Combinatorics of X -variables in finite type cluster algebras
CCombinatorics of X -variables in Finite Type Cluster Algebras Melissa Sherman-BennettFebruary 26, 2019
Abstract
We compute the number of X -variables (also called coefficients) of a cluster algebra offinite type when the underlying semifield is the universal semifield. For classical types, thesenumbers arise from a bijection between coefficients and quadrilaterals (with a choice of diagonal)appearing in triangulations of certain marked surfaces. We conjecture that similar resultshold for cluster algebras from arbitrary marked surfaces, and obtain corollaries regarding thestructure of finite type cluster algebras of geometric type. Cluster algebras were introduced by Fomin and Zelevinsky in the early 2000s [6], with the intent ofestablishing a general algebraic structure for studying dual canonical bases of semisimple groups andtotal positivity. A cluster algebra, or equivalently its seed pattern, is determined by an initial setof cluster variables (which we call A -variables) and coefficients (which we call X -variables), alongwith some additional data. As the terminology suggests, in the original definitions, A -variableswere the main focus. This is reflected in much of the research on cluster algebras to date, whichfocuses largely on A -variables and their dynamics. However, X -variables are important in totalpositivity, and X -variables over the universal semifield have recently appeared in the context ofscattering amplitudes in N = 4 Super Yang-Mills theory [9]. Moreover, in the setting of clustervarieties, introduced by Fock and Goncharov [3], the A - and X -varieties (associated with A - and X -variables, respectively) are on equal footing. Fock and Goncharov conjectured that a dualityholds between the two varieties [3, Conjecture 4.3], which was later shown to be true under fairlygeneral assumptions [10]. This duality suggests that studying X -variables could be fruitful both inits own right and in furthering our understanding of cluster algebras.A study along these lines was undertaken by Speyer and Thomas in the case of acyclic cluster al-gebras with principal coefficients (that is, the semifield P is the tropical semifield P =Trop( t , . . . , t k )and the X -variables of the initial cluster are ( t , . . . , t k )) [14] . Using methods from quiver repre-sentation theory, they found that the X -variables are in bijection with roots of an associated rootsystem and give a combinatorial description of which roots can appear in the same X -cluster. Sevenfound that in this context, mutation of X -seeds roughly corresponds to reflection across hyperplanesorthogonal to roots [13]. However, the above results do not address X -variables over the universalsemifield, and the numerology in the case of principal coefficients is quite different from what weobtain here. Our proofs are also completely combinatorial.We investigate the combinatorics of X -variables for seed patterns of finite type, particularly inthe case when the underlying semifield is the universal semifield. The combinatorics of A -variables1 a r X i v : . [ m a t h . C O ] F e b or finite type seed patterns is particularly rich, with connections to finite root systems [7] andtriangulations of certain marked surfaces [5, Chapter 5]. Parker [11] conjectures, and Scherlis [12]gives a partial proof, that in type A , X -variables over the universal semifield are in bijection withthe quadrilaterals of these triangulations. We generalize and prove this statement for all classicaltypes ( ABCD ). Theorem 1.1.
Let S be an X -seed pattern of classical type Z n over the universal semifield suchthat one X -cluster consists of algebraically independent elements. Let P be the marked polygonassociated to type Z n . Then the X -variables of S are in bijection with the quadrilaterals (with achoice of diagonal) appearing in triangulations of P . We also obtain the following corollary, which follows from Theorem 1.1 in classical types andwas verified by computer in exceptional types.
Corollary 1.2.
Let R be a finite type A -seed pattern. There is a bijection between ordered pairs ofexchangeable A -variables in R and X ( S sf ) . As another corollary, we compute the number of X -variables in a classical type X -seed pattern S sf over the universal semifield. We also compute the number |X ( S sf ) | of X -variables over theuniversal semifield for exceptional types using a computer algebra system. Note that all other X -seed patterns with the same exchange matrices have at most as many X -variables as S sf . Thenumbers |X ( S sf ) | are listed in the second row of the following table (the numbers for A n for n ≤ D , and E were also computed in [11]). For comparison, the third row gives the number of X -variables in a finite type X -seed pattern S pc with principal coefficients, a corollary of the results in[14]. Type A n B n , C n D n E E E F G |X ( S sf ) | (cid:0) n +34 (cid:1) n ( n + 1)( n + 2) n ( n − n + 4 n −
6) 770 2100 6240 196 16 |X ( S pc ) | n ( n + 1) 2 n n ( n −
1) 72 126 240 48 12
We largely follow the conventions of [8].
We begin by fixing a semifield ( P , · , ⊕ ), a multiplicative abelian group ( P , · ) equipped with an(auxiliary) addition ⊕ , a binary operation which is associative, commutative, and distributive withrespect to multiplication. Example 2.1.
Let t , . . . , t k be algebraically independent over Q . The universal semifield Q sf ( t , . . . , t k )is the set of all rational functions in t , . . . , t k that can be written as subtraction-free expressions in t , . . . , t k . This is a semifield with respect to the usual multiplication and addition of rational ex-pressions. Note that any (subtraction-free) identity in Q sf ( t , . . . , t k ) holds in an arbitrary semifieldfor any elements u , . . . , u k [1, Lemma 2.1.6]. Example 2.2.
The tropical semifield
Trop( t , . . . , t k ) is the free multiplicative group generated by t , . . . , t k , with auxiliary addition defined by 2 (cid:89) i =1 t a i i ⊕ k (cid:89) i =1 t b i i = k (cid:89) i =1 t min( a i ,b i ) i . Let QP denote the field of fractions of the group ring ZP . We fix an ambient field F , isomorphicto QP ( t , . . . , t n ). Definition 2.3. A labeled X -seed in P is a pair ( x , B ) where x = ( x , . . . , x n ) is a tuple of elementsin P and B = ( b ij ) is a skew-symmetrizable n × n integer matrix, that is there exists a diagonalinteger matrix D with positive diagonal entries such that DB is skew-symmetric.A labeled A -seed in F is a triple ( a , x , B ) where ( x , B ) is a labeled X -seed in P and a =( a , . . . , a n ) is a tuple of elements of F which are algebraically independent over QP and generate F . We call x the (labeled) X -cluster , a the (labeled) A -cluster , and B the exchange matrix of thelabeled seed ( a , x , B ).The elements of an X - (respectively A -)cluster are called X - (respectively A -)variables. In thelanguage of Fomin and Zelevinsky, the X -cluster is the coefficient tuple, the A -cluster is the cluster,and the X - and A -variables are coefficients and cluster variables, respectively. The notation here ischosen to parallel Fock and Goncharov’s A - and X - cluster varieties. Note that an X -seed consistsonly of an exchange matrix and an X -cluster, but an A -seed consists of an exchange matrix, an A -cluster and an X -cluster. For simplicity, we use “cluster”, “seed”, etc. without a prefix when astatement holds regardless of prefix.One moves from labeled seed to labeled seed by a process called mutation . Definition 2.4 ([8, Definition 2.4]) . Let ( a , x , B ) be a labeled A -seed in F . The A -seed mutation in direction k , denoted µ k , takes ( a , x , B ) to the labeled A -seed ( a (cid:48) , x (cid:48) , B (cid:48) ) where • The entries b (cid:48) ij of B (cid:48) are given by b (cid:48) ij = − b ij if i = k or j = kb ij + b ik | b kj | if b ik b kj > b ij else . (2.1) • The A -cluster a (cid:48) = ( a (cid:48) , . . . , a (cid:48) n ) is obtained from a by replacing the k th entry a k with anelement a (cid:48) k ∈ F satisfying the exchange relation a (cid:48) k a k = x k (cid:81) b ik > a b ik i + (cid:81) b ik < a − b ik i x k ⊕ . (2.2) • The X -cluster x (cid:48) = ( x (cid:48) , . . . , x (cid:48) n ) is given by x (cid:48) j = (cid:40) x − j if j = kx j ( x sgn ( − b kj ) k ⊕ − b kj else (2.3)where sgn ( x ) = 0 for x = 0 and sgn ( x ) = | x | /x otherwise.3imilarly, the X -seed mutation µ k in direction k takes the labeled X -seed ( x , B ) in P to the X -seed ( x (cid:48) , B (cid:48) ) in P and matrix mutation takes B to B (cid:48) . Two skew-symmetrizable integer matricesare mutation equivalent if some sequence of matrix mutations takes one to the other.Note that µ k ( a , x , B ) is indeed another labeled A -seed, as B (cid:48) is skew-symmetrizable and a (cid:48) again consists of algebraically independent elements generating F . One can check that µ k is aninvolution. We organize all seeds obtainable from each other by a sequence of mutations in a seed pattern . Let T n denote the (infinite) n -regular tree with edges labeled with 1 , . . . , n so that no vertex is in twoedges with the same label. Definition 2.5.
A rank n A -seed pattern (respectively, X -seed pattern ) S is an assignment oflabeled A -seeds (respectively X -seeds) Σ t to the vertices t of T n so that if t and t (cid:48) are connectedby an edge labeled k , then Σ t = µ k (Σ t (cid:48) ).Since mutation is involutive, a seed pattern S is completely determined by the choice of a singleseed Σ; we write S (Σ) for the seed pattern containing Σ. Note that in the language of Fomin andZelevinsky, an A -seed pattern is a “seed pattern” and an X -seed pattern is a “ Y -pattern”.Given an A -seed pattern S ( a , x , B ) , one can obtain two X -seed patterns. The first is S| X := S ( x , B ), the X -seed pattern in P obtained by simply ignoring the A -clusters of every seed. Thesecond is an X -seed pattern in F whose construction is outlined in the following proposition. Proposition 2.6 ([8, Proposition 3.9]) . Let S = { Σ t } t ∈ T n be an A -seed pattern in F . For a seed Σ t = ( a , x , B ) of S with a = ( a , . . . , a n ) , x = ( x , . . . , x n ) , and B = ( b ij ) let ˆΣ t = (ˆ x , B ) where ˆ x = (ˆ x , . . . , ˆ x n ) is the n -tuple of elements of F given by ˆ x j = x j (cid:89) i a b ij i . Then ˆ S := { ˆΣ t } t ∈ T n is an X -seed pattern in F . In other words, if µ k (Σ t ) = Σ t (cid:48) , then µ k ( ˆΣ t ) =ˆΣ t (cid:48) . One can think of ˆ S as recording the “exchange information” of S ( a , x , B ); indeed, the X -variables of ˆ S are rational expressions whose numerators and denominators are, up to multiplicationby an element of P , the two terms on the right hand side of an exchange relation of S .Note that seed patterns may carry redundant information, in that the same seed can be assignedto multiple vertices of T n . Further, two labeled seeds in a seed pattern may be the same up torelabeling. We remedy this by calling two labeled seeds Σ = ( a , x , B ) and Σ (cid:48) = ( a (cid:48) , x (cid:48) , B (cid:48) ) equivalent(and writing Σ ∼ Σ (cid:48) ) if one can obtain Σ (cid:48) by simultaneously reindexing a , x , and the rows andcolumns of B . We define an analogous equivalence relation for X -seeds, also denoted ∼ .An A -seed in F (respectively X -seed in P ) is an equivalence class of labeled A -seeds in F (respectively labeled X -seeds in P ) with respect to ∼ . The seed represented by the labeled seed Σis denoted [Σ]. We mutate a seed [Σ] by applying a mutation µ k to Σ and taking its equivalenceclass. 4 efinition 2.7. The exchange graph of a seed pattern S is the ( n -regular connected) graph whosevertices are the seeds in S and whose edges connect seeds related by a single mutation. Equivalently,the exchange graph is the graph one obtains by identifying the vertices t, t (cid:48) of T n such that Σ t ∼ Σ t (cid:48) .Exchange graphs were defined for A -seed patterns in [6, Definition 7.4], but can equally bedefined for X -seed patterns as we do here. It is conjectured that the exchange graph of an A -seedpattern S = S ( a , x , B ) depends only on B [8, Conjecture 4.3], meaning that S| X does not influencethe combinatorics of S . The exchange graphs of S| X and ˆ S can be obtained by identifying somevertices of the exchange graph of S , as passing to either X -seed pattern preserves mutation and theequivalence of labeled seeds. It is not known in general when any pair of these exchange graphs isequal. We now restrict our attention to seed patterns of finite type.
Definition 3.1. An A -seed pattern is of finite type if it has finitely many seeds.Finite type seed patterns were classified completely in [7]; they correspond exactly to finite(reduced crystallographic) root systems, or equivalently, finite type Cartan matrices (see for example[2, Chapter 5]).For a skew-symmetrizable integer matrix B = ( b ij ), its Cartan counterpart is the matrix A ( B ) =( a ij ) defined by a ii = 2 and a ij = −| b ij | for i (cid:54) = j . Theorem 3.2 ([7, Theorems 1.5-1.7]) . (i.) An A -seed pattern is of finite type if and only if theCartan counterpart of one of its exchange matrices is a finite type Cartan matrix.(ii.) Suppose B , B (cid:48) are skew-symmetrizable integer matrices such that A ( B ) , A ( B (cid:48) ) are finite typeCartan matrices. Then A ( B ) and A ( B (cid:48) ) are of the same Cartan-Killing type if and only if B and B (cid:48) are mutation equivalent (modulo simultaneous relabeling of rows and columns.). In light of this theorem, we refer to a finite type A -seed pattern as type A n , B n , etc. Definition 3.3. An X -seed pattern S is of type Z n if the Cartan counterpart of one of its exchangematrices is a type Z n Cartan matrix.We will call such X -seed patterns Dynkin type rather than finite type, since not all X -seedpatterns with finitely many seeds are of this form. For example, let P be any tropical semifield, and B the matrix (cid:20) − (cid:21) . Then the X -seed pattern S ((1 , , B ) in P has a single unlabeled seed, but the Cartan counterpartof B is not finite type. In general, if B is an n × n skew-symmetrizable matrix with finite mutationequivalence class, then the X -seed pattern S ((1 , . . . , , B ) in a tropical semifield will have finitelymany seeds. Indeed, in this case, mutation of the initial seed ((1 , . . . , , B ) in direction k results inthe seed ((1 , . . . , , µ k ( B )). It follows that any sequence of mutations results in a seed with every X -variable equal to 1, so seeds are distinguished from each other only by their exchange matrices.As B is mutation equivalent to finitely many matrices, there are only finitely many seeds in theseed pattern. 5igure 1: Triangulations of P and P • are shown in solid lines; the dashed arc is the flip of γ . Onthe right, the quadrilateral q T ( γ ) = { α, β, δ, (cid:15) } is shown in bold. A and D The material in this section is part of a more general theory of A -seed patterns from surfaces,developed in [4].Let P n denote a convex n -gon and P • n denote a convex n -gon with a distinguished point p (a puncture ) in the interior. For P ∈ { P n , P • n } , the vertices and puncture of P are called markedpoints . An arc of P is a non-self-intersecting curve γ in P such that the endpoints of γ are distinctmarked points, the relative interior of γ is disjoint from ∂ P ∪ { p } , and γ does not cut out anunpunctured digon. An arc incident to the puncture p is a radius . Arcs are considered up toisotopy.A tagged arc of P is either an ordinary arc between two vertices or a radius that is labeledeither “notched” or “plain.” Two tagged arcs γ , γ (cid:48) are compatible if their untagged versions donot cross (or to be precise, there are two noncrossing arcs isotopic to γ and γ (cid:48) ) with the followingmodification: if γ is a notched radius and γ (cid:48) is plain, they are compatible if and only if theiruntagged versions coincide.A tagged triangulation T is a maximal collection of pairwise compatible tagged arcs. All taggedtriangulations of P consist of the same number of arcs. Definition 3.4.
Let T be a tagged triangulation, γ an arc in T , and γ (cid:48) either an arc in T or aboundary segment. Then γ and γ (cid:48) are adjacent in T if they are adjacent in a triangle of T or ifthere is a third arc α in T such that { α, γ, γ (cid:48) } form a once-punctured digon with a radius (seeFigure 2).In particular, if T contains two arcs forming a once-punctured digon, the two sides of the digonare adjacent, each side is adjacent to each radius inside the digon, and the two radii inside the digonare not adjacent. Definition 3.5.
Let T be a tagged triangulation, and γ an arc in T . The quadrilateral q T ( γ ) ofan arc γ in T consists of the arcs of T and boundary segments adjacent to γ if γ is not a radiusin a once-punctured digon (see Figure 1); if γ is a radius in a once-punctured digon, q T ( γ ) consists6 L O Q R S T U Figure 2: Triangulations of P and P • and the associated quivers Q ( T ).of the arcs and boundary segments adjacent to γ together with the two radii compatible with γ inthe once-punctured digon (see Figure 4, lower right). The arc γ is a diagonal of its quadrilateral.Note that if γ is a radius in a once-punctured digon, q T ( γ ) is not part of any tagged triangulation.The following result gives us a local move on tagged triangulations. Proposition 3.6 ([4, Theorem 7.9]) . Let T = { γ , . . . , γ n } be a tagged triangulation of P . Forall k , there exists a unique tagged arc γ (cid:48) k (cid:54) = γ k such that µ k ( T ) := T \ { γ k } ∪ { γ (cid:48) k } is a taggedtriangulation of P . The arc γ (cid:48) in the above proposition is called the flip of γ with respect to T (or with respect to q T ( γ ), since γ and γ (cid:48) are exactly the two diagonals of q T ( γ )).We define the flip graph of P to be the graph whose vertices are tagged triangulations of P andwhose edges connect triangulations that can be obtained from each other by flipping a single arc.The flip graph of P is connected.We can encode a tagged triangulation T = ( γ , . . . , γ n ) in a skew-symmetric n × n integer matrix B ( T ). The nonzero entries in B ( T ) correspond to pairs of adjacent arcs; the sign of these entriesrecords the relative orientation of the arcs.To find the entries of B ( T ), we define a quiver Q ( T ). Place a vertex i in the interior of each arc γ i . Then put an arrow from i to j if γ i and γ j are two sides of a triangle in T and i immediatelyprecedes γ j moving clockwise around this triangle, unless γ i and γ j are arcs in a triangle containingthe notched and plain versions of the same radius. In this case, add arrows as shown in Figure 2.Finally, delete a maximal collection of 2-cycles. If there is an arrow from i to j in Q ( T ), we write i → j .Let b ij ( T ) := { arrows i → j in Q ( T ) } − { arrows j → i in Q ( T ) } . We define B ( T ) :=( b ij ( T )).Flips of arcs are related to matrix mutation in the following way: for a tagged triangulation T = { γ , . . . , γ n } of P , µ k ( B ( T )) = B ( µ k ( T )), or, in words, flipping γ k changes B ( T ) by mutationin direction k . 7igure 3: Triangulations of P G and P • G are shown in solid lines. The orbit [ γ ] of γ is { γ, γ (cid:48) } andthe flip of [ γ ] is dashed. On the right, q T ([ γ ]) = { α, β, γ, γ (cid:48) } .As the following theorem shows, these triangulations entirely encode the combinatorics of type A and D A -seed patterns. Theorem 3.7 ([4]) . Let P = P n +3 (resp. P = P • n ). Consider an A -seed pattern S such that someexchange matrix is B ( T ) for some triangulation T of P . Then S is type A n (resp. D n ) and thereis a bijection γ (cid:55)→ a γ between arcs of P and A -variables of S . Further, if Σ = ( a , x , B ) is a seed of S , there is a unique triangulation T such that a = { a γ } γ ∈ T and B = B ( T ) . Finally, mutation indirection k takes the seed corresponding to T to the seed corresponding to µ k ( T ) , implying that theexchange graph of S is isomorphic to the flip graph of P . B and C To obtain triangulations whose adjacency matrices are exchange matrices of type B n and C n A -seedpatterns, we “fold” triangulations of P n +2 and P • n +1 . This is part of a larger theory of foldedcluster algebras (see [5, Chapter 4]).Let G = Z / Z . We write P G n for P n equipped with the G -action taking vertex i to vertex i (cid:48) := i + n (with labels considered modulo 2 n ). This induces an action of G on the arcs of P n . Thetriangulations of P G n are defined to be the triangulations of P n fixed by the G -action, commonlycalled centrally symmetric triangulations.We write P • Gn for P • n equipped with the G -action switching the notched and plain version of aradius. Again, the triangulations of P • Gn are defined to be the triangulations of P • n fixed under the G -action, which are exactly those triangulations containing both the notched and plain versions ofthe same radius.Let P ∈ { P n , P • n } , and let T be a triangulation of P G . For γ ∈ T , let [ γ ] denote the G -orbitof γ . The quadrilateral of [ γ ], denoted q T ([ γ ]), is all of the arcs and boundary segments adjacentto arcs in [ γ ] (which is exactly the G -orbit of q T ( γ ) as long as γ is not a radius in a once-punctureddigon). Flipping the arcs in [ γ ] results in another triangulation of P G (which does not depend onthe order of arc flips, since arcs in the same orbit are pairwise not adjacent). We define the flipgraph of P G in direct analogy to that of P ; again, it is connected.8e associate to each triangulation T of P G a skew-symmetrizable integer matrix B G ( T ), whoserows and columns are labeled by G -orbits of arcs of T . Let T = { γ , . . . , γ m } , and let I, J be theindices of arcs in the G -orbit of γ i and γ j . Then b GIJ ( T ) = (cid:88) i ∈ I b ij ( T ) (3.1)where B ( T ) = ( b ij ( T )) is the usual signed adjacency matrix of T . In terms of Q ( T ), b GIJ ( T ) isthe total number of arrows from all representatives of I to a fixed representative of J . The entriesof B G ( T ) are well-defined: for g ∈ G , if g ( γ i ) = γ (cid:48) i and g ( γ j ) = γ (cid:48) j , then b ij = b i (cid:48) j (cid:48) , so the value of b GIJ does not change if a different element of J is used to compute (3.1).Just as with usual triangulations, (orbits of) arc flips and matrix mutation interact nicely: if T is a triangulation containing arc γ , flipping the arcs in [ γ ] corresponds to mutating B G ( T ) in thedirection labeled by [ γ ]. Further, we have the following theorem. Theorem 3.8 ([5, Section 5.5]) . Let P = P • n +1 (resp. P = P n +2 ). Consider an A -seed pattern S such that some exchange matrix is B G ( T ) for some triangulation T of P G . Then S is type B n (resp. C n ) and there is a bijection [ γ ] (cid:55)→ a [ γ ] between arcs of P and A -variables of S . Further,if Σ = ( a , x , B ) is a seed of S , there is a unique triangulation T such that a = { a [ γ ] } γ ∈ T and B = B G ( T ) . Finally, mutation in direction K takes the seed corresponding to T to the seedcorresponding to µ K ( T ) , implying that the exchange graph of S is isomorphic to the flip graph of P G . X -seed patterns Let S be an X -seed pattern of type Z n ( Z ∈ { A, B, C, D } ) over an arbitrary semifield P , andlet X ( S ) denote the set of X -variables of S . Let P be the surface whose triangulations encodethe combinatorics of type Z n A -seed patterns ( P = P n +3 for Z n = A n , P = P • n for Z n = D n , P = P G n +2 for Z n = C n , P = P • Gn +1 for Z n = B n ). In this section, we relate the X -variables of S to the triangulations of P , and show a bijection between X ( S ) and quadrilaterals (with a choiceof diagonal) of P in the case when P is the universal semifield. Note that in what follows, “arc”should usually be understood to mean “orbit of arc” if S is type B or C .First, notice that Theorems 3.7 and 3.8 imply that one can associate to each triangulation of P a seed of S such that mutation of seeds corresponds to flips of arcs. Indeed, consider any A -seedpattern R with R| X = S ; if the triangulation T corresponds to the A -seed ( a , x , B ) with arc γ k corresponding to A -variable a k , then we associate to T the X -seed ( x , B ) and to the arc γ k the X -variable x k . We write Σ T to indicate this association, and write x T,γ for the X -variable associatedto arc γ in Σ T .Note that a priori two distinct triangulations may be associated to the same X -seed, and an X -variable may be associated to a number of different arcs (though we will see that this is not thecase). Further, an arc may be associated to different X -variables in different triangulations.The next observation follows immediately from the definition of seed mutation. Remark 4.1.
Consider an X -seed ( x , B ), and let B (cid:48) = ( b (cid:48) ij ) be the mutation of B in direction k .For j (cid:54) = k , if b jk = 0, then mutating at k will not change x j . Further, b (cid:48) ji = b ji and b (cid:48) ij = b ij for all i , since the skew-symmetrizability of B implies b kj = 0 as well. Thus, if k , . . . , k t are indices suchthat b k s j = 0, then the mutation sequence µ k t ◦ · · · ◦ µ k ◦ µ k leaves x j unchanged.9n other words, consider x T,γ , an X -variable in Σ T . Any sequence of flips of arcs not in q T ( γ ) ∪{ γ } will result in a triangulation S such that γ ∈ S and x S,γ = x T,γ . Proposition 4.2.
Let Q (cid:48) = { q T ( γ ) ∪{ γ }| T a triangulation of P , γ ∈ T } be the set of quadrilaterals(with a choice of diagonal) of P . Then the following map is a surjection: f : Q (cid:48) → X ( S ) q T ( γ ) ∪ { γ } (cid:55)→ x T,γ . We remark that this proposition in fact holds in the generality of X -seed patterns from markedsurfaces, by the discussion following Proposition 9.2 in [4]. If one considers a collection A ofcompatible tagged arcs of a marked surface ( S, M ), the simplicial complex of tagged arcs compatiblewith A is the tagged arc complex of another (possibly disconnected) surface ( S (cid:48) , M (cid:48) ), so its dualgraph (that is, the flip graph of ( S (cid:48) , M (cid:48) )) is connected by [4, Proposition 7.10]. This implies thatany two triangulations containing A are connected by a series of flips of arcs not in A . However,Proposition 4.2 is not hard to see directly in this case, so we provide a proof here. Proof. If f is well-defined, it is clearly surjective. By Remark 4.1, to show f is well-defined, itsuffices to show that all triangulations T with γ ∈ T and q T ( γ ) = q can be obtained from oneanother by flipping arcs not in q ∪ { γ } .Observe that removing the interior of q from P gives rise to a new surface P (cid:48) , whose connectedcomponents are polygons, once-punctured polygons, or unions of several of these that intersectonly at vertices. The triangulations of P (cid:48) do not include the arcs in q , and the flip graph of P (cid:48) is connected. There is a bijection from triangulations T (cid:48) of P (cid:48) to triangulations of P containing q ∪ { γ } : T (cid:48) (cid:55)→ T (cid:48) ∪ q ∪ { γ } (up to changing the tagging of radii in T ). This bijection respects flips,so this shows the desired result.We have the immediate corollary: Corollary 4.3.
Let q ( P ) denote the number of quadrilaterals of P . Then |X ( S ) | ≤ q ( P ) . The above statements hold regardless of the choice of P and initial X -cluster. However, |X ( S ) | can vary for different choices of P and the X -cluster of a fixed seed, as the following example shows. Example 4.4.
Let B be the following matrix with Cartan counterpart of type A : − − . If P is any tropical semifield, then the X -seed pattern S ((1 , , , B ) in P has a single X -variable,which is equal to 1.If P = Q sf ( x , x , x ), the X -seed pattern S := S ((1 /x , x /x , x ) , B ) in P has fewer X -variables than S := S (( x , x , x ) , B ). For example, in S , we have (cid:18)(cid:18) x , x x , x (cid:19) , B (cid:19) µ −→ (cid:18)(cid:18) x , x x (1 + x ) , x (cid:19) , µ ( B ) (cid:19) . Note that x appears twice in µ ((1 /x , x /x , x ) , B ).10n S , we have (( x , x , x ) , B ) µ −→ (cid:18)(cid:18) x , x x (1 + x ) , x (cid:19) , µ ( B ) (cid:19) . So |X ( S ) | ≤ |X ( S ) | − |X ( S ) | = 18,while |X ( S ) | = 30.If we fix an exchange matrix B and allow P and the X -cluster of the seed ( x , B ) to vary, S ( x , B )will have the largest number of X -variables when P = Q sf ( t , . . . , t n ) and x consists of elementsthat are algebraically independent over Q ( t , . . . , t n ). Indeed, let S sf be such a seed pattern, and S an arbitrary seed pattern in a semifield P containing the exchange matrix B . The X -variablesof S can be obtained from the X -variables of S sf by replacing “+” with “ ⊕ ” and evaluating atthe appropriate elements of P , so we have |X ( S ) | ≤ |X ( S sf ) | . As can be seen in Example 4.4, theassumption that the elements of x are algebraically independent is necessary. Note also that ifthe elements of x are algebraically independent, so are the elements of an arbitrary X -seed in S sf ,since X -seed mutation in this case simply has the effect of multiplying the X -variables by rationalfunctions.In light of this observation, we now focus on X ( S sf ). To show that the surjection f of Proposition4.2 is a bijection for S sf , it suffices to show that f is injective for some X -seed pattern of type Z n .To do this, we use specific examples of A -seed patterns of type Z n given in [5, Chapter 5] and [7,Section 12.3],which we denote by R ( Z n ). These seed patterns are over a tropical semifield P ; the A -variables and the generators of P are SL - or SO -invariant polynomials in the entries of a 2 × m matrix. Proposition 4.5.
Consider the A -seed pattern R = R ( Z n ) . Then the surjection f : Q (cid:48) → X ( ˆ R ) of Proposition 4.2 is injective. We delay the description of R ( Z n ) and the proof of this proposition to Section 5.Theorem 1.1 is an immediate corollary of Proposition 4.5, as is the number of X -variables in S sf . Corollary 4.6. |X ( S sf ) | = 2 q ( P ) . We label the vertices of P ∈ { P n , P • n , P Gn , P • Gn } clockwise with 1 , . . . , n . Proposition 4.7.
The number of quadrilaterals of each surface P are listed in the following table. P P n +3 P G n +2 , P • Gn +1 P • n q ( P ) (cid:0) n +34 (cid:1) n ( n + 1)( n + 2) n ( n − n + 4 n − P (cid:54) = P G n +2 , q ( P ) follow from a fairly straightforward inspection of the triangulations of theappropriate surfaces. It is clear that q ( P n +3 ) = (cid:0) n +34 (cid:1) , as quadrilaterals in a polygon are uniquelydetermined by their vertices. Proposition 4.8. q ( P • n ) = 4 (cid:0) n (cid:1) + 9 (cid:0) n (cid:1) + 2 (cid:0) n (cid:1) = n ( n − n + 4 n − T Figure 4: The 4 types of quadrilaterals in a triangulation of P • n , listed in clockwise order fromthe upper left. The arcs of the quadrilateral are solid; diagonals are dashed. The cut, the segmentfrom p to the boundary arc between vertices 1 and n , is shown in bold. Proof.
We first introduce a “cut” from p to the boundary arc between vertices 1 and n (see Figure4). For any two vertices i and j , there are two distinct arcs between i and j , one crossing the cutand the other not. For convenience, we here refer to boundary segments between adjacent verticesas “arcs,” though they are not arcs of any triangulation.Let q be a quadrilateral in some triangulation of P • n . There are 4 possibilities (see Figure 4):(i) The quadrilateral q consists of arcs between four vertices of S ; the diagonals of q are arcsbetween vertices. Then q cannot enclose the puncture, since its diagonals are arcs betweenvertices. This places restrictions on the possible configurations of arcs. If 1 ≤ i < j < k Let Q be the set of quadrilaterals of P n +2 , Q be the set of quadrilateralscontained in a closed δ i -half-disk some i , and Q the subset of Q \ Q of quadrilaterals whosevertices are not { i, j, i (cid:48) , j (cid:48) } . There is a bijection α : Q → Q .Proof. We consider P n +2 as a disk with 2 n + 2 vertices on the boundary. Let i, j, k be vertices.We denote by i | j a segment of the boundary with endpoints i and j . To specify a particularsegment, we give an orientation (e.g. “clockwise segment i | j ”) or an interior point k (e.g. i | k | j ). Ifa number of interior points are given, they are listed respecting the orientation of the segment. If aquadrilateral is in Q , we list its vertices in clockwise order, starting from any point in a half-diskdisjoint from the quadrilateral. Note that in this order, a quadrilateral ( a, b, c, d ) ∈ Q is containedin a closed δ a -half-disk and a closed δ d -half-disk. If the quadrilateral is not in Q , we list its verticesin clockwise order from an arbitrary starting point.Let ( a, b, c, d ) ∈ Q . The map α “flips” b to b (cid:48) , sending ( a, b, c, d ) to ( a, c, d, b (cid:48) ). The map α iswell-defined, as { a, c, d, b (cid:48) } (cid:54) = { i, i (cid:48) , j, j (cid:48) } for any i, j and for all i ∈ { a, b, c, d } no closed δ i -half-diskcontains all 4 vertices of α ( a, b, c, d ).We now show that α has an inverse, similarly given by flipping a vertex. Let q = ( a, b, c, d ) ∈ Q .Note that flipping a vertex of q results in a quadrilateral in Q only if it is the only vertex in somehalf-disk; one may take this half-disk to be a δ i -half-disk for some i ∈ { a, b, c, d } .Suppose for some i ∈ { a, b, c, d } , i (cid:48) is also in { a, b, c, d } . Wlog, i = a and thus i (cid:48) = c (otherwise q would be contained in a half-disk). Flipping a or c would result in 3 vertices, so they cannot beflipped to produce an element of Q . Since Q does not contain quadrilaterals of the form ( i, j, i (cid:48) , j (cid:48) ), b (cid:48) (cid:54) = d . The clockwise segment a | c necessarily includes d (cid:48) , since for all i, j , the segment i | i (cid:48) includesexactly one of j and j (cid:48) . So this segment is either a | d (cid:48) | b | c or a | b | d (cid:48) | c ; the clockwise segment c | a is13 | d | b (cid:48) | a or c | b (cid:48) | d | a , respectively. In the first case, flipping d gives a preimage of q , and flipping b does not; vice-versa in the second case.Now, suppose no vertices in q are flips of each other. There are two vertices of q in one δ a -half-disk and one, say i , in the other. Similarly, there are two vertices of q in one δ i -half disk and one,say j , in the other.We claim that i and j are the only vertices alone in δ k -half-disks for k ∈ { a, b, c, d } .If j = a , then the segments a | i | a (cid:48) and i | a | i (cid:48) contain no other vertices of q , implying that i (cid:48) | a | i | a (cid:48) is a segment containing no other vertices of q . Let k be one of the other vertices of q . Either k or k (cid:48) must lie on i (cid:48) | a | i | a (cid:48) between a and a (cid:48) , and between i and i (cid:48) . Thus k (cid:48) in fact lies between a and i onthis segment, so δ k separates a and i . Since one δ k -half-disk contains a single vertex, the statementfollows.If j (cid:54) = a , then the segments a | i | a (cid:48) and i | j | i (cid:48) contain no other vertices of q , so a | i | a (cid:48) | j | i (cid:48) is asegment containing no other vertices of q . Further, j (cid:48) must lie somewhere on a | i | a (cid:48) and cannot lieon i | j | i (cid:48) , implying that a | j (cid:48) | i | a (cid:48) | j | i (cid:48) is a segment. Since no other vertices of q appear in this segment, i is the single vertex in one δ j -half-disk. Let k be the other vertices of q . As in the above case,one of k and k (cid:48) must lie between a and a (cid:48) , and i and i (cid:48) . Thus, k (cid:48) lies between i and a (cid:48) in the arc a | i | a (cid:48) | j | i (cid:48) , implying δ k separates i and j . The statement follows.As noted above, flipping i will result in a quadrilateral q (cid:48) in Q . The vertex i (cid:48) will not be thefirst or last vertex in q (cid:48) , as by assumption the vertices of q are not contained in a δ i -half-disk.However, j will be, since all vertices of q (cid:48) are contained in one δ j -half-disk. The vertex of q (cid:48) closestto j is i (cid:48) , as if there is a vertex k between i (cid:48) and j , then j is not alone in a δ i -half-disk. Soflipping i gives a preimage of q only if i | j | i (cid:48) is a clockwise arc. Reversing the roles of i and j in theabove argument yields that flipping j gives a preimage of q only if j | i | j (cid:48) is clockwise (i.e. i | j | i (cid:48) iscounterclockwise).Since | Q | = | Q | + | Q | + |{ q ∈ Q : q = ( i, j, i (cid:48) , j (cid:48) ) for 1 ≤ i < j ≤ n + 1 }| , we have the followingcorollary. Corollary 4.12. | Q | = ( (cid:0) n +24 (cid:1) − (cid:0) n +12 (cid:1) )Note that the quadrilaterals of centrally symmetric triangulations of P n +2 are either in Q orhave vertex set { i, j, i (cid:48) , j (cid:48) } for 1 ≤ i < j ≤ n + 1. The latter quadrilaterals are fixed by the actionof G ; the quadrilaterals in Q have G -orbits of size two. So we have that q ( P G n +2 ) = 12 | Q | + (cid:18) n + 12 (cid:19) = 16 n ( n + 1)( n + 2) . (4.1) Let Z ∈ { E , E , E , F , G } and let S sf be a type Z X -seed pattern over Q sf with one (equiva-lently every) X -cluster consisting of algebraically independent elements. The value of |X ( S sf ) | wascomputed using a computer algebra system (Mathematica), by generating all possible X -seeds viamutation. The code is available at https://math.berkeley.edu/~msb . Proof of Proposition 4.5 We proceed type by type. The general recipe is as follows. As usual, for non-simply laced types,“arc” should be read as “orbit of arc.”Let V be a vector space and k the field of rational functions on V . For P ∈ { P n +3 , P • n , P • Gn +1 , P G n +2 } ,we assign a function P γ ∈ k to each arc and boundary segment γ of P .Let T be a triangulation of P . We define seeds Σ T = ( a , x , B ( T )), where the clusters andexchange matrix are indexed by arcs in T . The matrix B ( T ) is the exchange matrix as defined inSections 3.1 and 3.2, whose entries are based on Q ( T ). The A -variable associated to γ ∈ T is just P γ .To describe the X -variables, we first extend our construction of Q ( T ) to the boundary segmentsof P to create the quiver Q ( T ). That is, we place a vertex at the midpoint of every arc and boundarysegment, put an arrow from γ to γ (cid:48) if γ precedes γ (cid:48) moving clockwise around some triangle of T ,and delete a maximal collection of 2-cycles (making slight modifications for punctured polygons).If there is an arrow from γ to γ (cid:48) in Q ( T ), we write γ → γ (cid:48) . For simply laced types, the X -variableassociated to γ is (cid:81) γ (cid:48) ∈ ∂ P P { arrows γ (cid:48) → γ } γ (cid:48) (cid:81) γ (cid:48) ∈ ∂ P P { arrows γ → γ (cid:48) } γ (cid:48) . (5.1)For non-simply laced types, the X -variable associated to [ γ ] is (cid:81) γ (cid:48) ∈ ∂ P P { arrows [ γ (cid:48) ] → γ } γ (cid:48) (cid:81) γ (cid:48) ∈ ∂ P P { arrows γ → [ γ (cid:48) ] } γ (cid:48) where { arrows [ γ (cid:48) ] → γ } = (cid:80) τ ∈ [ γ (cid:48) ] { arrows τ → γ in Q ( T ) } and { arrows γ → [ γ (cid:48) ] } = (cid:80) τ ∈ [ γ (cid:48) ] { arrows γ → τ in Q ( T ) } .These seeds form R ( Z n ), an A -seed pattern of type Z n over QP where P =Trop( P γ : γ ⊆ ∂ P ).The reader familiar with cluster algebras of geometric type should note that we could equally definethis seed pattern by declaring P γ a frozen variable for γ a boundary arc, defining an extendedexchange matrix B ( T ) from the quiver Q ( T ), and then finding the X -variable associated to eacharc via the formula [8, Equation 2.13].In (cid:92) R ( Z n ), the X -variables of ˆΣ T are also indexed by arcs of T . To emphasize that the X -variableassociated to γ in ˆΣ T depends only on the quadrilateral q := q T ( γ ) of γ , we denote it by ˆ x q,γ . Insimply laced types, ˆ x q,γ = (cid:81) γ (cid:48) ∈ T ∪ ∂ P P { arrows γ (cid:48) → γ } γ (cid:48) (cid:81) γ (cid:48) ∈ T ∪ ∂ P P { arrows γ → γ (cid:48) } γ (cid:48) (5.2)and in non-simply laced types,ˆ x q, [ γ ] = (cid:81) γ (cid:48) ∈ T ∪ ∂ P P { arrows [ γ (cid:48) ] → γ } γ (cid:48) (cid:81) γ (cid:48) ∈ T ∪ ∂ P P { arrows γ → [ γ (cid:48) ] } γ (cid:48) . To show that ˆ x q,γ and ˆ x q (cid:48) ,γ (cid:48) are distinct for q (cid:54) = q (cid:48) , we show that they evaluate differently atspecific elements of V . 15 Figure 5: A quadrilateral and its associated quiver.In the subsequent sections, we again label the m vertices of P clockwise with 1 , . . . , m . A n Let V =Mat ,n +3 ( C ) be the vector space of 2 × ( n + 3) complex matrices. Let k be the field ofrational functions on V , written in terms of the coordinates of the column vectors v , . . . , v n +3 . Letthe Pl¨ucker coordinate P ij ∈ k be the determinant of the 2 × v i and v j .Pl¨ucker coordinates will be the basis of all of the following constructions. They give an embedding ofthe Grassmannian Gr ,n +3 of 2-planes in C n +3 into complex projective space of dimension (cid:0) n +32 (cid:1) − z ∈ V (up to simultaneous rescaling) depend only on therowspan of z . The Pl¨ucker coordinates also generate the coordinate ring of the affine cone over theGrassmannian Gr ,n +3 in the Pl¨ucker embedding.If γ is an arc or boundary segment of P n +3 between vertices i and j ( i < j ), we define P γ := P ij .In other words, the A -variables of R ( A n ) are exactly the Pl¨ucker coordinates P ij where i < i +1 < j .The semifield P is the tropical semifield generated by all consecutive Pl¨ucker coordinates P i,i +1 . For γ an arc with quadrilateral q in some triangulation, the X -variable x q,γ records which boundaryarcs τ are in triangles with γ . By the construction of Q ( T ) and (5.1), if τ precedes γ moving aroundthe triangle clockwise, P τ appears in the numerator of x q,γ . If τ instead follows γ , P τ appears inthe denominator.Let T be a triangulation of P n +3 containing an arc γ with vertices i and k . Suppose q := q T ( γ )has vertices 1 ≤ i < j < k < l ≤ n + 3 (see Figure 5). Then (5.2) becomesˆ x q,γ = (cid:81) τ : τ → γ P τ (cid:81) τ : γ → τ P τ = P il P jk P ij P kl . (5.3)The X -variable associated to the other diagonal of the quadrilateral is ˆ x − q,γ . Clearly, ˆ x q,γ (cid:54) = ˆ x − q,γ ,so the X -variables associated to the two diagonals of the same quadrilateral are distinct.Consider another quadrilateral q (cid:48) with vertices 1 ≤ a < b < c < d ≤ n +3 and diagonal γ . Choose s ∈ { a, b, c, d }\{ i, j, k, l } . The Pl¨ucker coordinate ± P st appears in ˆ x ± q (cid:48) ,γ for some t ∈ { a, b, c, d } , Let z be any matrix such that { v i ( z ) , v j ( z ) , v k ( z ) , v l ( z ) } = { (1 , , (0 , , (1 , , ( − , } and v s ( z ) = v t ( z ).Then ˆ x q,γ ( z ) is nonzero and ˆ x ± q (cid:48) ,γ ( z ) is either zero or undefined. This completes the proof ofProposition 4.5 for type A n . 16 .2 Type B n Let V = Mat ,n +2 ( C ) be the vector space of 2 × ( n + 2) complex matrices. Again, let k be the fieldof rational functions on V , written in terms of the coordinates of the column vectors v , . . . , v n +3 and P ij ∈ k be the determinant of the 2 × v i and v j . We define a modifiedPl¨ucker coordinate P ij := P i,n +2 P j,n +2 − P ij .Let γ be an arc or boundary segment of P • n +1 . If γ has endpoints i, j ∈ { , . . . , n + 1 } , let P [ γ ] := P ij (respectively P [ γ ] := P ij ) if it does not (respectively, does) cross the cut. If γ is a radiuswith endpoints i and p , then let P [ γ ] := P i,n +2 .Let q be a quadrilateral with vertices 1 ≤ i < j < k < l ≤ n + 1. Thenˆ x q, [ γ ] ∈ (cid:18) P il P jk P ij P kl (cid:19) ± , (cid:18) P il P jk P ij P kl (cid:19) ± , (cid:32) P il P jk P ij P kl (cid:33) ± , (cid:18) P il P jk P ij P kl (cid:19) ± . (5.4)Consider a quadrilateral q with two vertices i and j ( i < j ) and let γ be the plain radius withendpoints i and p . Then ˆ x q, [ γ ] = P ij P ij . (5.5)Finally, given a quadrilateral q with three vertices i < j < k and diagonal γ ,ˆ x q, [ γ ] ∈ (cid:32) P ij P k,n +2 P ik P jk (cid:33) ± , (cid:32) P ik P j,n +2 P ij P jk (cid:33) ± , (cid:32) P jk P i,n +2 P ik P ij (cid:33) ± . (5.6)We would like to show that all of these expressions are distinct. Note that in general, expressionsfrom quadrilaterals on different vertices are definitely distinct (as long as none are identically zeroon their domains, which follows easily from arguments below). Indeed, if a is a vertex of q (cid:48) and not q , then one can find a matrix z such that ˆ x q,γ ( z ) (cid:54) = 0 and this will not depend on v a ( z ). However,the index a appears in the expression ˆ x q (cid:48) ,γ (cid:48) , and thus one can freely choose v a ( z ) in order to makeˆ x q (cid:48) ,γ (cid:48) ( z ) either zero or undefined.The only instance in which this is not clear is when ˆ x ± q,γ = P il P jk P ij P kl and q (cid:48) is a quadrilateral withvertices j , k , and l . Because i appears only in modified Pl¨ucker coordinates, it is not immediatethat one can choose columns to make the expressions differ. The following remark gives a way todo this. Remark 5.1. P ab = 0 for v a = (1 , , v b = (0 , , v n +2 = ( − , v c = (1 , , v d = ( − , P s,t (cid:54) = 0 for all other pairs s, t ∈ { a, b, c, d } with s (cid:54) = t .So there exists a matrix z such that P ik ( z ) = 0 and all other (usual or modified) Pl¨uckercoordinates involving the indices i, j, k, l are nonzero, which covers the problematic case.Remark 5.1 also gives us that two X -variables associated to different quadrilaterals on the same4 vertices are distinct, as each expression in (5.4) involves a different modified Pl¨ucker coordinate.In the case when the quadrilaterals are on the same 3 vertices, if { v i , v j , v k } = { (1 , , (1 , , ( − , } ,choosing v n +2 = v a for a = i, j, k makes a unique expression in (5.6) zero or undefined.There are no two quadrilaterals on the same two vertices, and the expressions in (5.5) are clearlynot identically zero, so no further argument is needed.17 .3 Type C n Let V =Mat ,n +1 ( C ). Let M ∈ SO ( C ) be (cid:20) − 11 0 (cid:21) . Again, let k be the field of rational functions on V , written in terms of the coordinates of thecolumn vectors v , . . . , v n +1 and P ij ∈ k be the determinant of the 2 × v i and v j . We define the modified Pl¨ucker coordinate P ij as the determinant of the 2 × v j and whose second column is M v i . More explicitly, if v i = ( v i, , v i, ), then P ij = v j, v i, + v j, v i, .Let γ be an arc or boundary segment of P n +2 . We define P [ γ ] := P ij if [ γ ] contains an arc from i to j with i < j ≤ n + 1, i.e. if one element of [ γ ] is contained in a δ -half-disk (see Definition4.10). We define P [ γ ] := P ij if [ γ ] contains an arc from i to j + n + 1 with i ≤ j ≤ n + 1, i.e. if oneelement of [ γ ] is not contained in a δ -half-disk.In a departure from earlier notation, we will use i solely to denote a vertex in { , . . . , n + 1 } ; i (cid:48) := i + n + 1, as before. We will also include the vertices 1 (respectively n + 2) in the δ -half-diskcontaining 2 (respectively, n + 3). Recall that in P G n +2 , q T ([ γ ]) = [ q T ( γ )] for some arc γ andtriangulation T containing γ . If [ q T ( γ )] contains 2 quadrilaterals of P n +2 , we will choose as arepresentative the quadrilateral containing the smallest vertex; if this does not determine one of thequadrilaterals, we will choose the quadrilateral containing the two smallest vertices. Quadrilateralsare given as tuples of their vertices, listed in clockwise order starting with the smallest vertex.If q = [ q T ( γ )] consists of a single quadrilateral, then q T ( γ ) = ( i, j, i (cid:48) , j (cid:48) ). Thenˆ x q, [ γ ] ∈ (cid:32) P ij P ij (cid:33) ± . For the remaining cases, q = [ q T ( γ )] contains 2 quadrilaterals. If q T ( γ ) = ( i, j, k, l ) (i.e. allvertices are in one δ -half-disk), thenˆ x q, [ γ ] ∈ (cid:40)(cid:18) P il P jk P ij P kl (cid:19) ± (cid:41) . If q T ( γ ) has 3 vertices in one δ -half-disk, q T ( γ ) is ( i, j, k, l (cid:48) ) for k < l , ( i, j (cid:48) , k (cid:48) , l (cid:48) ) for i < j ,( i, j, k, i (cid:48) ), or ( i, j, k, k (cid:48) ) (see Figure 6). Then ˆ x q, [ γ ] is, respectively (cid:18) P il P jk P ij P kl (cid:19) ± , (cid:32) P il P jk P ij P kl (cid:33) ± , (cid:18) P ii P jk P ij P ik (cid:19) ± , (cid:18) P ik P jk P ij P kk (cid:19) ± . If q T ( γ ) has 2 vertices in each δ -half-disk, q T ( γ ) is ( i, j, k (cid:48) , l (cid:48) ) for j < k , or ( i, j, j (cid:48) , k (cid:48) ) (seeFigure 6). Then ˆ x q, [ γ ] is, respectively (cid:18) P il P jk P ij P kl (cid:19) ± or (cid:18) P ik P jj P ij P jk (cid:19) ± . As long as none are identically zero, X -variables from quadrilaterals involving different indicesare distinct by a similar argument as the type B n case. Note that M is invertible, so modifiedPl¨ucker coordinates P ij can be made zero by choosing either c i = M − c j or c j = M c i .18igure 6: The 6 types of quadrilaterals of P G n +2 that are not contained in δ -half-disks but arecontained in some closed half-disk. The thick vertical line is δ . The distinguished representativeof each quadrilateral is solid; the other representative is dashed.Consider a matrix with v a = (1 , , v b = (0 , , v c = (1 , , v d = ( − , P ab and P ba . Note also that P aa > 0. This showsthat no X -variable is identically zero, and also that X -variables involving the 4 same indices, butcorresponding to different quadrilaterals, are distinct.To see that X -variables containing the same 3 indices i, j, k and corresponding to differentquadrilaterals are distinct, note that they each have different values under the specialization v i =(1 , , v j = ( − , , v k = (0 , D n Let V =Mat ,n ( C ) and A := (cid:20) − (cid:21) . The eigenvalues of A are λ = 1 and λ = − a = (1 , 1) and a (cid:46)(cid:47) = (0 , − 1) are eigenvectors for λ and λ respectively. For i < j , we define a modified Pl¨ucker coordinate P ij as the determinant ofthe 2 × v j and Av i . We also use the shorthand P ia (respectively P ia (cid:46)(cid:47) ) forthe determinant of the 2 × v i and a (respectively a (cid:46)(cid:47) ).Let γ be an arc or boundary segment of P • n . Then19 γ = P ij if γ has endpoints i, j with i < j and does not cross the cut. P ij if γ has endpoints i, j with i < j and crosses the cut. P ia if γ is a plain radius with endpoints p and i.P ia (cid:46)(cid:47) if γ is a notched radius with endpoints p and i. We make a slight modification to the usual recipe for producing an A -seed pattern from thisinformation. We add two additional vertices to Q ( T ), one labeled with λ and the other with λ .Let T be the triangulation of P • n consisting only of radii. In Q ( T ), we add an arrow from λ tothe radius with endpoints 1 , p , and an arrow from the radius with endpoints n, p to λ . This isenough to determine the arrows involving λ and λ for the remaining triangulations (for example,by performing X -seed mutation on ˆΣ T ).Let i and j > i be vertices of P • n , γ be the plain radius from j to p and γ (cid:48) be the notchedversion of the same radius. Let q be the quadrilateral on i, j with γ as a diagonal, and q (cid:48) be thequadrilateral on i, j with γ (cid:48) as a diagonal. Then, by [5, Proposition 5.4.11], ˆ x q,γ and ˆ x q (cid:48) ,γ (cid:48) are,respectively, λP ij P ij , λP ij P ij . (5.7)Their inverses are the X -variables corresponding to plain and notched diagonals from i to p , re-spectively.For all other quadrilaterals, it suffices to consider the associated X -variables up to some Laurentmonomial in λ, λ . Since A is full rank, this ignored Laurent monomial will not impact the X -variables being well-defined or nonzero, and we omit it for the sake of brevity.If q has vertices i < j < k < l , it does not enclose the puncture. Let γ be a diagonal of q . Thenˆ x q,γ ∈ (cid:18) P il P jk P ij P kl (cid:19) ± , (cid:18) P il P jk P ij P kl (cid:19) ± , (cid:32) P il P jk P ij P kl (cid:33) ± , (cid:18) P il P jk P ij P kl (cid:19) ± . (5.8)If q has vertices i < j < k and has one diagonal γ that is a radius, ˆ x q,γ is one of (cid:18) P ij P ka P ia P jk (cid:19) ± , (cid:18) P jk P ia P ik P ja (cid:19) ± , (cid:18) P ik P ja P ij P ka (cid:19) ± (5.9)or one of these expressions with a (cid:46)(cid:47) instead of a .If q has vertices i < j < k and both diagonals are arcs between vertices, thenˆ x q,γ ∈ (cid:40)(cid:18) P ij P ka P ka (cid:46)(cid:47) P ik P jk (cid:19) ± , (cid:18) P ik P ja P ja (cid:46)(cid:47) P ij P jk (cid:19) ± , (cid:18) P jk P ia P ia (cid:46)(cid:47) P ik P ij (cid:19) ± (cid:41) . (5.10)As with the other types, X -variables of quadrilaterals involving different vertices are distinct,as long as none are identically zero. The matrix A is of course invertible, so modified Pl¨uckercoordinates can be made zero regardless of which index is constrained. The rest of the argument isthe same as with the other types.To see that X -variables from quadrilaterals on the same 4 vertices are distinct, notice that if v a = (1 , v b = ( − , { v c , v d } = { (0 , , (1 , } , then the only (usual or modified) Pl¨ucker20oordinate that is zero is P ab . Since each expression in (5.8) contains a unique modified Pl¨uckercoordinate, the result follows. (It is also clear that no expression is identically zero.)For X -variables from quadrilaterals on the same 3 vertices, it suffices to consider { v i , v j , v k } = { (1 , , (0 , , (1 , } . Then P ik (cid:54) = 0. By making different columns equal to a or a (cid:46)(cid:47) , one candifferentiate between any 2 expressions. (Also, under the specialization v i = (0 , v j = ( − , v k = (2 , X -variables from quadrilaterals on the same 2 vertices, the presence of λ and λ serve todistinguish between them. For example, under the specialization v i = (1 , v j = (1 , X -variables in (5.7) have different (nonzero) values. We make a few remarks regarding the implications of these results to the larger theory of X -seedpatterns and cluster algebras, and conjectural extensions.First, we conjecture that Theorem 1.1 extends to seed patterns from arbitrary marked surfaces. Conjecture 6.1. Let S be an X -seed pattern from a marked surface ( S, M ) . Then the map from { q T ( γ ) ∪ { γ }| T a triangulation of ( S, M ) , γ ∈ T } to X ( S sf ) , which sends q ∪ { γ } to x q,γ , is abijection. As remarked upon previously, a surjection from quadrilaterals (with choice of diagonal) to X -variables holds by results of [4].Theorem 1.1 implies that, for S sf of classical type, the X -variables of an X -seed determine the X -seed. Indeed, the X -variables give the quadrilaterals of a tagged triangulation and a taggedtriangulations is uniquely determined by its set of quadrilaterals. Thus each X -seed corresponds toa unique triangulation, and we have the following corollary. Corollary 6.2. The exchange graph of the X -seed pattern S sf in classical types coincides with theexchange graph of any A -seed pattern of the same type. Recall that the diagonals of a quadrilateral can be uniquely associated to a pair of A -variables.These pairs are precisely those variables that appear together on the left hand side of an exchangerelation; such A -variables are called exchangeable . Clearly, there is a bijection from ordered pairsof exchangeable A -variables to quadrilaterals with a choice of diagonal. Composing this bijectionwith the bijection of Theorem 1.1 gives Corollary 1.2 for classical types, which we give again herefor the reader’s convenience. It was checked by computer for exceptional types. Corollary 1.2. Let R be a finite type A -seed pattern. There is a bijection between ordered pairs ofexchangeable A -variables in R and X ( S sf ) . Let R is a finite type A -seed pattern over the tropical semifield with one (equivalently, every[ GSV ]) extended exchange matrix of full rank. As is remarked in [8, Section 7], in this case the X -variables in the corresponding seed of ˆ R are algebraically independent. Thus, the number of X -variables in ˆ R is |X ( S sf ) | . (Note that without this condition, the number of X -variables in ˆ R could be smaller, as Example 4.4 shows.) Recall that in this setting, the X -variables of ˆ R exactlyrecord the two terms on the right hand side of an exchange relation. In the bijection of Corollary1.2, the pairs of exchangeable A -variables are mapped to the X -variables recording the exchangerelation that the A -variables satisfy. This implies the following corollary.21 orollary 6.3. Let R be an A -seed pattern of classical type over the tropical semifield such thatone (equivalently, every) extended exchange matrix is full rank. Then the two monomials on theright hand side of an exchange relation (2.2) uniquely determine the variables being exchanged. Lastly, in the original development of finite type A -seed patterns, seed patterns were connectedto root systems of the same type. In particular, there is a bijection between A -variables and almostpositive roots (positive roots and negative simple roots). Two variables are exchangeable if andonly if the corresponding roots α, β have ( α || β ) = ( β || α ) = 1, where ( −||− ) is the compatibilitydegree [7]. This, combined with Corollary 1.2, give a root theoretic interpretation of the number of X -coordinates of S sf . Corollary 6.4. For S sf of Dynkin type, |X ( S sf ) | is the number of pairs of almost-positive roots ( α, β ) such that ( α || β ) = ( β || α ) = 1 in the root system of the same type. The author would like to thank Lauren Williams for helpful conversations and her comments ondrafts of this paper, and Dan Parker and Adam Scherlis for making their undergraduate thesesavailable. The author would also like to thank Nathan Reading for computing the number of pairsof exchangeable A -variables for seed patterns of exceptional types, thereby extending Corollaries1.2 and 6.4 to exceptional types. This research did not receive any specific grant from fundingagencies in the public, commercial, or not-for-profit sectors.An extended abstract outlining these results was accepted for presentation at FPSAC 2018. References [1] Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky. “Parametrizations of canonicalbases and totally positive matrices”. In: Adv. Math. Lie groups and lie algebras : chapters 4-6 . Berlin: Springer, 2008. isbn :978-3-540-69171-6.[3] Vladimir V. Fock and Alexander B. Goncharov. “Cluster ensembles, quantization and thedilogarithm”. In: Ann. Sci. ´Ec. Norm. Sup´er. (4) Acta Math. issn : 0001-5962.[5] Sergey Fomin, Lauren Williams, and Andrei Zelevinsky. Introduction to Cluster Algebras.Chapters 4-5. Preprint, arXiv:1707.07190. 2017.[6] Sergey Fomin and Andrei Zelevinsky. “Cluster algebras. I. Foundations”. In: J. Amer. Math.Soc. Invent. Math. Compos. Math. Journal of High Energy Physics issn : 1029-8479. doi : .[10] Mark Gross, Paul Hacking, Sean Keel, and Maxim Kontsevich. Canonical bases for clusteralgebras . Preprint, arXiv:1411.1394. 2014.[11] Daniel Parker. “Cluster algebra structures for scattering amplitudes in N = 4 Super Yang-Mills”. Unpublished undergraduate thesis. 2015.[12] Adam Scherlis. “Triangulations, Polylogarithms, and Grassmannian cluster algebras in par-ticle physics”. Unpublished undergraduate thesis. 2015.[13] Ahmet I. Seven. “Cluster algebras and symmetric matrices”. In: Proc. Amer. Math. Soc.