . . , n + N and Φ + is a bijection between cluster variables and positions 1, . . . , n, n + 1, . . . , n + N and this bijection induces the bijection used in (2.1).
In particular, the polytope Asso β (M c ) naturally correspond to the cluster variable with d-vector β ∈ Φ + . This correspondence turns out to be a structural correspondence as discussed in Section 3 where we show that Asso β (M c ) is the Newton polytope of the F -polynomial associated to this cluster variable.
We state the following mild generalization of [PS15, Proposition 5.17] for the present context. The proof given there also applies in the present generality and indeed for all root independent subword complexes as briefly defined in Section 2.2.1 below.
Proposition 2.9. We have the Minkowski decomposition Proof. We may neglect the contributions of the shifts by w(I ag , ·), as these cancel in all considerations. By definition we have To obtain equality we show that every vertex of β∈X Asso β (M c ) is also a vertex of Asso X (M c ).
Consider a linear functional f : V → R. For two adjacent facets I \ i = J \ j of SC cw • (c) and a positive root β ∈ X we have by Lemma 2.8 that either f (w(I, β)) = f (w(J, β)) or f (w(I, β)) − f (w(J, β)) has the same sign as f (b X (I)) − f (b X (J)). Therefore a facet I f maximizes f (b X (·)) among all facets if and only if it maximizes f (w(·, β)) for every β ∈ X. Let now v be a vertex of the Minkowski sum β∈X Asso β (M c ) and let f : V → R be a linear functional maximized at v. Thus, v = β∈X v β such that v β maximizes f for Asso β (M c ).
On the other hand, f is also maximized by some vertex b X (I f ) of Asso X (M c ). By the previous consideration, f thus maximizes w(I f , β) for every β ∈ X and we obtain v β = w(I f , β). Hence v = β∈X w(I f , β) = b X (I f ).
The description of the Minkowski decomposition of the brick polytope in the previous proposition also yields the following corollary. For later reference we note that r(I g , 6) = 111 ∆ and For later reference we note that b X ({34}) = b X ({45}) = 01 ∆ and 12 ∆ = r(I g , 5).
We next introduce the following canonical long flip sequence in the subword complex SC cw • (c) from the greedy to the antigreedy facet, I g = I 0 ≺ I 1 ≺ · · · ≺ I N = I ag where I ℓ+1 is obtained from I ℓ by flipping the unique index i in I ℓ such that I ℓ+1 \ {ℓ + 1 + n} = I ℓ \ {i}. Indeed, up to commutation of consecutive commuting letters, the index i is the smallest index that yields an increasing flip. Indeed, there is some flexibility in defining this sequence-any sequence of flips corresponding to source mutations in the associated cluster algebra would work. Example 2.14 (B 2 -example). For cw • (c) = 121212 the canonical long flip sequence is given by This flip sequence already appeared in [PS15, Proposition 6.7] and in the proof of [BS18, Lemma 3.7], where in particular the following property was used.
Lemma 2.15. For every index j ∈ {n + 1, . . . , n + N } there exists a unique pair I ℓ ≺ I ℓ+1 in the canonical long flip sequence and an index i such that I ℓ \ i = I ℓ+1 \ j. Moreover, in this case the weight function w(I ℓ+1 , ·) is obtained from w(I ℓ , ·) by In particular, w(I ℓ , ·) and w(I ℓ+1 , ·) only differ for the index j.
This lemma yields an interesting combinatorial property of the polytopes Asso β (M c ) that we do not use further below.
Corollary 2.16. For every β ∈ Φ + the segment connecting 0 and β is an edge of Asso β (M c ).
Proof. As the brick polytope Asso (M c ) realizes SC cw • (c) its edges are in one-to-one correspondence to flips in SC cw • (c) . Combining Lemma 2.8 and Proposition 2.9 we obtain a similar result for Asso β (M c ) saying its edges are in one-to-one correspondence with flips that change the weight function w(·, β). Applying Lemma 2.15 to the canonical long flip sequence we obtain for β = r(I g , n + i) that w(I g , β) = w(I 1 , β) = . . . = w(I i−1 , β), and w(I i , β) = . . . = w(I N −1 , β) = w(I ag , β).

2.2.
Generators of the type cone. The following definitions mostly follow [PPPP19]. Let F be an essential complete simplicial fan in R d . A polytopal realization of F is a convex polytope in R d whose outer normal fan agrees with F . The space of all polytopal realizations of F is called the type cone of F , denoted by TC(F ), see also [McM73]. A parametrization of TC(F ) is commonly described as follows. Denote by G ∈ R m×d the matrix whose rows generate the rays of F . Each height vector h ∈ R m defines a polytope

Now the type cone of F can be parametrized as the open polyhedral cone
We write P h ∈ TC(F ) by identifying a polytope P h with its height vector h ∈ R m . With this definition, TC(F ) has d-dimensional lineality space corresponding to translations in R d . More specifically, for P h ∈ TC(F ) and a translation vector b ∈ R d we have Thus the lineality space of TC(F ) is given by the image of the matrix G. We identify TC(F ) with its pointed quotient TC(F )/GR d . The closure TC(F ) is called the closed type cone. The faces of TC(F ) correspond to (weak) Minkowki summands of P with the same normal fan (which are coarsenings of F ). In particular, the (extremal) generators of TC(P ) correspond to the indecomposable Minkowski summands of P .
We aim at the description of the type cone TC(F g (M c )) of the g-vector fan F g (M c ) given in Theorem 1.1. We first state the following lemma which we then use to understand the rays of the type cone.
Lemma 2.17. Let C ⊂ R m be a full-dimensional closed polyhedral cone and let x = x 1 + · · · + x m for x 1 , . . . , x m ∈ C with (i) x is an interior point of C and (ii) x − x i is contained in the boundary of C for every i ∈ {1, . . . , m}.
Proof. Write X = {x 1 , . . . , x m }. We first show that X is linearly independent. Assuming the contrary, one may express some x i in terms of X \ {x i }. This would mean that x = (x − x i ) + x i would be in the linear span of X \ {x i }. By condition (i), this point is in the interior of C, while it is on the boundary by condition (ii)-a contradiction. It follows that cone(X) is a simplicial full-dimensional cone inside C. As condition (ii) implies that its boundary is also contained in the boundary of C, we conclude the statement.
Proof of Theorem 1.1. The g-vector fan F g (M c ) is an essential complete simplicial fan in R n with n + N rays. Therefore, after passing to the quotient by its n-dimensional lineality space, the closed type cone TC(F g (M c )) is an N -dimensional pointed polyhedral cone. We aim at applying Lemma 2.17 using the points {Asso β (M c ) | β ∈ Φ + }. We have seen in (1.2) that is an interior point of TC(F g (M c )). Therefore, it suffices to show that for each γ ∈ Φ + the polytope Asso Φ + \{γ} (M c ) is contained in the boundary of TC(F g (M c )).
Let γ ∈ Φ + and let j ∈ {n + 1, . . . , n + N } be the unique index such that r(I g , j) = γ. Lemma 2.15 ensures the existence of a unique index ℓ such that j is contained in I ℓ+1 but not in I ℓ in the canonical long flip sequence I 0 ≺ · · · ≺ I N . Since w(I ℓ , ·) and w(I ℓ+1 , ·) only differ for the index j, it follows that b Φ + \{γ} (I ℓ ) = b Φ + \{γ} (I ℓ+1 ). Proposition 2.9 and the second part of Lemma 2.8 now show that the number of vertices of Asso Φ + \{γ} (M c ) is strictly less than the number of vertices of Asso (M c ). This means that it is a proper weak Minkowski summand and it is thus not contained in the interior of TC(F g (M c )). Invoking Lemma 2.17 yields the proposed statement and that the type cone is in particular simplicial.
2.2.1. Generators of the type cone for general spherical subword complexes. We close this section with a brief discussion of properties of type cones for examples of general subword complexes. It turns out that the situation for cluster complexes is particularly special. Most importantly, the conclusion of Lemma 2.15 does not hold in general for spherical subword complexes.
The complex SC cw • (c) is known to have the following properties. For a word Q, we call a spherical subword complex SC(Q) root-independent if the multiset R(I) = r(I, i) i ∈ I is linearly independent for any (and thus every) facet I and it is of full support if every position in Q is contained in some facet (meaning that all elements of the ground set are indeed vertices). Observe that spherical subword complexes of full support are also full-dimensional, meaning that R(I) generates V for any facet I. This is an immediate consequence of [PS15, Proposition 3.8]. We conjecture that these properties identify cluster complexes among spherical subword complexes.
Conjecture 2.18. Let Q be a word in S. The following statements are equivalent: (1) Up to commutations of consecutive commuting letters Q = cw • (c) for some Coxeter element c.
(2) SC(Q) is root-independent and of full support.
Remark that the first property was shown to be equivalent to the so-called SIN-property in [CLS14, Theorem 2.7]. Furthermore they conjecture these subword complexes to maximize the number of facets a subword complex SC(Q) with |Q| = n + N can have [CLS14, Conjecture 9.8].
In particular, the type cone of the brick polytope is not simplicial.
where P ? = conv{00 ∆ , 11 ∆ } is the missing generator of the type cone.

Newton polytopes of F -polynomials
Let A(M c ) be the finite type cluster algebra with acyclic initial mutation matrix M c with principal coefficients and denote by F g (M c ) its g-vector fan. We have seen in Theorem 1.1 that the type cone TC(F g (M c )) is generated by the natural Minkowski summands of the brick polytope Asso (M c ), A description of the generators of TC F g (M c ) was also obtained by combining results from [BMDM + 18, AHL20] and [PPPP19] as follows. In [BMDM + 18, Theorem 1] the authors provide polytopal realizations of F g (M c ). This construction produces a generalized associahedron X p for each p ∈ R Φ + >0 . It was then shown in [BMDM + 18, Theorem 3] (simply-laced types) and in [AHL20, Theorem 6.1] (multiply-laced types) that X p for p = e β and β ∈ Φ + equals the Newton polytope of the F -polynomial F β . In [PPPP19, Theorem 2.26], the authors explain that within the latter constuction R Φ + >0 can be understood as (a linear transformation of) the type cone TC F g (M c ) . In particular, this establishes the fact that the Newton polytopes of the F -polynomials generate the type cone, In order to prove Theorem 1.3, it remains to properly identify which Newton polytope of an F -polynomial corresponds to which Minkowski summand of the brick polytope. This is done using the following property of F -polynomials.
Now we are ready to proof our second main result.
Proof of Theorem 1.3. Since TC F g (M c ) is a simplicial cone of dimension N = |Φ + | we already know that the two sets of generators, are non-redundant and coincide up to scalar factors. Let β ∈ Φ + . By Proposition 3.1 the unique maximal and minimal vertices of Newton (F β ) are β and 0, respectively. Since β = b {β} (I g ) and 0 = b {β} (I ag ), these vectors are by Proposition 2.9 vertices of Asso β (M c ) as well. Applying Lemma 2.8 we see that they are the maximal and minimal vertices of Asso β (M c ), respectively. Thus the polytopes Newton (F β ) and Asso β (M c ) coincide.

The tropical positive cluster variety
In this section, we prove Theorem 1.4 starting from the type cone description (⋆⋆) on page 2 in terms of Newton polytopes of F -polynomials. It is independent of the subword complex description and does not make use of it.
Following [SW05], we start with the needed notions from tropical geometry. Let E ⊂ Z d ≥0 be non-empty and finite and let f = e∈E f e u e ∈ Q[u] with f e = 0 for all e ∈ E be a rational polynomial supported on E. For each weight w ∈ R d we define E(w) = arg max e∈E (w · e) .
That is, E(w) is the intersection of E with the face of Newton (f ) = conv(E) that is maximized in direction w. The tropical hypersurface Trop(f ) ⊂ R d is the collection of those weights w ∈ R d for which E(w) contains at least two elements. Trop(f ) naturally carries the structure of a polyhedral fan, whose cones are formed by those weights w ∈ Trop(f ) that yield the same E(w). This fan thus agrees with the codimension-one skeleton of the normal fan of Newton (f ).
The positive part Trop + (f ) of the tropical hypersurface was introduced in [SW05] and is defined as follows. Split E = E + f ⊔ E − f according to the signs of the coefficients of f . That is, Trop + (f ) is defined as the subfan of Trop(f ) consisting of those weights for which neither E(w) ∩ E + f nor E(w) ∩ E − f is empty,  Example 4.2 (B 2 -example). We continue Example 3.3 with initial mutation matrix M c from Example 2.4 on page 5. We denote by X ∆ = {x 1 , x 2 } the initial cluster variables and by Y = {y 1 , y 2 } the principle coefficient variables. This yields the non-initial cluster variables x 10∆ = x 2 2 + y 1 x 1 x 11∆ = x 1 y 1 y 2 + x 2 2 + y 1 x 1 x 2 x 12∆ = x 2 1 y 1 y 2 2 + 2x 1 y 1 y 2 + x 2 2 + y 1 x 1 x 2 2 x 01∆ = x 1 y 2 + 1 x 2 .
The domains of linearity of Trop Ψ define a complete four-dimensional polyhedral fan F Ψ in R X∆⊔Y = R 4 with two-dimensional lineality space. The coordinate projection R X∆⊔Y → R Y transforms F Ψ to the 2-dimensional g-vector fan F g (M c ). By Theorem 1.3, the latter is the common coarsening of the four normal fans of the Newton polytopes of the F -polynomials F 10∆ , F 11∆ , F 12∆ , F 01∆ from Example 3.3, see Figure 1.