Geometric structures of $G$-fans associated with rank $3$ cluster-cyclic exchange matrices

GEOMETRIC STR UCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EX CHANGE MA TRICES R YOT A AKA GI AND ZHICHAO CHEN Abstract. In this pap er, we in vestigate the geometric structures of G -fans associated with rank 3 real cluster-cyclic exc hange matrices. In this class, a simple recursion for tropical signs w as found, whic h enables us to study the detailed prop erties of c -, g -v ectors. W e introduce tw o kinds of upp er b ounds of the G -fans. The ﬁrst one is the global upp er b ound, which comes from a hyperb olic surface containing all g -vectors after an initial mutation. The second one is the lo cal upp er b ound, which reﬂects the internal separateness structure. As applications, we prov e that there is no p erio dicity among g -vectors, and we completely determine the sign of g -vectors. W e also prov e the monotonicity of g -vectors under the minimum assumption. Moreo ver, w e sho w that the three global upp er b ounds can b e simpliﬁed to a single uniform upp er b ound. Keyw ords: G -fan, global upp er b ound, lo cal upp er b ound, p erio dicity , monotonicity . 2020 Mathematics Sub ject Classiﬁcation: 13F60, 05E10, 05E45. Contents 1. In tro duction 2 1.1. Bac kground 2 1.2. Purp ose and related works 4 1.3. Con v entions 4 1.4. Main results 5 1.5. Structure of this pap er 7 2. Preliminaries 8 2.1. B -, C -, G -matrices 8 2.2. G -fan structure 10 2.3. Sk ew-symmetrizing metho d 12 2.4. Mo diﬁed c -, g -v ectors 12 2.5. Stereographic pro jection 13 3. Rank 3 cluster-cyclic framework 14 3.1. Mark o v constan t and cluster-cyclicity 14 3.2. Mutation of tropical signs 17 3.3. Mutation formulas 20 Date : March 18, 2026. 1 R YOT A AKAGI AND ZHICHA O CHEN 2 3.4. Initial setup 22 4. Global upp er b ound 23 4.1. Quadratic surface for g -vectors 23 4.2. General fact of the quadratic surface 24 4.3. Main theorem 29 5. S -mutations 30 5.1. F ormulas for S -m utations 30 5.2. Asymptotic structure 32 6. Supp ort of trunks 33 7. Lo cal upp er b ounds of branches 34 8. Separateness among lo cal upp er b ounds 39 8.1. Relations among maximal branches 39 8.2. Pro of of Prop osition 8.1 44 9. Applications to g -v ectors 45 9.1. Non-p erio dicity of g -v ectors 45 9.2. Sign structure of g -vectors 46 10. Under the minimum assumption 49 10.1. Minim um assumption 49 10.2. Monotonicit y of g -v ectors 52 10.3. Simpliﬁcation of global upp er b ounds 55 Ac kno wledgements 56 References 57 1. Introduction 1.1. Bac kground. Cluster algebras are a class of commutativ e rings equipp ed with cluster v ariables, which are rational functions, and mutations, which are their transformation rules. This concept w as introduced b y [ FZ02 ] and has been dev elop ed b y man y researc hers. Now ada ys, related structures hav e b een discov ered in a wide range of areas in mathematics. It is known that cluster algebras p ossess many in teresting com binatorial structures. One of the most imp ortant ob jects is the g -ve ctors . Roughly sp eaking, these are vectors deﬁned b y a certain “degree” of cluster v ariables. Although g -v ectors are deﬁned using the speciﬁc information from cluster v ariables, it is a surprising fact that they inherit information ab out the perio dicity of cluster v ariables [ CIKLFP13 , CL20 , Nak23 ]. Th us, they pro vide a natural parametrization of cluster v ariables. It is also known that the g -vectors form an imp ortant com binatorial structure called the G - fans (or the g -ve ctor fans ), whic h are simplicial fans in a Euclidean space. This is a geometric GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 3 Figure 1. G -fan corresp onding to a cluster-cyclic exchange matrix. realization of the cluster complex in [ FZ07 ]. The existence of this concept was conjectured by [ FZ07 ], which w as even tually pro v ed b y [ GHKK18 ], and it has b een studied b y man y researc hers (e.g. [ Rea14 , Nak23 , Nak24 , Y ur25 ]). This concept is strongly related to the p ointe d b ases , whic h are a class of bases in cluster algebras parametrized by Z n . According to [ Qin24 ], such bases m ust con tain all cluster monomials, and each pair of p ointed bases can b e transformed via certain linear com binations. Since all cluster monomials are parametrized by the p oints in the G -fan, the problem of constructing a p ointed basis can b e reduced to the problem of adding elemen ts parametrized by p oints in the complimen t of the G -fan. Recently , from this p oint of view, the dominanc e r e gion was deﬁned by using the G -fan, and studied in [ RS23 , RRS25 ]. Through the cluster sc attering diagr ams and the theta b ases in [ GHKK18 ], the G -fans play essen tial roles in the construction of a p ointed basis. The cluster scattering diagrams are wall- crossing structures that naturally contain the G -fans. The theta basis is one kind of the pointed bases. The construction of the theta basis strongly depends on the geometry of the cluster scattering diagrams. Hence, it is a central topic in cluster algebra theory to understand the G -fans. R YOT A AKAGI AND ZHICHA O CHEN 4 Although the G -fans hav e imp ortant information of cluster algebras, it seems that they app ear to b e highly complex and v aried. Due to this complexity , there has b een limited progress in the explicit understanding. In some sp eciﬁc cases, there are several nice results. F or example, for the rank 2 cluster algebras, all the G -fans are explicitly understo o d in [ Rea14 ]. Also, the relationships b etw een the G -fans (or scattering diagrams) and the ro ots of Kac-Mo o dy Lie algebras were studied in [ RS16 , RS18 , Rea20a , Rea20b , RS22 ] for cases with acyclic and aﬃne initial exchange matrices, and in [ Y ur20 , Rea23 ] for the case arising from the surface. Based on the observ ations from these results, it seems quite diﬃcult to connect the kno wn theory with the G -fans if the initial exchange matrix has an orien ted cycle. Recen tly , in [ Nak24 ], sev eral prototypical examples of rank 3 G -fans were presented, along with a conjecture for the classiﬁcation of G -fans. 1.2. Purp ose and related works. In this pap er, w e inv estigate the structure of the G -fan asso ciated with the cluster-cyclic exchange matrices of rank 3. This is an exc hange matrix whose mutation equiv alen t exchange matrices all hav e an oriented cycle. As stated ab ov e, this class is rather diﬃcult to study via existing theories. On the other hand, this class has an imp ortan t example called the Markov quiver , see ( 3.3 ). Hence, it is w orthwhile to explore a new approac h to this class. Although it is hard to establish the connections betw een this class and another theory , several unique and in teresting phenomena ha ve already b een found. Firstly , it is known that the clas- siﬁcation of these exchange matrices can b e done via the Markov c onstant [ BBH11 , Ak a24 ], see Prop osition 3.4 . In [ Sev12 ], another classiﬁcation w as pro vided by the distribution of eigen v alues of the corresp onding pseudo Cartan companion, and w e reveal their detailed prop erty in this pap er, see Lemma 4.5 . Secondly , in [ LL24 ], it was shown that all g -vectors are con tained in a quadratic surface arising from the pseudo Cartan companion. W e reﬁne this theorem in Theo- rem 4.12 . Lastly , and most imp ortantly , in [ AC25b ], the sign-coherence of c -v ectors w as shown ev en when r e al en tries are allo w ed, and a recursion for their signs w as obtained, see Theorem 3.7 . The example of the G -fan or the scattering diagram asso ciated with the Marko v quiver is presen ted in [ Ch´ a12 , FG16 , Rea23 ]. In [ Nak24 , Section 6], it was conjectured that the class of cluster-cyclic exchange matrices provides one class for the classiﬁcation of the G -fans. Al- though w e ha ve not got the explicit deﬁnition how to classify the G -fans, our results provide an aﬃrmativ e answ er to this conjecture. 1.3. Con v en tions. A sequence w “ r k 1 , . . . , k r s of 1 , 2 , 3 is said to b e r e duc e d if k i ‰ k i ` 1 for an y i “ 1 , 2 , . . . , r ´ 1. By conv en tion, the empty sequence H “ r s is also reduced. W e write the set of all reduced sequences by T . Then, for eac h j “ 1 , 2 , 3 and w P T , we can assign c -v ectors c w j and g -v ectors g w j follo wing the standard deﬁnition, see Deﬁnition 2.2 . F or any w “ r k 1 , . . . , k r s P T and l “ 1 , 2 , 3, deﬁne w r l s P T as follows: w r l s “ r k 1 , . . . , k r , l s if k r ‰ l , and w r k s “ r k 1 , . . . , k r ´ 1 s if k r “ l . Rep eating this pro cess, w e deﬁne the pro duct of t w o reduced sequences w r l 1 , . . . , l m s “ w r l 1 sr l 2 s . . . r l m s . GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 5 W e deﬁne the partial order ď on T as follo ws: r k 1 , . . . , k r s ď r l 1 , . . . , l r 1 s ð ñ r ď r 1 and k i “ l i for any i “ 1 , . . . , r . (1.1) F or an y w P T , deﬁne the subset T ě w of T by T ě w “ t u P T | u ě w u . (1.2) W e sp ecify the following points, whic h sligh tly diﬀer from conv entions used in previous w orks: ‚ Gener alization to r e al entries : Since the sign-coherence has b een established for this class allo wing for real entries, w e handle the G -fan in this broader setting. This generalization w as in tro duced in [ A C25a ] with sev eral conjectures, though those conjectures hav e b een solv ed for the class of exc hange matrices studied in this pap er. ‚ Mo diﬁe d c -, g -ve ctors : Instead of usual c -, g -v ectors, we employ the mo diﬁe d c -ve ctors ˜ c w j and the mo diﬁe d g -ve ctors ˜ g w j in tro duced by [ A C25a ], see Deﬁnition 2.19 . They are deﬁned b y m ultiplying the original vectors by certain p ositive scalars; this mo diﬁcation do es not aﬀect the structure of the G -fan. How ev er, to reco v er the results for the original c -, g -v ectors, certain adjustments are required. ‚ K , S , T -lab eling : T o indicate the m utation directions and indices, we use the symbol K , S , T instead of 1, 2, 3, see Deﬁnition 3.9 . This lab eling is compatible with the recursion for the signs of c -v ectors obtained in [ AC25b ]. T o explain the main theorem, w e in tro duce some notations. Let M b e the free monoid generated b y S and T . W e introduce the right monoid action of M on T ztHu by ( 3.19 ). Then, every nonempt y reduced sequence w P T ztHu is uniquely expressed as w “ r i s X for some i “ 1 , 2 , 3 and X P M . See Deﬁnition 3.9 for the precise deﬁnition. In this expression, we hav e T ěr i s “ tr i s X | X P M u . W e decomp ose this set in to the following tw o parts: ‚ T runk : T ăr i s S 8 “ tr i s S n | n P Z ě 0 u . ‚ Maximal br anches : T ěr i s S n T “ tr i s S n T X | X P M u with n P Z ě 0 . See Figure 3 as an example. F or any w in a maximal branc h, the set T ě w is called a br anch . 1.4. Main results. Let B b e a cluster-cyclic initial exchange matrix of rank 3, and let ∆ p B q b e the G -fan asso ciated with B . F or an y reduced sequence w , we deﬁne the sub G -fan ∆ ě w p B q , whic h consists of the cones spanned b y ˜ g u 1 , ˜ g u 2 , and ˜ g u 3 with u P T ě w . When T ě w is a branch, the sub G -fan ∆ ě w p B q is also called a branch. The ﬁrst main theorem in troduces the glob al upp er b ound , which is an upp er b ound depending only on the initial mutation direction i “ 1 , 2 , 3. See the thick blue lines in Figure 1 . By [ LL24 ], it is known that all mo diﬁed g -v ectors with the initial mutation direction i are on a quadratic surface H i (Lemma 4.4 ). Note that this quadratic surface is deﬁned by the mutated exchange matrix B r i s “ µ i p B q , not the initial one B . W e pro v e that this quadratic surface is a h yp erb oloid of t wo sheets, or a degeneration thereof, which consists of tw o connected comp onen ts separated b y a plane orthogonal to a sign-coherent vector (Lemma 4.5 ). W e decomp ose H i in to the R YOT A AKAGI AND ZHICHA O CHEN 6 “p ositiv e” part H ` i and the “negative” part H ´ i . See Figure 6 and Figure 7 as examples. Let Q ` i b e the minimum conv ex cone including H ` i and 0 . See Deﬁnition 4.8 for the precise deﬁnition. Then, the ﬁrst main theorem is that Q ` i is an upp er b ound for ∆ ěr i s p B q . Theorem 1.1 (Theorem 4.12 ) . The fol lowing inclusion holds: | ∆ ěr i s p B q| Ă Q ` i . (1.3) In p articular, we have | ∆ p B q| Ă Q ` 1 Y Q ` 2 Y Q ` 3 . The second main theorem introduces the lo c al upp er b ound V w , which serves as an upp er b ound for each branch ∆ ě w p B q , see Deﬁnition 7.3 . This set can b e computed solely from lo cal information at w , such as the mo diﬁed g -vectors ˜ g w 1 , ˜ g w 2 , ˜ g w 3 and the exchange matrix B w . Let V w ˝ b e the interior of V w . Then, the following statements hold. Theorem 1.2 (Theorem 7.4 ) . F or any br anch ∆ ě w p B q , the fol lowing inclusion holds. | ∆ ě w p B q| Ă V w . (1.4) Mor e over, ab out the next lo c al upp er b ounds V w S , V w T , the fol lowing pr op erties hold. ( a ) Their interiors V w S ˝ and V w T ˝ ar e sep ar ate d by the plane H D p ¯ c w q . ( b ) The lo c al upp er b ounds ar e monotonic al ly de cr e asing V w Ą V w S , V w T . In Figure 1 , the blue dashed lines represen t the lo cal upper bounds for the ro ots of the maximal branc hes w “ r i s S n T , while the red lines corresp ond to the planes H D p ¯ c w q in this theorem. This theorem allo ws us to appro ximate the sub G -fan ∆ ě w p B q , whic h consists of inﬁnitely man y cones, by only one simplicial cone V w (excluding certain b oundaries). The claim ( b ) implies that the accuracy of this approximation improv es with m utations. This situation is illustrated in Figure 11 . Based on these tw o theorems and the preparatory result in Prop osition 8.1 , we derive several prop erties of g -vectors. W e sa y that an equality g w l “ g u m ( w , u P T , l , m “ 1 , 2 , 3) of g -vectors is trivial if l “ m and u “ w r k 1 , . . . , k r s with k i ‰ l . Roughly sp eaking, a nontrivial e quality me ans p erio dicity among g -ve ctors with r esp e ct to the mutations . The following theorem implies that there is no p erio dicit y among g -v ectors asso ciated with a cluster-cyclic exc hange matrix of rank 3. Theorem 1.3 (Theorem 9.1 ) . Al l e qualities g w l “ g u m of g -ve ctors ar e trivial. Since the cluster v ariables and the g -v ectors share the same p erio dicity , this theorem can b e restated as the one for cluster v ariables, see Corollary 9.2 . Next application is for the signs of g -vectors. In the ordinary cluster algebra theory , it is kno wn that ev ery G -cone is a subset of one orthan t [ GHKK18 ]. This prop erty is called the sign-c oher enc e of G -matric es , which implies that ev ery row vector of each G -matrix has the same sign, and for the cluster-cyclic exc hange matrix of rank 3, it still holds even when w e GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 7 consider the generalization to real entries [ A C25b ]. The sign of the j th row of G w is denoted b y τ w j P t˘ 1 u . In fact, as exp ected from Figure 1 , the explicit structure of these signs can b e completely expressed. Theorem 1.4 (Theorem 9.3 ) . The signs of G -matric es ar e given in T able 2 . Later, we obtain some speciﬁc phenomena under the minimum assumption , whic h means that the initial exc hange matrix is the minim um element in its mutation equiv alence class. It is known that there is the minimum element in the mutation equiv alence class of each inte ger cluster-cyclic exc hange matrix. See Lemma 10.3 for the precise statemen t. Hence, it is worth considering the structure under this assumption. Firstly , we establish the monotonicit y of g -v ectors. Theorem 1.5 (Theorem 10.7 ) . Under the minimum assumption, for any j “ 1 , 2 , 3 and u , w P T , if w ď u , it holds that abs p g w j q ď abs p g u j q , (1.5) wher e abs p g q “ p| g 1 | , | g 2 | , | g 3 |q J for g “ p g 1 , g 2 , g 3 q J P R 3 . Recall the global upp er b ounds Q i in Theorem 1.1 . As stated b efore Theorem 1.1 , this upp er b ound dep ends on the initial mutation direction i “ 1 , 2 , 3. On the other hand, w e can also deﬁne Q ` initial as the same wa y with resp ect to the initial exchange matrix. In general, this is not an upper b ound of the G -fan. How ev er, under the minim um assumption, w e can obtain the follo wing inclusion. Theorem 1.6 (Theorem 10.11 ) . Under the minimum assumption, the fol lowing inclusion holds. | ∆ p B q| Ă Q ` initial . (1.6) 1.5. Structure of this pap er. This pap er is organized as follows. In Section 2 , we recall the basic notations for the G -fans. In Section 3 , we recall the recursion for the signs of c -v ectors (Theorem 3.7 ), and simplify some notations b y using the sp eciﬁc structure of the rank 3 cluster-cyclic exchange matrices. In Section 4 , we introduce the global upp er b ounds (Theorem 4.12 ). In Section 5 , we fo cus on the prop erties of S -mutations. In Section 6 , w e give the explicit expressions in trunks. In Section 7 , w e introduce the lo cal upp er b ounds (Theorem 7.4 ). In Section 8 , w e prov e the separateness among lo cal upp er b ounds. In Section 9 , w e giv e some applications about exhibiting the non-perio dicity (Theorem 9.1 ) and the signs of G -matrices (Theorem 9.3 ). In Section 10 , we show some prop erties under the minimum assumption, including the monotonicity of g -vectors (Theorem 10.7 ) and the simpliﬁcation of the global upp er b ound (Theorem 10.11 ). R YOT A AKAGI AND ZHICHA O CHEN 8 2. Preliminaries In this pap er, w e alw a ys consider the matrices asso ciated with rank 3 cluster algebras unless stated otherwise, whose entries are usually integers but are allo wed to b e real num bers in more general cases. F ollo wing [ Nak23 ], we also in tro duce the basic notations for the G -fans. 2.1. B -, C -, G -matrices. A matrix B P M 3 p R q is said to b e skew-symmetrizable if there exists a p ositive diagonal matrix D “ diag p d 1 , d 2 , d 3 q ( d 1 , d 2 , d 3 ą 0) suc h that D B is skew-symmetric. F or a 3 ˆ 3 matrix B , it is kno wn that this condition is equiv alen t to the sign-skew-symmetry (sign p b ij q “ ´ sign p b j i q for any i, j ) and the following equality [ FZ03 , Lem. 7.4]: | b 12 b 23 b 31 | “ | b 21 b 32 b 13 | . (2.1) In the cluster algebra theory , a skew-symmetrizable matrix is called an exchange matrix , and w e also use this terminology dep ending on the context. F or any real num b er x P R , deﬁne r x s ` “ max p x, 0 q . Recall that T denotes the set of all reduced sequences of 1, 2, 3 (Subsection 1.3 ). Then, the mutation and the B -pattern are deﬁned as follows. Deﬁnition 2.1. Let B “ p b ij q P M 3 p R q b e a skew-symmetrizable matrix. F or any k “ 1 , 2 , 3, the k -dir e ction mutation µ k p B q “ p b 1 ij q of B is deﬁned as: b 1 ij “ # ´ b ij if i “ k or j “ k , b ij ` sign p b ik qr b ik b kj s ` if i ‰ k and j ‰ k . (2.2) Note that µ k is an inv olution. F or an y w “ r k 1 , . . . , k r s P T , we deﬁne B w “ µ k r ¨ ¨ ¨ µ k 1 p B q . The collection of all such matrices B p B q “ t B w u w P T is called the B -p attern or the mutation e quivalenc e class . F or this pattern, the giv en B is called the initial exchange matrix of B p B q . If B and B 1 form the same mutation equiv alence class B p B q “ B p B 1 q as a set, these tw o matrices are said to b e mutation e quivalent . Deﬁnition 2.2. Let B P M 3 p R q b e an initial exc hange matrix. W e deﬁne the C -matric es C w “ p c w ij q P M 3 p R q and the G -matric es G w “ p g w ij q P M 3 p R q by the initial conditions C H “ G H “ I (the identit y matrix) and the follo wing recursions: c w r k s ij “ $ & % ´ c w ik j “ k , c w ij ` c w ik r b w kj s ` ` r´ c w ik s ` b w kj j ‰ k , g w r k s ij “ $ & % ´ g w ik ` ř 3 l “ 1 g w il r b w lk s ` ´ ř 3 l “ 1 b H il r c w lk s ` j “ k , g w ij j ‰ k . (2.3) The collections of all C -matrices and G -matrices are called the C -p attern and the G -p attern , and w e denote them by C p B q “ t C w u w P T and G p B q “ t G w u w P T , resp ectively . Each column v ector GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 9 of C w (resp. G w ) is called a c -ve ctor (resp. g -ve ctor ), and w e denote them b y C w “ p c w 1 , c w 2 , c w 3 q and G w “ p g w 1 , g w 2 , g w 3 q . Remark 2.3. The c -, g -v ectors were introduced in [ FZ07 ] to record the mutation dynamics of co eﬃcients through the tropical semiﬁeld and describ e the Z n -grading on cluster v ariables resp ectiv ely . They play a central role in understanding the sign-coherence, separation formulas, and the parameterization of cluster v ariables. W e introduce a partial order ď on R 3 as follo ws: p u 1 , u 2 , u 3 q J ď p v 1 , v 2 , v 3 q J if and only if u i ď v i for any i “ 1 , 2 , 3. The follo wing is a key notion for C -matrices and G -matrices. Deﬁnition 2.4. Let B P M 3 p R q b e an initial exc hange matrix. Then, a C -matrix C w is said to b e sign-c oher ent if either c w i ě 0 or c w i ď 0 holds for any i “ 1 , 2 , 3. The C -pattern C p B q is called sign-c oher ent if every C w is sign-coheren t. When C w is sign-coheren t, the tr opic al sign ε w i P t 0 , ˘ 1 u of c w i is deﬁned as follows: ε w i “ 0 if c w i “ 0 , otherwise ε w i P t˘ 1 u such that ε w i c w i ě 0 . Remark 2.5. In [ GHKK18 ], it was shown that every C -pattern corresp onding to an inte ger sk ew-symmetrizable matrix is sign-coheren t. How ev er, when real entries are allo w ed, it is known that there exists a sign-incoheren t C -pattern [ AC25a , Ex. 5.4], and it is an op en problem when the sign-coherence holds. Under this assumption, we can obtain the recursion for c -, g -vectors. Prop osition 2.6 ([ FZ07 ]) . L et B P M 3 p R q b e an initial exchange matrix. If C p B q is sign- c oher ent, we obtain a r e cursion for c -, g -ve ctors as c w r k s i “ $ & % ´ c w k i “ k , c w i ` r ε w k b w ki s ` c w k i ‰ k , (2.4) g w r k s i “ $ & % ´ g w k ` ř 3 j “ 1 r´ ε w k b w j k s ` g w j i “ k , g w i i ‰ k . (2.5) As a consequence of this recursion, we obtain the following imp ortant prop erties. Prop osition 2.7 ([ FZ07 ]) . Supp ose that C p B q is sign-c oher ent. F or any w P T , we have | C w | “ | G w | “ p´ 1 q | w | . In p articular, the sets of c -ve ctors t c w 1 , c w 2 , c w 3 u and g -ve ctors t g w 1 , g w 2 , g w 3 u ar e b ases of R 3 . When w e discuss real C -, G -matrices, the following c onjecture is important but not shown in general. How ev er, for our fo cusing matrices which are said to b e cluster-cyclic of rank 3, it has already b een prov ed in [ AC25b ], see also Prop osition 3.8 . Conjecture 2.8 ([ A C25a , Conj. 6.1 & Conj. 6.3]) . L et B P M 3 p R q b e an initial exchange matrix. Supp ose that its C -p attern C p B q is sign-c oher ent. R YOT A AKAGI AND ZHICHA O CHEN 10 ( a ) F or any w P T , al l C -p atterns C p B w q and C pp B w q J q ar e sign-c oher ent. ( b ) F or any w P T , let B 1 “ B w . Consider its C -p attern C p B 1 q “ t C u B 1 u u P T . If its c -ve ctor c u i ; B 1 is expr esse d as c u i ; B 1 “ α e j for some α P R and j P t 1 , . . . , n u , then we have α “ ˘ b d i d ´ 1 j . 2.2. G -fan structure. In this section, we ﬁx an initial exchange matrix B P M 3 p R q and its sk ew-symmetrizer D “ diag p d 1 , d 2 , d 3 q . W e supp ose that C p B q is sign-coherent. In this case, there are some go o d geometric structures in c -, g -vectors. W e in tro duce an inner pro duct x , y D in R 3 as x a , b y D “ a J D b . (2.6) Then, the following duality b et ween c -, g -v ectors holds. Prop osition 2.9 ([ Nak23 , Prop. 2.16]) . Supp ose that C p B q is sign-c oher ent. Then, for any w P T and i, j “ 1 , 2 , 3 , we have x g w i , c w j y D “ $ & % d i if i “ j , 0 if i ‰ j . (2.7) Motiv ated b y this equality , w e alwa ys ﬁx one skew-symmetrizer D and the corresp onding inner pro duct x , y D in R 3 . Then, we introduce the follo wing notions. Deﬁnition 2.10 (Conv ex cone) . A nonempty set C Ă R 3 is called a c onvex c one if λ a ` λ 1 b P C holds for any a , b P C and λ, λ 1 P R ą 0 . F or any con v ex cone C Ă R 3 , we write its r elative interior b y C ˝ , which is the interior of C in the subspace x C y vec Ă R 3 . F or any con vex cone C Ă R 3 , its dimension dim p C q is deﬁned by the dimension dim x C y vec of the vector subspace spanned by C . F or an y ﬁnite set of vectors a 1 , . . . , a r P R 3 , we deﬁne the follo wing t w o con v ex cones: C p a 1 , . . . , a r q “ # r ÿ i “ 1 λ i a i ˇ ˇ ˇ ˇ ˇ λ i ě 0 + , C ˝ p a 1 , . . . , a r q “ # r ÿ i “ 1 λ i a i ˇ ˇ ˇ ˇ ˇ λ i ą 0 + . (2.8) Note that C ˝ p a 1 , . . . , a r q is the relativ e interior of C p a 1 , . . . , a r q . In con v en tion, w e write C pHq “ t 0 u . W e say that a conv ex cone of the form C p a 1 , . . . , a r q is a p olyhe dr al c one , or simply a c one . If a 1 , . . . , a r are linearly indep enden t, then the cone C p a 1 , . . . , a r q said to b e simplicial . In this case, r ď 3 holds. F or an y v ector v P R 3 , set H D p v q “ t x P R 3 | x x , v y D “ 0 u , H ` D p v q “ t x P R 3 | x x , v y D ą 0 u , H ´ D p v q “ t x P R 3 | x x , v y D ă 0 u . (2.9) Note that H D p 0 q “ R 3 . Also, w e set H ` D p v q “ H ` D p v q Y H D p v q and H ´ D p v q “ H ´ D p v q Y H D p v q . When v ‰ 0 , they represen t the corresp onding closures. GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 11 When d 1 “ d 2 “ d 3 “ 1, that is, when the inner pro duct coincides with the Euclidean inner pro duct, w e omit D and write H p v q , H ` p v q , and so on. Deﬁnition 2.11 (F ace) . Let C Ă R 3 b e a con v ex cone. Let v P R 3 and supp ose that C Ă H ` D p v q . Then, the following set face v p C q Ă C is called a fac e of C asso ciated with a normal ve ctor v . face v p C q “ C X H D p v q . (2.10) Note that C “ face 0 p C q holds. Hence, eac h cone is a face of itself. Let C “ C p a 1 , . . . , a r q b e a simplicial cone generated by linearly indep endent vectors a 1 , . . . , a r , where r ď 3. F or any J Ă t 1 , . . . , r u , we write C J p a 1 , . . . , a r q “ # ÿ j P J λ j a j ˇ ˇ ˇ ˇ ˇ λ j ě 0 + . (2.11) Then, every face of a simplicial cone C “ C p a 1 , . . . , a r q may b e expressed as the ab ov e form. In fact, for any v P R 3 satisfying C Ă H ` D p v q , by setting J “ t j P t 1 , 2 , . . . , r u | a j P H D p v qu , we ha v e face v p C q “ C J p a 1 , . . . , a r q . Deﬁnition 2.12 (F an) . A nonempt y set ∆ of cones in R 3 is called a fan if it satisﬁes the follo wing t w o conditions: ‚ F or an y C P ∆, every face of C is also con tained in ∆. ‚ F or an y C 1 , C 2 P ∆, then C 1 X C 2 is a face of b oth C 1 and C 2 . F or any fan ∆, its supp ort | ∆ | is deﬁned b y Ť C P ∆ C . Moreov er, a fan ∆ is said to b e simplicial if every cone C P ∆ is simplicial. In the cluster algebra theory , it is known that g -vectors form a fan structure. Deﬁnition 2.13 ( G -cone, G -fan) . Let B P M 3 p R q b e an initial exchange matrix. Supp ose that C p B q is sign-coherent. Then, we write C p G w q “ C p g w 1 , g w 2 , g w 3 q , (2.12) and call it a G -c one . Moreo ver, for any J Ă t 1 , 2 , 3 u , we write C J p G w q “ C J p g w 1 , g w 2 , g w 3 q . The set of all G -cones and their faces ∆ p B q “ t C J p G w q | J Ă t 1 , 2 , 3 u , w P T u (2.13) is called the G -fan asso ciated with B . F or an y w 0 P T , we deﬁne the sub G -fan ∆ ě w 0 p B q by ∆ ě w 0 p B q “ t C J p G w q | J Ă t 1 , 2 , 3 u , w ě w 0 u . (2.14) Prop osition 2.14 ([ FZ07 , Rea14 , Nak23 ], [ AC25a , Thm. 8.3]) . L et B P M 3 p R q b e an initial exchange matrix. Supp ose that C p B q is sign-c oher ent and Conje ctur e 2.8 holds for this B . Then, the G -fan ∆ p B q is r e al ly a fan. R YOT A AKAGI AND ZHICHAO CHEN 12 2.3. Sk ew-symmetrizing metho d. The matrix patterns are deﬁned b y the sk ew-symmetrizable matrix B . On the other hand, the matrix patterns corresp onding to B are closely related to the ones corresp onding to the follo wing sk ew-symmetric matrix ˜ B . Deﬁnition 2.15. F or an y skew-symmetrizable matrix B P M 3 p R q with a skew-symmetrizer D “ diag p d 1 , d 2 , d 3 q , let ˜ B “ D 1 2 B D ´ 1 2 , where D 1 2 “ diag p ? d 1 , ? d 2 , ? d 3 q and D ´ 1 2 “ p D 1 2 q ´ 1 . Then, for any w “ r k 1 , k 2 , . . . , k r s P T , w e deﬁne ˜ B w “ µ k r ¨ ¨ ¨ µ k 1 p ˜ B q . These matrices can easily b e calculated by the follo wing rule. Lemma 2.16 ([ FZ03 , Lem. 8.3]) . L et B P M 3 p R q b e an initial exchange matrix with a skew- symmetrizer D . Set B w “ p b w ij q P M 3 p R q and ˜ B w “ p ˜ b w ij q P M 3 p R q . Then, we have ˜ B w “ D 1 2 B w D ´ 1 2 , ˜ b w ij “ sign p b w ij q b | b w ij b w j i | . (2.15) In p articular, ˜ B w is indep endent of the choic e of D and it is always skew-symmetric. Based on this fact, w e write Sk p B q “ D 1 2 B D ´ 1 2 . Since this matrix is indep enden t of the c hoice of D , the op erator Sk can b e seen as a function on the set of all skew-symmetrizable matrices. No w, we ma y consider the tw o C -, G -patterns. One is C p B q “ t C w u and G p B q “ t G w u , and the other is C p Sk p B qq “ t ˆ C w u and G p Sk p B qq “ t ˆ G w u . Let ˆ C w “ p ˆ c w 1 , ˆ c w 2 , ˆ c w 3 q and ˆ G w “ p ˆ g w 1 , ˆ g w 2 , ˆ g w 3 q . Then, the following relations hold. Prop osition 2.17 ([ AC25a ]) . F or any w P T and i “ 1 , 2 , 3 , we have ˆ C w “ D 1 2 C w D ´ 1 2 , ˆ G w “ D 1 2 G w D ´ 1 2 , ˆ c w i “ 1 ? d i D 1 2 c w i , ˆ g w i “ 1 ? d i D 1 2 g w i . (2.16) In p articular, for the G -fans, ther e is a one-to-one c orr esp ondenc e ∆ p B q Ñ ∆ p Sk p B qq given by C J p G w q ÞÑ C J p ˆ G w q “ D 1 2 C J p G w q p J Ă t 1 , 2 , 3 u , w P T q . (2.17) Remark 2.18. Thanks to Prop osition 2.17 , when we consider the C -, G -matrices asso ciated with a r e al exc hange matrix B , it suﬃces to fo cus on the ones asso c iated with the sk ew- symmetric matrix Sk p B q . Ho wev er, since the standard theory of cluster algebras is formulated for skew-symmetrizable matrices, w e provide the subsequen t pro ofs and statements in the skew- symmetrizable setting to ensure broader applicability . 2.4. Mo diﬁed c -, g -v ectors. When w e discuss the G -fan structure, the following mo diﬁed v ectors are more helpful than the ordinary c -, g -vectors. Deﬁnition 2.19. Let B P M 3 p R q b e an initial exc hange matrix with a skew-symmetrizer D “ diag p d 1 , d 2 , d 3 q . W e deﬁne the mo diﬁe d C -matrix ˜ C w and the mo diﬁe d G -matrix ˜ G w b y ˜ C w “ C w D ´ 1 2 , ˜ G w “ G w D ´ 1 2 . (2.18) GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 13 Their column vectors are called mo diﬁe d c -ve ctors ˜ c w i and mo diﬁe d g -ve ctors ˜ g w i , whic h are giv en b y ˜ c w i “ 1 ? d i c w i , ˜ g w i “ 1 ? d i g w i . (2.19) As in Deﬁnition 2.13 , w e may consider the cone C J p ˜ G w q ( J Ă t 1 , 2 , 3 u ) spanned by its modiﬁed g -vectors. How ev er, by ( 2.19 ), it holds that C J p G w q “ C J p ˜ G w q . (2.20) W e deﬁne the mo diﬁe d standar d ve ctors ˜ e i “ b d ´ 1 i e i for any i “ 1 , 2 , 3. Then, we ma y obtain the following recursion. Prop osition 2.20 ([ A C25a ]) . L et B P M 3 p R q b e an initial exchange matrix. Supp ose that C p B q is sign-c oher ent. L et ˜ B w “ p ˜ b w ij q P M 3 p R q b e the skew-symmetric matrix deﬁne d in Deﬁnition 2.15 . Then, the mo diﬁe d c -, g -ve ctors may b e obtaine d by the fol lowing r e cursions. ‚ ˜ c H i “ ˜ g H i “ ˜ e i . ‚ F or any w P T and k “ 1 , 2 , 3 , we have ˜ c w r k s i “ $ & % ´ ˜ c w k i “ k , ˜ c w i ` r ε w k ˜ b w ki s ` ˜ c w k i ‰ k , ˜ g w r k s i “ $ & % ´ ˜ g w k ` ř 3 j “ 1 r´ ε w k ˜ b w j k s ` ˜ g w j i “ k , ˜ g w i i ‰ k . (2.21) One adv antage of using mo diﬁed v ectors is that certain formulas b ecome simpler. F or example, b y ( 2.7 ), the standard duality holds as follows. Prop osition 2.21 ([ AC25a ]) . The fol lowing e quality holds: x ˜ g w i , ˜ c w j y D “ $ & % 1 if i “ j , 0 if i ‰ j . (2.22) 2.5. Stereographic pro jection. T o draw the G -fan in R 3 , we often use its stereographic pro- jection. W e explain how to dra w such pictures, whic h can also b e referred to [ Mul16 , Rea23 , Nak23 , Nak24 ]. (1) Firstly , w e consider the pro jection of G -fan to the unit sphere S 2 in R 3 . (2) Secondly , w e use the stereographic pro jection to the tangent plane P of S 2 at p ? 3 3 , ? 3 3 , ? 3 3 q from the antipo de N “ p´ ? 3 3 , ´ ? 3 3 , ´ ? 3 3 q . This map is a con tinuous bijection betw een a certain pro jectivization p R 3 z C p N qq{ R ą 0 – S 2 zt N u and the plane P – R 2 . In the con text of the G -fan, the eliminated p oint C p N q do es not aﬀect the essential structure. This is b ecause the negative orthant R 3 ď 0 is known to b e either a single c ham b er or the compliment of the G -fan (e.g. [ NZ12 ], [ AC25a , Prop. 5.4]). In this image, eac h half line in R 3 is sent to a p oint, and each plane is sent to a circle. One imp ortan t fact for the G -fan is that each 3-dimensional cone app ears as a triangle. R YOT A AKAGI AND ZHICHAO CHEN 14 p` , ´ , ´q p´ , ` , ´q p´ , ´ , `q e 1 e 2 e 3 ´ e 1 ´ e 2 ´ e 3 p` , ` , `q p` , ´ , `q p` , ` , ´q p´ , ` , `q Figure 2. The orthants in the stereographic pro jection F or example, the three planes H p e 1 q , H p e 2 q , and H p e 3 q are illustrated as the three circles in Figure 2 , and eac h connected comp onen t in this picture represents an orthan t. The center triangle is the cone C p e 1 , e 2 , e 3 q . This is a cham b er of every G -fan b ecause the initial matrix is the identit y . 3. Rank 3 cluster-cyclic framework In this section, w e in tro duce rank 3 cluster-cyclic exc hange matrices and its classiﬁcation, and presen t the corresp onding mutation form ulas, which are simpliﬁed due to the sp eciﬁc nature of the cluster-cyclic structure. 3.1. Mark o v constant and cluster-cyclicit y. F rom now on, w e alwa ys fo cus on rank 3 cluster-cyclic exchange matrices throughout this pap er, which is deﬁned as follows. GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 15 Deﬁnition 3.1. Let B P M 3 p R q b e a skew-symmetrizable matrix. This matrix is said to b e cyclic if it holds that sign p B q “ ˘ ¨ ˚ ˝ 0 ` ´ ´ 0 ` ` ´ 0 ˛ ‹ ‚ . (3.1) (W e assume that all oﬀ-diagonal en tries are nonzero.) F or any initial exchange matrix B P M 3 p R q , if every B w ( w P T ) is cyclic, B is said to b e cluster-cyclic . Let B b e a cluster-cyclic matrix. When we consider B w r k s , the signs on the k th row and the k th column are changed. Thus, it holds that sign p B w r k s q “ ´ sign p B w q . (3.2) Example 3.2. One important example of the cluster-cyclic matrices is the sk ew-symmetric matrix corresp onding to the Markov quiver , which is giv en by ˘ ¨ ˚ ˝ 0 ´ 2 2 2 0 ´ 2 ´ 2 2 0 ˛ ‹ ‚ . (3.3) Let B b e one of the ab ov e matrices. In this case, the m utation is given by µ i p B q “ ´ B for eac h i “ 1 , 2 , 3. In view of Lemma 2.16 and Proposition 2.17 , this prop erty can be generalized to the following sk ew-symmetrizable matrix: B “ ˘ ¨ ˚ ˝ 0 ´ p 1 r p 0 ´ q 1 ´ r 1 q 0 ˛ ‹ ‚ , (3.4) where p, q , r , p 1 , q 1 , r 1 P R ą 0 satisfy pp 1 “ q q 1 “ r r 1 “ 4 and pq r “ p 1 q 1 r 1 . In this case, Sk p B q shap es as ( 3.3 ) and the m utation of B is giv en b y µ i p B q “ ´ B for eac h i “ 1 , 2 , 3. The following num ber plays an imp ortan t role in the classiﬁcation of cluster-cyclicity . Deﬁnition 3.3. F or any cyclic matrix B “ p b ij q P M 3 p R q , deﬁne the Markov c onstant C p B q b y C p B q “ | b 12 b 21 | ` | b 23 b 32 | ` | b 31 b 13 | ´ | b 12 b 23 b 31 | . (3.5) It w as in tro duced by [ BBH11 , Ak a24 ]. At ﬁrst glance, the expression of the Marko v constant seems not symmetric due to the last term b 12 b 23 b 31 . How ev er, by ( 2.1 ), the last term can b e replaced with | b 21 b 32 b 13 | , which ensures the symmetry of this deﬁnition. In particular, b y setting p ij “ a | b ij b j i | , we hav e | b 12 b 23 b 31 | “ p 12 p 23 p 31 . Thus, w e ma y obtain the following expression. C p B q “ p 2 12 ` p 2 23 ` p 2 31 ´ p 12 p 23 p 31 . (3.6) R YOT A AKAGI AND ZHICHAO CHEN 16 Prop osition 3.4 ([ BBH11 , Ak a24 ]) . L et B “ p b ij q P M 3 p R q b e a skew-symmetrizable and cyclic matrix. Then, the fol lowing two c onditions ar e e quivalent. ‚ B is cluster-cyclic. ‚ It holds that | b ij b j i | ě 4 for any i ‰ j and C p B q ď 4 . Remark 3.5. Another classiﬁcation is known b y [ Sev12 ] based on an admissible quasi-Cartan c omp anion . In this pap er, we take a sp eciﬁc one called the pseudo Cartan c omp anion , see ( 4.1 ), and state this fact in Remark 4.2 . W e now give some other inequalities derived from Prop osition 3.4 . F or a real num ber p ě 2, set α p p q “ 1 2 p p ` a p 2 ´ 4 q . By a direct calculation, w e v erify α p p q 2 ´ pα p p q ` 1 “ 0. Lemma 3.6. L et B “ p b ij q P M 3 p R q b e a cluster-cyclic matrix. Set p ij “ a | b ij b j i | and α ij “ α p p ij q . Then, for any t i, j, k u “ t 1 , 2 , 3 u , the fol lowing ine qualities hold. 1 2 ´ p ik p kj ´ b p p 2 ik ´ 4 qp p 2 kj ´ 4 q ¯ ď p ij ď 1 2 ´ p ik p kj ` b p p 2 ik ´ 4 qp p 2 kj ´ 4 q ¯ , (3.7) α ik p kj ´ p ij ě 0 , (3.8) α ik α kj ´ α ij ě 0 . (3.9) In [ FT19 , Lem. 4.3], an equiv alent expression of ( 3.9 ) is derived, and is called the triangle ine quality . F or the reader’s conv enience, we also giv e a pro of. Pr o of. F or simplicit y , set p “ p ij , q “ p ik , r “ p kj , and α x “ α p x q for any x ě 2. By Prop osition 3.4 , we hav e p 2 ` q 2 ` r 2 ´ pq r ď 4. This inequality ma y b e rearranged to " p ´ 1 2 ´ q r ` a p q 2 ´ 4 qp r 2 ´ 4 q ¯ * " p ´ 1 2 ´ q r ´ a p q 2 ´ 4 qp r 2 ´ 4 q ¯ * ď 0 . (3.10) Th us, ( 3.7 ) holds. Moreo v er, we hav e α q r ´ p ě 1 2 ´ q ` a q 2 ´ 4 ¯ r ´ 1 2 ´ q r ` a p q 2 ´ 4 qp r 2 ´ 4 q ¯ “ 1 2 a q 2 ´ 4 ´ r ´ a r 2 ´ 4 ¯ ě 0 . (3.11) Th us, ( 3.8 ) holds. Set p 0 “ 1 2 p q r ` a p q 2 ´ 4 qp r 2 ´ 4 qq . Then, p ď p 0 implies α p p q ď α p p 0 q . Th us, w e hav e α q α r ´ α p ě 1 4 ´ q ` a q 2 ´ 4 ¯ ´ r ` a r 2 ´ 4 ¯ ´ 1 2 ˆ p 0 ` b p 2 0 ´ 4 ˙ . (3.12) Since p 2 0 ´ 4 “ 1 4 ´ 2 q 2 r 2 ´ 4 q 2 ´ 4 r 2 ` 2 q r a p q 2 ´ 4 qp r 2 ´ 4 q ¯ “ 1 4 ´ q a r 2 ´ 4 ` r a q 2 ´ 4 ¯ 2 , (3.13) GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 17 w e ha v e p 0 ` b p 2 0 ´ 4 “ 1 2 ´ q r ` a p q 2 ´ 4 qp r 2 ´ 4 q ` q a r 2 ´ 4 ` r a q 2 ´ 4 ¯ “ 1 2 ´ q ` a q 2 ´ 4 ¯ ´ r ` a r 2 ´ 4 ¯ . (3.14) By substituting it into ( 3.12 ), we obtain ( 3.9 ). □ 3.2. Mutation of tropical signs. When we concentrate on the cluster-cyclic case, the recur- sion for the tropical signs ε w j can b e obtained as follo ws. Theorem 3.7 ([ A C25b , Thm. 3.4]) . L et B P M 3 p R q b e a cluster-cyclic initial exchange matrix. Then, its C -p attern is sign-c oher ent. Mor e over, for any w P T ztHu , let k b e the last index of w . Then, the fol lowing statements hold. ( a ) Ther e exists a unique s P t 1 , 2 , 3 uzt k u such that ε w s ‰ ε w k , ε w s sign p b w ks q “ ´ 1 . (3.15) ( b ) L et s P t 1 , 2 , 3 uzt k u b e the ab ove index, and let t P t 1 , 2 , 3 uzt k , s u b e the other index. Then, we have p ε w r s s k , ε w r s s s , ε w r s s t q “ p´ ε w k , ´ ε w s , ε w t q , p ε w r t s k , ε w r t s s , ε w r t s t q “ p ε w k , ε w s , ´ ε w t q . (3.16) In p articular, al l the tr opic al signs ε w j may b e obtaine d by the ab ove r e cursion and the fol lowing initial c onditions. ε H j “ 1 , ε r i s j “ $ & % ´ 1 i “ j, 1 i ‰ j. (3.17) Moreo v er, the following claim ab out the fan structure holds. Prop osition 3.8 ([ A C25b , Prop. 4.11]) . Conje ctur e 2.8 holds for any cluster-cyclic exchange matrix of r ank 3 . In p articular, by Pr op osition 2.14 , its G -fan is r e al ly a fan. Thanks to Prop osition 3.8 , we may consider the G -fan structure without an y assumption. Beforehand, we introduce the follo wing notation for later use. Deﬁnition 3.9. Let K, S, T : T ztHu Ñ t 1 , 2 , 3 u b e the maps deﬁned b y K p w q “ k , S p w q “ s , and T p w q “ t , where k , s, t are the indices deﬁned in Theorem 3.7 . F or any w P T ztHu and M , M 1 “ K , S, T , we write ε w M “ ε w M p w q , b w M M 1 “ b w M p w q ,M 1 p w q , g w M “ g w M p w q , (3.18) and so on. Let M b e the free monoid generated by tw o letters S and T . W e deﬁne the righ t monoid action of M on T ztHu by w S “ w r S p w qs , w T “ w r T p w qs . (3.19) R YOT A AKAGI AND ZHICHAO CHEN 18 Fix one initial m utation direction i “ 1 , 2 , 3, and let us fo cus on the subset T ěr i s . W e deﬁne the trunk T ăr i s S 8 of T ěr i s b y T ăr i s S 8 “ tr i s S n | n P Z ě 0 u . (3.20) F or each X P M , the subset T ěr i s X is deﬁned by ( 1.2 ). This subset T ěr i s X is called a br anch of T ěr i s if X has at least one letter T . In particular, for eac h n P Z ě 0 , T ěr i s S n T is called the n th maximal br anch of T ěr i s . Note that each T ěr i s is decomp osed into the trunk T ăr i s S 8 and the maximal branches T ěr i s S n T with n P Z ě 0 , and each branch T ěr i s X is a subset of some maximal branch. These indices K p w q , S p w q , and T p w q ob ey the following recursive rules. Lemma 3.10 ([ AC25b , Lem. 6.4]) . The fol lowing r e curr enc e formulas of indic es hold: p K p w S q , S p w S q , T p w S qq “ p S p w q , K p w q , T p w qq , (3.21) p K p w T q , S p w T q , T p w T qq “ $ & % p T p w q , S p w q , K p w qq if w is in a trunk , p T p w q , K p w q , S p w qq if w is in a branch . (3.22) Example 3.11. W e giv e an example how the tropical signs p ε w 1 , ε w 2 , ε w 3 q are obtained and these indices k “ K p w q , s “ S p w q , t “ T p w q change. Consider sign p B q “ ¨ ˚ ˝ 0 ´ ` ` 0 ´ ´ ` 0 ˛ ‹ ‚ , i “ 1 . (3.23) By ( 3.17 ) and sign p B r i s q “ ¨ ˚ ˝ 0 ` ´ ´ 0 ` ` ´ 0 ˛ ‹ ‚ , (3.24) w e obtain K pr 1 sq “ 1, S pr 1 sq “ 3, and T pr 1 sq “ 2. After the single m utation, b y using ( 3.16 ) and Lemma 3.10 , w e can obtain the tropical signs p ε w 1 , ε w 2 , ε w 3 q and k “ K p w q , s “ S p w q , t “ T p w q in Figure 3 . Let k “ K p w q and s “ S p w q . By ( 3.21 ), rep eating S -m utation means rep eating m utations in tw o directions s and k . Namely , it holds that w S n “ w r s, k , s, k , . . . n terms s . (3.25) In particular, for eac h i “ 1 , 2 , 3, b y setting k 0 “ K pr i sqp“ i q and s 0 “ S pr i sq , the trunk T ăr i s S 8 consists of the ﬁnite sequence of the form r k 0 , s 0 , k 0 , . . . s . The mutation formulas for c -, g -v ectors dep end on ε w M b w M 1 M . W e summarize these signs in the following lemma. GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 19 p` , ` , `q initial p´ k , ` t , ` s q p` s , ` t , ´ k q p´ k , ` t , ` s q p` s , ` t , ´ k q p´ t , ´ k , ` s q p´ k , ` s , ´ t q p` t , ´ s , ` k q p´ t , ´ k , ` s q p` k , ` t , ´ s q p´ s , ` k , ` t q p` s , ´ t , ´ k q p` s , ´ k , ´ t q p´ t , ` s , ´ k q p` k , ´ s , ` t q p´ t , ´ k , ` s q T runk T ăr i s S 8 i “ 1 S “ 3 S “ 1 S “ 3 T “ 2 T “ 2 T “ 2 S “ 3 T “ 1 S “ 1 T “ 3 S “ 2 T “ 1 S “ 2 T “ 3 Figure 3. T ropical signs p ε 1 , ε 2 , ε 3 q in the case of ( 3.23 ). The b elow subscripts k , s , t are the ones in Theorem 3.7 . Lemma 3.12. L et w P T ztHu . Then, the fol lowing statements hold. ( a ) Supp ose that w is in a trunk. Then, ε w M sign p b w M 1 M q ar e given in the fol lowing list. p M , M 1 q p K, S q p K, T q p S, K q p S, T q p T , K q p T , S q ε w M sign p b w M 1 M q ´ 1 1 ´ 1 1 1 ´ 1 (3.26) ( b ) Supp ose that w is in a br anch. Then, ε w M sign p b w M 1 M q ar e given in the fol lowing list. p M , M 1 q p K, S q p K, T q p S, K q p S, T q p T , K q p T , S q ε w M sign p b w M 1 M q ´ 1 1 ´ 1 1 ´ 1 1 (3.27) T o pro ve this lemma, we mention the following basic facts. Lemma 3.13 ([ AC25b , Lem. 6.5, Prop. 6.7]) . Fix an initial mutation dir e ction i “ 1 , 2 , 3 . ( a ) If w P T ăr i s S 8 is in a trunk, then we have ε w K “ ´ 1 , ε w S “ ε w T “ 1 . (3.28) ( b ) Supp ose that w P T ztHu is in a br anch. We expr ess w “ r i s X , wher e X P M has at le ast one letter T . Then, we have ε r i s X K “ ε r i s X T “ p´ 1 q # T p X q , ε r i s X S “ ´p´ 1 q # T p X q , (3.29) wher e # T p X q P Z ě 1 is the numb er of the letter T app e aring in X . Lemma 3.14. F or any distinct M , M 1 , M 2 P t K , S, T u , the fol lowing statements hold. ( a ) F or any w P T ztHu , we have sign p b w M M 1 q “ ´ sign p b w M 1 M q . ( b ) F or any w P T ztHu , we have sign p b w M M 1 q “ ´ sign p b w M M 2 q and sign p b w M 1 M q “ ´ sign p b w M 2 M q . R YOT A AKAGI AND ZHICHAO CHEN 20 Pr o of. The claim ( a ) follows from the fact that B w is sign-sk ew-symmetric. T o prov e that sign p b w M M 1 q “ ´ sign p b w M M 2 q , w e consider the M p w q th row of ( 3.1 ). Since M 1 p w q , M 2 p w q ‰ M p w q , exactly one of the corresp onding entries is p ositiv e and the other is negative. Hence, this implies sign p b w M M 1 q “ ´ sign p b w M M 2 q . By fo cusing on the M p w q th column instead, w e similarly obtain sign p b w M 1 M q “ ´ sign p b w M 2 M q . □ Pr o of of L emma 3.12 . In the case of p M , M 1 q “ p S, K q for any w , this follows from the second condition of ( 3.15 ). If we know this sp ecial case, the others are automatically determined by considering the following tw o facts. ( A ) By Lemma 3.13 , if w is in a trunk, w e hav e ε w K “ ´ ε w S “ ´ ε w T . If w is in a branch, we ha v e ε w K “ ´ ε w S “ ε w T . ( B ) By Lemma 3.14 , for any distinct three indices M , M 1 , M 2 P t K , S, T u , we obtain that sign p b w M M 1 q “ ´ sign p b w M 1 M q , sign p b w M M 1 q “ ´ sign p b w M M 2 q , and sign p b w M 1 M q “ ´ sign p b w M 2 M q . F or example, ε w K sign p b w T K q “ 1 can b e sho wn as follows: ε w K sign p b w T K q p A q “ ´ ε w S sign p b w T K q p B q “ ε w S sign p b w S K q p B q “ ´ ε w S sign p b w K S q “ 1 , (3.30) where the last equality ma y b e shown by the result of p M , M 1 q “ p S, K q . □ 3.3. Mutation formulas. T o study G -fan structures, we introduce mo diﬁed c –v ectors ˜ c w i and mo diﬁed g -v ectors ˜ g w i . F or this purp ose, the skew-symmetric matrices ˜ B deﬁned in Deﬁni- tion 2.15 are more con v enient and useful than the original ones. W e write p w ij “ b | b w ij b w j i | “ | ˜ b w ij | , α w ij “ α p p w ij q “ 1 2 ´ p w ij ` b p p w ij q 2 ´ 4 ¯ , (3.31) where ˜ b w ij is the p i, j q th entry of ˜ B w . By deﬁnition, it holds that p α w ij q 2 ´ p w ij α w ij ` 1 “ 0 . (3.32) When we consider w “ H , for brevity , we omit it. Namely , we write b ij “ b H ij , p ij “ p H ij “ a | b ij b j i | and α ij “ α H ij . As in Deﬁnition 3.9 , for any M , M 1 “ K , S, T and w P T ztHu , w e write p w M M 1 “ p w M p w q M 1 p w q , α w M M 1 “ α w M p w q M 1 p w q , ˜ g w M “ ˜ g w M p w q , and so on. Under these notations, the mutation rules ( 2.2 ) and ( 2.21 ) can b e expressed as follows. Lemma 3.15 ( S -mutation) . L et w P T ztHu . Then, the S -mutation data ar e given by p w S M M 1 “ $ ’ ’ & ’ ’ % p w S K p M , M 1 q “ p K , S q , p S, K q , p w K S p w S T ´ p w K T p M , M 1 q “ p S, T q , p T , S q , p w S T p M , M 1 q “ p K , T q , p T , K q , (3.33) GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 21 ˜ c w S M “ $ ’ ’ & ’ ’ % ´ ˜ c w S M “ K , ˜ c w K ` p w S K ˜ c w S M “ S, ˜ c w T M “ T , ˜ g w S M “ $ ’ ’ & ’ ’ % ´ ˜ g w S ` p w S K ˜ g w K M “ K , ˜ g w K M “ S, ˜ g w T M “ T . (3.34) Lemma 3.16 ( T -mutation) . L et w P T ztHu . Then, the T -mutation data ar e given by the fol lowing rules. ( a ) If w is in a trunk, we have p w T M M 1 “ $ ’ ’ & ’ ’ % p w T S p M , M 1 q “ p K , S q , p S, K q , p w S T p w T K ´ p w S K p M , M 1 q “ p S, T q , p T , S q , p w K T p M , M 1 q “ p K , T q , p T , K q , (3.35) ˜ c w T M “ $ ’ ’ & ’ ’ % ´ ˜ c w T M “ K , ˜ c w S ` p w S T ˜ c w T M “ S, ˜ c w K M “ T , ˜ g w T M “ $ ’ ’ & ’ ’ % ´ ˜ g w T ` p w S T ˜ g w S M “ K , ˜ g w S M “ S, ˜ g w K M “ T . (3.36) ( b ) If w is in a br anch, we have p w T M M 1 “ $ ’ ’ & ’ ’ % p w T K p M , M 1 q “ p K , S q , p S, K q , p w K T p w T S ´ p w K S p M , M 1 q “ p S, T q , p T , S q , p w S T p M , M 1 q “ p K , T q , p T , K q , (3.37) ˜ c w T M “ $ ’ ’ & ’ ’ % ´ ˜ c w T M “ K , ˜ c w K ` p w K T ˜ c w T M “ S, ˜ c w S M “ T , ˜ g w T M “ $ ’ ’ & ’ ’ % ´ ˜ g w T ` p w K T ˜ g w K M “ K , ˜ g w K M “ S, ˜ g w S M “ T . (3.38) Pr o of of L emma 3.15 and L emma 3.16 . By using Lemma 3.10 and Lemma 3.12 , we may c hec k eac h form ula. Here, we show p w S S T “ p w K S p w S T ´ p w K T and ˜ c w S S “ ˜ c w K ` p w S K ˜ c w S , and the others can b e sho wn by a similar argumen t. By Lemma 3.10 , we hav e S p w S q “ K p w q and T p w S q “ T p w q . Let τ “ sign p ˜ b w K T q . Then, we ha v e p w K T “ τ ˜ b w K T . By ( 3.2 ), it implies that sign p ˜ b w S S T q “ sign p ˜ b w S K p w q ,T p w q q “ ´ sign p ˜ b w K T q “ ´ τ and p w S S T “ ´ τ ˜ b w S S T . Moreov er, by Lemma 3.14 , it holds that sign p ˜ b w K S q “ ´ τ and sign p ˜ b w S T q “ ´ τ . In particular, ˜ b w K S ˜ b w S T ě 0 holds. By ( 2.2 ), we hav e p w S S T “ ´ τ ˜ b w S S T “ ´ τ ˜ b w r S p w qs K p w q ,T p w q “ ´ τ p ˜ b w K T ` sign p ˜ b w K S q| ˜ b w K S ˜ b w S T |q “ ´ τ p τ p w K T ´ τ p w K S p w S T q “ p w K S p w S T ´ p w K T . (3.39) By Lemma 3.10 , we ha v e S p w S q “ K p w q . Thus, b y ( 2.21 ), it implies that ˜ c w S S “ ˜ c w r S p w qs K p w q “ ˜ c w K ` r ε w S ˜ b w S K s ` ˜ c w S . (3.40) R YOT A AKAGI AND ZHICHAO CHEN 22 Since ˜ b w S K “ ´ ˜ b w K S , we hav e r ε w S ˜ b w S K s ` “ r´ ε w S ˜ b w K S s ` . By Lemma 3.12 in the case of p M , M 1 q “ p S, K q , we obtain ε w S ˜ b w K S ă 0. It implies r´ ε w S ˜ b w K S s ` “ | ˜ b w K S | “ p w K S . Th us, we hav e ˜ c w S S “ ˜ c w K ` p w S K ˜ c w S . □ Example 3.17. In the stereographic pro jection, according to Lemma 3.15 and Lemma 3.16 , the m utation rules of g -vectors can b e understo o d as in Figure 4 and Figure 5 . Note that the thic k colored lines are on the same plane. By ( 2.22 ), each plane is orthogonal to a c -v ector. K S T w K S T w S K S T w T Figure 4. In trunks. K S T w K S T w S K S T w T Figure 5. In branches. 3.4. Initial setup. W e ﬁx the initial mutation i P t 1 , 2 , 3 u . After a single mutation r i s , the m utation rule is described in Lemma 3.15 and Lemma 3.16 . In contrast, the ﬁrst m utation should b e computed directly . W e give some relev an t formulas for the initial mutation as follows. Let k 0 “ K pr i sq , s 0 “ S pr i sq , and t 0 “ T pr i sq . By the same argument in Example 3.11 , these indices are listed in T able 1 . B t 0 ´ 0 ´ ` ` 0 ´ ´ ` 0 ¯ ´ 0 ` ´ ´ 0 ` ` ´ 0 ¯ i 1 2 3 1 2 3 p k 0 , s 0 , t 0 q p 1 , 3 , 2 q p 2 , 1 , 3 q p 3 , 2 , 1 q p 1 , 2 , 3 q p 2 , 3 , 1 q p 3 , 2 , 1 q T able 1. The list of k 0 “ K pr i sq , s 0 “ S pr i sq , t 0 “ T pr i sq . F or a given initial exc hange matrix B , if we consider the relation b et w een tw o diﬀerent initial m utation directions i ‰ j , the follo wing fact is useful: S pr i sq ‰ S pr j sq , T pr i sq ‰ T pr j sq . (3.41) F or example, set k 0 “ K pr i sq , s 0 “ S pr i sq , and t 0 “ T pr i sq . F or the initial mutation j “ s 0 , since T pr j sq ‰ t 0 , we hav e K pr j sq “ s 0 , S pr j sq “ t 0 , T pr j sq “ k 0 . (3.42) Similarly , for j “ t 0 , since S pr j sq ‰ s 0 , we hav e K pr j sq “ t 0 , S pr j sq “ k 0 , T pr j sq “ s 0 . (3.43) As explained in Subsection 3.3 , we omit the empty sequence H “ r s in the sup erscript. By using the data at H , we give the expressions for p r i s M M 1 , ˜ g r i s M , and ˜ c r i s M . GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 23 Lemma 3.18. The fol lowing e qualities hold: $ ’ ’ ’ & ’ ’ ’ % p r i s S T “ p k 0 s 0 p k 0 t 0 ´ p s 0 t 0 , p r i s K T “ p k 0 t 0 , p r i s K S “ p k 0 s 0 , $ ’ ’ ’ & ’ ’ ’ % ˜ g r i s K “ ´ ˜ e k 0 ` p k 0 s 0 ˜ e s 0 , ˜ g r i s S “ ˜ e s 0 , ˜ g r i s T “ ˜ e t 0 , $ ’ ’ ’ & ’ ’ ’ % ˜ c r i s K “ ´ ˜ e k 0 , ˜ c r i s S “ ˜ e s 0 ` p k 0 s 0 ˜ e s 0 , ˜ c r i s T “ ˜ e t 0 . (3.44) Pr o of. This follo ws from the same argument in the pro ofs of Lemma 3.15 and Lemma 3.16 . □ 4. Global upper bound F rom now on, we ﬁx an cluster-cyclic initial exchange matrix B P M 3 p R q together with its sk ew-symmetrizer D “ diag p d 1 , d 2 , d 3 q . In this section, w e exhibit the global upp er b ounds for the G -fan, which can b e referred to the thic k blue lines in Figure 1 . 4.1. Quadratic surface for g -v ectors. In [ LL24 ], a restriction on g -vectors was established. T o state it, we introduce the follo wing notion. Deﬁnition 4.1. F or an y cyclic matrix B “ p b ij q P M 3 p R q , we deﬁne the pseudo Cartan c om- p anion of B as A “ ¨ ˚ ˝ 2 | b 12 | | b 13 | | b 21 | 2 | b 23 | | b 31 | | b 32 | 2 ˛ ‹ ‚ . (4.1) F or an y cluster-cyclic initial exchange matrix B and an y reduced sequence w P T , we write the pseudo Cartan companion of B w as A w . Remark 4.2. In [ Sev12 , Thm. 2.6], the following classiﬁcation of the cluster-cyclicit y was es- tablished: a given cyclic matrix B P M 3 p R q is cluster-cyclic if and only if the corresp onding pseudo Cartan companion A has exactly one p ositiv e eigenv alue and t wo non-p ositive eigen- v alues. Moreo v er, the follo wing relationship b etw een the pseudo Cartan companion and the Mark o v constan t w as found. det A “ 2 p 4 ´ C p B qq . (4.2) Eac h pseudo Cartan companion A is a symmetrizable matrix with a symmetrizer D . No w, w e can exhibit the following restriction. Note that in [ LL24 ], it is constructed under the skew- symmetric case. Here, we can generalize this result to the skew-symmetrizable case. Lemma 4.3 (cf. [ LL24 ]) . L et B P M 3 p R q b e a cluster-cyclic initial exchange matrix, and let A w b e the pseudo Cartan c omp anion of B w . Fix an initial mutation dir e ction i “ 1 , 2 , 3 . Then, for any w P T ěr i s , we have p G w q J D A r i s G w “ D A w K , (4.3) wher e w K “ w r K p w qs is the se quenc e obtaine d by deleting the last index of w . In p articular, every g -ve ctor g w l “ p x 1 , x 2 , x 3 q J ( l “ 1 , 2 , 3) satisﬁes d 1 x 2 1 ` d 2 x 2 2 ` d 3 x 2 3 ` d 1 | b r i s 12 | x 1 x 2 ` d 2 | b r i s 23 | x 2 x 3 ` d 3 | b r i s 31 | x 3 x 1 “ d l . (4.4) R YOT A AKAGI AND ZHICHAO CHEN 24 Pr o of. By doing the same argumen t in [ LL24 ], w e ma y obtain this result for an y r e al sk ew- symmetric matrix with d 1 “ d 2 “ d 3 “ 1. Let ˜ B “ D 1 2 B D ´ 1 2 . W e write the g -vector with the initial exc hange matrix ˜ B by ˆ G w . Then, for any w P T ěr i s , by the result for the sk ew-symmetric case, we hav e p ˆ G w q J ˜ A r i s ˆ G w “ ˜ A w K , (4.5) where ˜ A r i s and ˜ A w K are the pseudo Cartan companion of ˜ B r i s and ˜ B w K . By Lemma 2.16 , w e ha v e ˜ B r i s “ D 1 2 B r i s D ´ 1 2 and ˜ B w K “ D 1 2 B w K D ´ 1 2 . Th us, we obtain ˜ A r i s “ D 1 2 A r i s D ´ 1 2 and ˜ A w K “ D 1 2 A w K D ´ 1 2 . By ( 2.16 ), we ha v e ˆ G w “ D 1 2 G w D ´ 1 2 . By substituting these equalities in to ( 4.5 ), w e obtain p G w q J D A r i s G w “ D A w K . By fo cusing on the diagonal e n tries of this equalit y , we ma y obtain ( 4.4 ). □ W e state this result by using mo diﬁed g -vectors. T o this end, we deﬁne ˜ A w “ D 1 2 A w D ´ 1 2 as the pseudo Cartan companion of ˜ B w “ D 1 2 B w D ´ 1 2 . Lemma 4.4. Fix an initial mutation dir e ction i “ 1 , 2 , 3 . F or any w P T ěr i s , every mo diﬁe d G -matrix ˜ G w satisﬁes p D 1 2 ˜ G w q J ˜ A r i s p D 1 2 ˜ G w q “ ˜ A w K . (4.6) In p articular, every mo diﬁe d g -ve ctor ˜ g w l ( l “ 1 , 2 , 3) satisﬁes p D 1 2 ˜ g w l q J ˜ A r i s p D 1 2 ˜ g w l q “ 2 . (4.7) Pr o of. By substituting G w “ ˜ G w D 1 2 in to ( 4.3 ), we obtain the claim. □ The equalit y ( 4.7 ) claims that, after the initial mutation i “ 1 , 2 , 3, every mo diﬁed g -v ectors are on the same quadratic surface. F or later use, w e start by studying more detailed prop erties of this surface. 4.2. General fact of the quadratic surface. In this subsection, we restrict our attention to the region indicated by the quadratic equation x 2 1 ` x 2 2 ` x 2 3 ` p 12 x 1 x 2 ` p 23 x 2 x 3 ` p 31 x 3 x 1 “ 1 , (4.8) and w e temp orarily set aside the cluster algebraic structures, such as B -matrices and g -v ectors. Namely , we ﬁx p 12 , p 23 , p 31 ě 2 satisfying C p p 12 , p 23 , p 31 q “ p 2 12 ` p 2 23 ` p 2 31 ´ p 12 p 23 p 31 ď 4. T o simplify the statement, we set p j i “ p ij and ˜ A “ ¨ ˚ ˝ 2 p 12 p 13 p 21 2 p 23 p 31 p 32 2 ˛ ‹ ‚ . (4.9) Note that the equalit y ( 4.8 ) ma y b e expressed as 1 2 x J ˜ A x “ 1. (This ˜ A corresp onds to the pseudo Cartan companion of ˜ B .) Then, the region corresp onding to the equality ( 4.8 ) can b e depicted as follows. GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 25 Lemma 4.5. ( a ) The matrix ˜ A has thr e e r e al eigenvalues λ ą ν 1 ě ν 2 , wher e λ is strictly p ositive and ν 1 , ν 2 ar e non-p ositive al lowing ν 1 “ ν 2 . Mor e over, λ satisﬁes λ ą 2 ` b p 2 12 ` p 2 23 ` p 2 31 . (4.10) ( b ) One eigenve ctor of ˜ A with r esp e ct to λ may b e expr esse d as v “ ¨ ˚ ˝ p λ ´ 2 q 2 ` p p 12 ` p 13 qp λ ´ 2 q ` p 12 p 23 ` p 13 p 32 ´ p 2 23 p λ ´ 2 q 2 ` p p 23 ` p 21 qp λ ´ 2 q ` p 23 p 31 ` p 21 p 13 ´ p 2 31 p λ ´ 2 q 2 ` p p 31 ` p 32 qp λ ´ 2 q ` p 31 p 12 ` p 32 p 21 ´ p 2 12 ˛ ‹ ‚ , (4.11) and every c omp onent of this ve ctor is strictly p ositive. ( c ) The r e gion of 1 2 x J ˜ A x “ 1 is any of hyp erb oloid of two she ets (when ν 1 , ν 2 ă 0 ), hyp erb olic cylinder (when ν 1 “ 0 , ν 2 ă 0 ), or r e al p ar al lel planes (when ν 1 “ ν 2 “ 0 ) in the sense of [ Zwi95 , § 4.20] . In p articular, for e ach ˜ A , ther e ar e two c onne cte d c omp onents, and they ar e sep ar ate d by the plane H p v q ortho gonal to v . x 1 x 2 x 3 O H ` H ´ Figure 6. Real parallel planes. x 1 x 2 x 3 O H ` H ´ Figure 7. Hyp erb oloid of tw o sheets. Remark 4.6. According to the classiﬁcation of quadratic forms, the quadratic form ( 4.8 ) falls in to one of the three types describ ed in Lemma 4.5 ( c ). How ev er, we do not care ab out these diﬀerences in this paper. The only prop erty we require is that the region arising from the quadratic form ( 4.8 ) has t w o connected comp onen ts separated by the plane H p v q . As stated in Remark 4.2 , the signs of λ , ν 1 , and ν 2 can b e obtained by [ Sev12 ]. How ev er, for the later purp ose, we need to know more detailed prop erties suc h as ( 4.10 ) and ( b ). Hence, we pro vide an alternative pro of here. Pr o of. Let f p t q “ det p tI ´ ˜ A q . Then, b y using p ij “ p j i , we may expand it as f p t q “ p t ´ 2 q 3 ´ p p 2 12 ` p 2 23 ` p 2 31 qp t ´ 2 q ´ 2 p 12 p 23 p 31 . (4.12) Consider f 1 p t q “ 3 p t ´ 2 q 2 ´ p p 2 12 ` p 2 23 ` p 2 31 q . Then, b y p 12 , p 23 , p 31 ě 2, we ha ve f 1 p 0 q “ 12 ´ p p 2 12 ` p 2 23 ` p 2 31 q ď 0 . (4.13) R YOT A AKAGI AND ZHICHAO CHEN 26 Since f 1 p t q has a p ositiv e leading co eﬃcien t, one ro ot of f 1 p t q “ 0 is non-negativ e and the other ro ot is non-p ositive. In particular, f p t q is unimo dal when t ě 0. By a direct calculation with ( 2.1 ), we hav e f p 0 q “ ´ 2 p 4 ´ C p p 12 , p 23 , p 31 qq ď 0 . (4.14) Th us, we ma y sho w that there exists precisely one positive solution λ ą 0. Since ˜ A is symmetric, there should exist three real eigenv alues, counting multiplicit y . Thus, there exist other t wo eigen v alues ν 1 , ν 2 ď 0. Moreo v er, b y f p 2 ` a p 2 12 ` p 2 23 ` p 2 31 q “ ´ 2 p 12 p 23 p 31 ă 0, the unique p ositiv e eigen v alue λ should satisfy 2 ` a p 2 12 ` p 2 23 ` p 2 31 ă λ . Next, w e sho w that an eigenv ector of λ may b e giv en b y ( 4.11 ). This is b ecause we can calculate the following equatity: ¨ ˚ ˝ λ ´ 2 ´ p 12 ´ p 13 ´ p 21 λ ´ 2 ´ p 23 ´ p 31 ´ p 32 λ ´ 2 ˛ ‹ ‚ v “ 0 . (4.15) F or example, the equality of the ﬁrst comp onent may b e v eriﬁed as follows. p λ ´ 2 , ´ p 12 , ´ p 13 q v “ p λ ´ 2 q 3 ` p p 12 ` p 13 qp λ ´ 2 q 2 ` p p 12 p 23 ` p 13 p 32 ´ p 2 23 qp λ ´ 2 q ´ p 12 p λ ´ 2 q 2 ´ p 12 p p 23 ` p 21 qp λ ´ 2 q ´ p 12 p p 23 p 31 ` p 21 p 13 ´ p 2 31 q ´ p 13 p λ ´ 2 q 2 ´ p 13 p p 31 ` p 32 qp λ ´ 2 q ´ p 13 p p 31 p 12 ` p 32 p 21 ´ p 2 12 q “ p λ ´ 2 q 3 ´ p p 2 12 ` p 2 23 ` p 2 31 qp λ ´ 2 q ´ 2 p 12 p 23 p 31 “ 0 , (4.16) where the last equality follo ws from f p λ q “ 0. Last, we show that all entries of v are p ositive. F or example, w e may show that the ﬁrst comp onent is strictly p ositiv e by λ ą 2 and p λ ´ 2 q 2 ´ p 2 23 ą b p 2 12 ` p 2 23 ` p 2 31 2 ´ p 2 23 “ p 2 12 ` p 2 31 ą 0 , (4.17) where the ﬁrst inequality follo ws from λ ´ 2 ą a p 2 12 ` p 2 23 ` p 2 31 . □ Remark 4.7. F rom ( 4.13 ) and ( 4.14 ), we hav e the following classiﬁcation of these three kinds of surface type. ‚ ν 1 “ ν 2 “ 0 ð ñ p 12 “ p 23 “ p 31 “ 2. ‚ ν 1 “ 0 , ν 2 ă 0 ð ñ C p p 12 , p 23 , p 31 q “ 4 and max p p 12 , p 23 , p 31 q ą 2. ‚ ν 1 , ν 2 ă 0 ð ñ C p p 12 , p 23 , p 31 q ă 4. In particular, the real parallel planes case ( ν 1 “ ν 2 “ 0) happ ens only when the matrix B is giv en b y ( 3.4 ). F or later purp ose, we include information ab out a skew-symmetrizer D “ diag p d 1 , d 2 , d 3 q . (Ho w ever, w e treat it as a triple of p ositive num b ers d 1 , d 2 , d 3 ą 0 in this subsection.) GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 27 Deﬁnition 4.8. F or eac h ˜ A and D “ diag p d 1 , d 2 , d 3 q , let v b e the p ositiv e eigen vector of ˜ A giv en b y ( 4.11 ). Deﬁne the following sets: Q “ t x P R 3 | p D 1 2 x q J ˜ A p D 1 2 x q ą 0 u Y t 0 u , H “ ! x P R 3 ˇ ˇ ˇ p D 1 2 x q J ˜ A p D 1 2 x q “ 2 ) , Q ` “ t x P Q | x D 1 2 x , v y ě 0 u , H ` “ t x P H | x D 1 2 x , v y ě 0 u , Q ´ “ t x P Q | x D 1 2 x , v y ď 0 u , H ´ “ t x P H | x D 1 2 x , v y ď 0 u . (4.18) Later, we tak e D to b e a skew-symmetrizer of a giv en skew-symmetrizable matrix B . In this case, the three sets Q, Q ` , and Q ´ are indep endent of the c hoice of D since all skew-symmetrizers of B are identical up to a scalar. Ho w ev er, H , H ` , and H ´ dep end on D due to this scalar indeﬁniteness. Since v is an eigen vector with respect to the unique p ositive eigen v alue λ of ˜ A , we hav e Q ` “ t x P Q | x D 1 2 x , v y ą 0 u Y t 0 u and Q ´ “ t x P Q | x D 1 2 x , v y ă 0 u Y t 0 u . Lemma 4.9. The sets Q ` and Q ´ ar e b oth c onvex c ones. Pr o of. W e pro v e the case for Q ` . Let v 1 “ v | v | . Since ˜ A is symmetric, there exists an or- thogonal eigenbasis r v 1 , u 1 , u 2 s corresp onding to the eigenv alues p λ, ν 1 , ν 2 q of ˜ A . Note that r D ´ 1 2 v 1 , D ´ 1 2 u 1 , D ´ 1 2 u 2 s is also a basis of R 3 . T ake an y elements x 1 , x 2 P Q ` and a 1 , a 2 P R ą 0 . If either x 1 or x 2 is 0 , we can prov e a 1 x 1 ` a 2 x 2 P Q ` b y a direct calculation. Th us, we may assume x 1 , x 2 ‰ 0 . F or each j “ 1 , 2, w e express x j “ r j D ´ 1 2 v 1 ` α j D ´ 1 2 u 1 ` β j D ´ 1 2 u 2 “ D ´ 1 2 p v 1 , u 1 , u 2 q ˆ r j α j β j ˙ . (4.19) for some r j , α j , β j P R . Since x D 1 2 x j , v 1 y “ r j , we hav e r j ě 0. Th us, we may sho w that x D 1 2 p a 1 x 1 ` a 2 x 2 q , v 1 y “ a 1 r 1 ` a 2 r 2 ě 0. No w, it suﬃces to prov e that ! D 1 2 p a 1 x 1 ` a 2 x 2 q ) J ˜ A ! D 1 2 p a 1 x 1 ` a 2 x 2 q ) ą 0 . (4.20) In fact, we may express ! D 1 2 p a 1 x 1 ` a 2 x 2 q ) J ˜ A ! D 1 2 p a 1 x 1 ` a 2 x 2 q ) “ a 2 1 p D 1 2 x 1 q J ˜ A p D 1 2 x 1 q ` 2 a 1 a 2 p D 1 2 x 1 q J ˜ A p D 1 2 x 2 q ` a 2 2 p D 1 2 x 2 q J ˜ A p D 1 2 x 2 q ą 2 a 1 a 2 p D 1 2 x 1 q J ˜ A p D 1 2 x 2 q . (4.21) Hence, w e only need to prov e that p D 1 2 x 1 q J ˜ A p D 1 2 x 2 q ě 0. Since p v 1 , u 1 , u 2 q J ˜ A p v 1 , u 1 , u 2 q “ diag p λ, ν 1 , ν 2 q , by using ( 4.19 ), we obtain that p D 1 2 x 1 q J ˜ A p D 1 2 x 2 q “ λr 1 r 2 ` ν 1 α 1 α 2 ` ν 2 β 1 β 2 . (4.22) R YOT A AKAGI AND ZHICHAO CHEN 28 No w, our desired inequality is reduced to λr 1 r 2 ě p´ ν 1 q α 1 α 2 ` p´ ν 2 q β 1 β 2 . (4.23) This inequality can b e sho wn as follows. Note that p D 1 2 x j q J ˜ A p D 1 2 x j q ě 0. Then, we hav e λr 2 j ě p´ ν 1 q α 2 j ` p´ ν 2 q β 2 j . Since b oth sides are non-negativ e and r j ě 0, w e hav e ? λr j ě b p´ ν 1 q α 2 j ` p´ ν 2 q β 2 j . In particular, we obtain that λr 1 r 2 “ p ? λr 1 qp ? λr 2 q ě b p´ ν 1 q α 2 1 ` p´ ν 2 q β 2 1 b p´ ν 1 q α 2 2 ` p´ ν 2 q β 2 2 . (4.24) Consider the vectors a j “ p ? ´ ν 1 α j , ? ´ ν 2 β j q J for j “ 1 , 2. Since ν 1 , ν 2 ď 0, these vectors lie in R 2 . Thus, b y the Cauc hy-Sc hw arz inequality | a 1 || a 2 | ě x a 1 , a 2 y with resp ect to the Euclidean inner pro duct, we hav e b p´ ν 1 q α 2 1 ` p´ ν 2 q β 2 1 b p´ ν 1 q α 2 2 ` p´ ν 2 q β 2 2 ě p´ ν 1 q α 1 α 2 ` p´ ν 2 q β 1 β 2 . (4.25) By com bining these t w o inequalities ( 4.24 ) and ( 4.25 ), w e obtain ( 4.23 ) as desired, whic h implies that a 1 x 1 ` a 2 x 2 P Q ` . □ By Lemma 4.5 , H is decomp osed in to the connected comp onents H ` and H ´ suc h that H ` “ H X H ` ´ D 1 2 v ¯ , H ´ “ H X H ´ ´ D 1 2 v ¯ . (4.26) By Lemma 4.9 , Q is the minim um conv ex cone containing H and the origin 0 . Hence, the ab o ve situation also o ccurs in Q zt 0 u . More precisely , Q ` zt 0 u and Q ´ zt 0 u are the connected comp onen t of Q zt 0 u such that Q ` zt 0 u “ p Q zt 0 uq X H ` ´ D 1 2 v ¯ , Q ´ zt 0 u “ p Q zt 0 uq X H ´ ´ D 1 2 v ¯ . (4.27) F or later use, we establish the follo wing preliminary lemma. Lemma 4.10. F or any x P H ` and y P H ´ , we have x ` y R Q zt 0 u . Pr o of. It suﬃces to sho w that t D 1 2 p x ` y qu J ˜ A t D 1 2 p x ` y qu ď 0 . (4.28) Let r v 1 , u 1 , u 2 s b e an orthogonal eigenbasis corresp onding to the eigenv alues p λ, ν 1 , ν 2 q of ˜ A . Then, we may express x “ αD ´ 1 2 v 1 ` a 1 D ´ 1 2 u 1 ` a 2 D ´ 1 2 u 2 , y “ β D ´ 1 2 v 1 ` b 1 D ´ 1 2 u 1 ` b 2 D ´ 1 2 u 2 (4.29) for some α, β , a i , b i P R . Since x P H ` and y P H ´ , w e hav e β ă 0 ă α and p D 1 2 x q J ˜ A p D 1 2 x q “ p D 1 2 y q J ˜ A p D 1 2 y q “ 2. Thus, we ha v e t D 1 2 p x ` y qu J ˜ A t D 1 2 p x ` y qu “ 4 ` 2 p D 1 2 x q J ˜ A p D 1 2 y q “ 4 ` 2 p λαβ ` ν 1 a 1 b 1 ` ν 2 a 2 b 2 q , (4.30) GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 29 where the last equality follo ws from the fact that r v 1 , u 1 , u 2 s is an orthogonal eigen basis of ˜ A . Th us, our desired inequality is reduced to ´ λαβ ě 2 ` ν 1 a 1 b 1 ` ν 2 a 2 b 2 . (4.31) Since p D 1 2 x q J ˜ A p D 1 2 x q “ 2, we ha v e λα 2 “ 2 ´ ν 1 a 2 1 ´ ν 2 a 2 2 . (4.32) Since α ě 0 and λ ą 0, this implies ? λα “ a 2 ´ ν 1 a 2 1 ´ ν 2 a 2 2 . Using a similar argument, w e ma y obtain ´ ? λβ “ a 2 ´ ν 1 b 2 1 ´ ν 2 b 2 2 . (Note that β ď 0.) Thus, the inequality ( 4.31 ) is equiv alent to b 2 ´ ν 1 a 2 1 ´ ν 2 a 2 2 b 2 ´ ν 1 b 2 1 ´ ν 2 b 2 2 ě 2 ` ν 1 a 1 b 1 ` ν 2 a 2 b 2 . (4.33) In fact, this inequalit y can b e shown as follo ws. Let a “ p ? 2 , ? ´ ν 1 a 1 , ? ´ ν 2 a 2 q J and b “ p ? 2 , ´ ? ´ ν 1 b 1 , ´ ? ´ ν 2 b 2 q J . Since ν 1 , ν 2 ď 0, b oth vectors lie in R 3 . By the Cauch y-Sc h w arz inequalit y | a || b | ě x a , b y with respect to the Euclidean inner pro duct, w e ma y obtain the desired inequalit y ( 4.33 ). □ 4.3. Main theorem. In the previous subsection, we hav e in vestigated sev eral geometric prop- erties of the quadratic form. In this subsection, we apply these prop erties to g -vectors. Deﬁnition 4.11. Let B P M 3 p R q b e a cluster-cyclic initial exchange matrix. F or eac h initial m utation direction i “ 1 , 2 , 3, deﬁne Q i , Q ` i , Q ´ i , H i , H ` i , and H ´ i as the corresp onding sets in ( 4.18 ) with resp ect to ˜ A “ ˜ A r i s and a skew-symmetrizer D of B r i s . W e call Q ` i the glob al upp er b ound for the direction i . Note that these sets Q i , Q ` i , and Q ´ i are indep endent of the choice of D . Thanks to Lemma 4.3 , all g -vectors b elong to Q i . The main result of this section sho ws that the up- p er b ound can b e reﬁned as follo ws. Theorem 4.12 (Global upp er b ound) . L et B P M 3 p R q b e a cluster-cyclic initial exchange matrix. Fix any initial mutation dir e ction i “ 1 , 2 , 3 . Then, it holds that | ∆ ěr i s p B q| Ă Q ` i . (4.34) In p articular, we have | ∆ p B q| Ă Q ` 1 Y Q ` 2 Y Q ` 3 . Remark 4.13. More strongly , by Lemma 4.4 , all the mo diﬁed g -vectors g w l ( w P T ěr i s ) are on the same plane H ` i . Pr o of. Let w ě r i s . Since Q ` i is a con vex cone by Lemma 4.9 , it suﬃces to show that every mo diﬁed g -v ector ˜ g w l b elongs to H ` i Ă Q ` i . By Lemma 4.4 , w e hav e already kno wn that ˜ g w l P H i . Supp ose that the claim do es not hold. Then, there exists a mo diﬁed g -vector ˜ g w l with w ě r i s and l P t 1 , 2 , 3 u , suc h that it do es not b elong to H ` i . Without loss of generality , w e ma y assume that suc h w is a minimal one. That is to sa y , every prop er subsequence u ď w satisﬁes ˜ g u j P H ` i . R YOT A AKAGI AND ZHICHAO CHEN 30 Note that ˜ g r i s l P H ` i ( i, l “ 1 , 2 , 3). It implies that | w | ě 2. Let u b e the mutation subsequence that is one step shorter than w . Hence, there exists M P t S, T u , suc h that w “ u M . The key p oint is the following three conditions. ˜ g u M P H ` i , ˜ g w K P H ´ i , ˜ g w K ` ˜ g u M P Q i zt 0 u . (4.35) F or simplicity , we might assume that M “ S and we can do a similar argumen t for M “ T . The ﬁrst condition is direct since u is a prop er subsequence of w . By Lemma 3.15 , ˜ g w S and ˜ g w T ha v e already app eared in t ˜ g u K , ˜ g u S , ˜ g u T u . Since we assumed that Theorem 4.12 breaks at this w , ˜ g w K should do so. Namely , the ﬁrst t w o conditions of ( 4.35 ) hold. By Lemma 3.15 , we hav e ˜ g w K ` ˜ g u S “ p u S K ˜ g u K . Since ˜ g u K P Q i zt 0 u , we hav e the third condition of ( 4.35 ). Ho w ever, due to Lemma 4.10 , the ﬁrst and the second conditions should imply that ˜ g w K ` ˜ g u M R Q i zt 0 u , whic h is a contradiction. Hence, all the mo diﬁed g -vectors lie on H ` i and | ∆ ěr i s p B q| Ă Q ` i . Moreov er, it is direct that ˜ g H j “ ˜ e j P Q ` i for any i, j “ 1 , 2 , 3, whic h implies that | ∆ p B q| Ă Q ` 1 Y Q ` 2 Y Q ` 3 . □ Example 4.14. W e can resp ectively refer to the thick blue lines in Figure 12 as the global upp er b ounds for the case of ν 1 “ ν 2 “ 0 and for the general cluster-cyclic type ( ν 1 , ν 2 ă 0). In general, the three global upp er b ounds Q ` 1 , Q ` 2 , and Q ` 3 are diﬀerent. How ev er, when an initial exc hange matrix B is giv en b y ( 3.4 ), it holds that Q ` 1 “ Q ` 2 “ Q ` 3 “ t x 1 ˜ e 1 ` x 2 ˜ e 2 ` x 3 ˜ e 3 P R 3 | x 1 ` x 2 ` x 3 ą 0 u Y t 0 u . (4.36) 5. S -mut a tions F ollowing Deﬁnition 3.9 , for an y M “ K, S, T , we write the modiﬁed c -v ectors as ˜ c w M “ ˜ c w M p w q and the mo diﬁed g -v ectors as ˜ g w M “ ˜ g w M p w q , and similarly for the remaining data. In Subsection 3.3 , we hav e seen that the m utations can b e distinguished b y S and T . In this section, we fo cus on the structure of rep eating S -mutations. Note that the b eha vior of w S n is detailed in ( 3.25 ). 5.1. F orm ulas for S -m utations. Firstly , we give an explicit S -mutation formula. In fact, it is closely related to the Chebyshev p olynomials U n p p q of the second kind, whic h is deﬁned b y the follo wing recursion: U ´ 2 p p q “ ´ 1 , U ´ 1 p p q “ 0 , U n ` 1 p p q “ 2 pU n p p q ´ U n ´ 1 p p q p n P Z ě´ 1 q . (5.1) Set u n p p q “ U n ` p 2 ˘ . Then, it ob eys the recursion u n ` 1 p p q “ pu n p p q ´ u n ´ 1 p p q . (5.2) GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 31 F or a given p ě 2, an imp ortan t fact is that t u n p p qu is monotonically increasing for n . W e also obtain the following formula for u n p p q . u n p p q “ $ & % n ` 1 p “ 2 , 1 ? p 2 ´ 4 p α n ` 1 p ´ α ´ n ` 1 p q p ą 2 , (5.3) where α p “ p ` ? p 2 ´ 4 2 and α ´ 1 p “ p ´ ? p 2 ´ 4 2 . In particular, we hav e lim n Ñ8 a p 2 ´ 4 α n p u n p p q “ α p p p ą 2 q , lim n Ñ8 1 n u n p 2 q “ α 2 “ 1 . (5.4) By using u n p p q , we can express the S -m utation formulas as follo ws. Lemma 5.1. L et w P T ztHu and n P Z ě 0 . Then, we have the fol lowing formulas: (1) F ormulas for p w S n M M 1 : p w S n S T “ p w S n T S “ ´ u n ´ 1 p p w S K q p w K T ` u n p p w S K q p w S T , p w S n K T “ p w S n T K “ ´ u n ´ 2 p p w S K q p w K T ` u n ´ 1 p p w S K q p w S T , p w S n S K “ p w S n K S “ p w S K . (5.5) (2) F ormulas for mo diﬁe d g -ve ctors ˜ g w S n M : ˜ g w S n K “ ´ u n ´ 1 p p w S K q ˜ g w S ` u n p p w S K q ˜ g w K , ˜ g w S n S “ ´ u n ´ 2 p p w S K q ˜ g w S ` u n ´ 1 p p w S K q ˜ g w K , ˜ g w S n T “ ˜ g w T . (5.6) (3) F ormulas for mo diﬁe d c -ve ctors ˜ c w S n M : ˜ c w S n K “ ´ u n ´ 2 p p w S K q ˜ c w K ´ u n ´ 1 p p w S K q ˜ c w S , ˜ c w S n S “ u n ´ 1 p p w S K q ˜ c w K ` u n p p w S K q ˜ c w S , ˜ c w S n T “ ˜ c w T . (5.7) Pr o of. They can b e established b y the induction on n , where the inductive step follo ws from the form ulas in Lemma 3.15 . More precisely , w e exhibit the pro of for ˜ g w S n ` 1 K and others are similar. Note that by ( 5.2 ) and the induction conditions for n , w e hav e ˜ g w S n ` 1 K “ ´ ˜ g w S n S ` p w S n S K ˜ g w S n K “ u n ´ 2 p p w S K q ˜ g w S ´ u n ´ 1 p p w S K q ˜ g w K ` p w S K r´ u n ´ 1 p p w S K q ˜ g w S ` u n p p w S K q ˜ g w K s “ ´ u n p p w S K q ˜ g w S ` u n ` 1 p p w S K q ˜ g w K . (5.8) □ R YOT A AKAGI AND ZHICHAO CHEN 32 5.2. Asymptotic structure. In this subsection, we exhibit the asymptotic phenomenon about S -mutations. Now, we consider the inner pro duct x a , b y D “ a J D b on R 3 and write its norm by } a } D “ a x a , a y D . W e introduce the follo wing equiv alence relation „ on R 3 zt 0 u : a „ b ð ñ b “ λ a for s ome λ ą 0 . (5.9) Then, the sphere S 2 D “ t a P R 3 | } a } D “ 1 u Ă R 3 is a representativ e of this equiv alence class. By identifying p R 3 zt 0 uq{„ with S 2 D , we in tro duce the top ology . Namely , for any sequence of nonzero vectors v n P R 3 zt 0 u , if lim n Ñ8 v n } v n } D exists, we write Ă lim n Ñ8 v n “ lim n Ñ8 v n } v n } D . (5.10) Since the sphere S 2 D is a closed set,    Ă lim n Ñ8 v n    D “ 1 holds if it exists. W e abuse this symbol „ to express the following equiv alence relation on R : a „ b ð ñ sign p a q “ sign p b q . (5.11) F or an y a , b P R 3 zt 0 u and x P R 3 , the relation a „ b on R 3 zt 0 u implies x a , x y D „ x b , x y D on R . Firstly , w e calculate Ă lim n Ñ8 ˜ c w S n S and Ă lim n Ñ8 ˜ g w S n K , whic h pla y essen tial roles in the later argumen t. W e hav e already kno wn that ˜ c w S n S and ˜ g w S n K can b e expressed as in ( 5.6 ) and ( 5.7 ). T o calculate their limits, we show the following lemma. Lemma 5.2. L et a , b P R 3 b e line arly indep endent ve ctors, and let p ě 2 . Set v n “ u n ´ 1 p p q a ` u n p p q b . Then, we have Ă lim n Ñ8 v n „ a ` α p b , (5.12) wher e α p “ p ` ? p 2 ´ 4 2 . Pr o of. Supp ose p ą 2. Then, by ( 5.4 ), we ha v e lim n Ñ8 a p 2 ´ 4 α n p v n “ a ` α p b . (5.13) Since a and b are linearly indep endent, the righ t hand side is a nonzero v ector. Th us, the claim holds. If p “ 2, then it holds that u n p p q “ n ` 1 by ( 5.3 ). Th us, we obtain v n “ n p a ` b q ` b . Note that α 2 “ 1. Hence, we ha ve lim n Ñ8 v n n “ a ` b . (5.14) □ As in Deﬁnition 3.9 , w e deﬁne p w M M 1 “ p w M p w q ,M 1 p w q and α w M M 1 “ α w M p w q ,M 1 p w q . Based on these lemmas, we may obtain the limit of S -m utations. Lemma 5.3. F or any w P T , we have Ă lim n Ñ8 ˜ g w S n K „ α w S K ˜ g w K ´ ˜ g w S , Ă lim n Ñ8 ˜ c w S n S „ α w S K ˜ c w S ` ˜ c w K . (5.15) GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 33 Pr o of. This is immediately sho wn by Lemma 5.1 and Lemma 5.2 . □ Example 5.4. Fix w P T ztHu . Then, all cones C p G w S n q obtained by S -mutations can b e illustrated as in Figure 8 in the stereographic pro jection. ˜ g w S ˜ g w T “ ˜ g w S n T H D p ˜ c w T q Ă lim n Ñ8 ˜ g w S n K „ α w S K ˜ g w K ´ ˜ g w S ˜ g w K ˜ g w S K ˜ g w S 2 K w Figure 8. G -cones obtained by S -mutations. 6. Suppor t of trunks Fix an initial mutation direction i P t 1 , 2 , 3 u . Recall from Deﬁnition 3.9 that we deﬁned the t w o kinds of subsets of the subtree T ěr i s : the trunk T ăr i s S 8 and the branches T ěr i s X , where X con tains at least one letter T . Corresp ondingly , for the G -fan, w e deﬁne the trunk of ∆ ěr i s p B q b y ∆ ăr i s S 8 p B q “ t C J p G w q | w P T ăr i s S 8 , J Ă t 1 , 2 , 3 uu , (6.1) and a br anch by ∆ ěr i s X p B q “ t C J p G w q | w P T ěr i s X , J Ă t 1 , 2 , 3 uu , (6.2) where X P M con tains at least one letter T . In this section, we give an explicit description of the trunks ∆ ăr i s S 8 p B q . Lemma 6.1. Fix an initial mutation dir e ction i “ 1 , 2 , 3 . Set k 0 , s 0 , t 0 P t 1 , 2 , 3 u as in T able 1 . F or e ach n P Z ě 0 , we have the fol lowing formulas. (1) F ormulas for p r i s S n M M 1 : p r i s S n S T “ p r i s S n T S “ ´ u n p p k 0 s 0 q p s 0 t 0 ` u n ` 1 p p k 0 s 0 q p k 0 t 0 , p r i s S n K T “ p r i s S n T K “ ´ u n ´ 1 p p k 0 s 0 q p s 0 t 0 ` u n p p k 0 s 0 q p k 0 t 0 , p r i s S n S K “ p r i s S n K S “ p k 0 s 0 . (6.3) (2) F ormulas for mo diﬁe d g -ve ctors ˜ g r i s S n M : ˜ g r i s S n K “ ´ u n p p k 0 s 0 q ˜ e k 0 ` u n ` 1 p p k 0 s 0 q ˜ e s 0 , ˜ g r i s S n S “ ´ u n ´ 1 p p k 0 s 0 q ˜ e k 0 ` u n p p k 0 s 0 q ˜ e s 0 , ˜ g r i s S n T “ ˜ e t 0 . (6.4) R YOT A AKAGI AND ZHICHAO CHEN 34 (3) F ormulas for mo diﬁe d c -ve ctors ˜ c r i s S n M : ˜ c r i s S n K “ ´ u n ´ 1 p p k 0 s 0 q ˜ e s 0 ´ u n p p k 0 s 0 q ˜ e k 0 , ˜ c r i s S n S “ u n p p k 0 s 0 q ˜ e s 0 ` u n ` 1 p p k 0 s 0 q ˜ e k 0 , ˜ c r i s S n T “ ˜ e t 0 . (6.5) Pr o of. This can b e directly prov ed according to Lemma 3.18 and Lemma 5.1 . □ Prop osition 6.2. Fix an initial mutation dir e ction i “ 1 , 2 , 3 , and set k 0 “ K pr i sq , s 0 “ S pr i sq , and t 0 “ T pr i sq . Then, we have Ă lim n Ñ8 ˜ g r i s S n K „ α s 0 k 0 ˜ e s 0 ´ ˜ e k 0 , Ă lim n Ñ8 ˜ c r i s S n S „ α s 0 k 0 ˜ e k 0 ` ˜ e s 0 . (6.6) Mor e over, we have | ∆ ăr i s S 8 p B q| “ C ˝ p ˜ e t 0 , ˜ e s 0 , α k 0 s 0 ˜ e s 0 ´ ˜ e k 0 q Y C p ˜ e s 0 , ˜ e t 0 q Y C ˝ p ˜ e s 0 , α k 0 s 0 ˜ e s 0 ´ ˜ e k 0 q “ ´ H ´ D p ˜ e k 0 q X H ` D p ˜ e t 0 q X H ` D p α s 0 k 0 ˜ e k 0 ` ˜ e s 0 q ¯ Y C p ˜ e t 0 q . (6.7) Pr o of. This can b e sho wn b y Lemma 5.3 . □ Example 6.3. The G -cones in the trunks ∆ ăr i s S 8 p B q are illustrated as in Figure 9 . Figure 9. T runks. 7. Local upper bounds of branches In this section, w e introduce the lo cal upp er b ound for eac h branc h ∆ ě w p B q that p ossesses diﬀeren t prop erties from the global upp er b ound introduced in Section 4 . Let ∆ ě w p B q b e a branch. Namely , w e assume that w “ r i s X , where X P M contains at least one letter T . In this case, b y ( 3.38 ) and ( 5.15 ), we hav e Ă lim n Ñ8 ˜ g w T S n K „ p p w T K α w T K ´ 1 q ˜ g w K ´ α w T K ˜ g w T , Ă lim n Ñ8 ˜ c w T S n S „ p p w T K α w T K ´ 1 q ˜ c w T ` α w T K ˜ c w K . (7.1) By ( 3.32 ), the equality p w T K α w T K ´ 1 “ p α w T K q 2 holds. Thus, w e hav e Ă lim n Ñ8 ˜ g w T S n K „ α w T K ˜ g w K ´ ˜ g w T , Ă lim n Ñ8 ˜ c w T S n S „ α w T K ˜ c w T ` ˜ c w K . (7.2) GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 35 No w, we fo cus on the tw o planes H D p α w S K ˜ c w S ` ˜ c w K q and H D p α w T K ˜ c w T ` ˜ c w K q . The ﬁrst one app ears as the limit of ˜ c w S n S , and the second one app ears as the limit of ˜ c w T S n S . Since such tw o v ectors α w S K ˜ c w S ` ˜ c w K and α w T K ˜ c w T ` ˜ c w K are linearly indep endent, their in tersection is spanned b y one nonzero vector, which will b e denoted by ¯ g w later. This is illustrated in Figure 10 . ¨ ¨ ¨ ˜ g w S n K ¨ ¨ ¨ ˜ g w T S n K w H D ´ Ă lim n Ñ8 ˜ c w S n S ¯ H D ´ Ă lim n Ñ8 ˜ c w T S n S ¯ ¯ g w O Figure 10. ¯ g w and V w . W e ﬁrst giv e an explicit expression for such a vector ¯ g w . Lemma 7.1. L et ∆ ě w p B q b e a br anch. L et ¯ g w “ ˜ g w K ´ p α w S K q ´ 1 ˜ g w S ´ p α w T K q ´ 1 ˜ g w T . (7.3) Then, we have H D p α w S K ˜ c w S ` ˜ c w K q X H D p α w T K ˜ c w T ` ˜ c w K q “ x ¯ g w y vec . (7.4) Pr o of. Note that H D p α w S K ˜ c w S ` ˜ c w K q and H D p α w T K ˜ c w T ` ˜ c w K q are 2-dimensional subspaces of R 3 . Since α w S K ˜ c w S ` ˜ c w K and α w T K ˜ c w T ` ˜ c w K are linearly indep enden t, these tw o subspaces H D p α w S K ˜ c w S ` ˜ c w K q and H D p α w T K ˜ c w T ` ˜ c w K q are diﬀerent. In particular, H D p α w S K ˜ c w S ` ˜ c w K q X H D p α w T K ˜ c w T ` ˜ c w K q is a prop er subspace of H D p α w S K ˜ c w S ` ˜ c w K q . Hence, the dimension of their intersection is one. By ( 2.22 ), the vector ¯ g w is orthogonal to b oth α w S K ˜ c w S ` ˜ c w K and α w T K ˜ c w T ` ˜ c w K . Thus, this v ector is in H D p α w S K ˜ c w S ` ˜ c w K q X H D p α w T K ˜ c w T ` ˜ c w K q . Since it is nonzero v ector, the subspace H D p α w S K ˜ c w S ` ˜ c w K q X H D p α w T K ˜ c w T ` ˜ c w K q is spanned by ¯ g w . □ Lemma 7.2. L et ∆ ě w p B q b e a br anch. L et ¯ g w “ ˜ g w K ´ p α w S K q ´ 1 ˜ g w S ´ p α w T K q ´ 1 ˜ g w T . Then, one of the normal ve ctors of C p g w K , ¯ g w q is α w S K ˜ c w S ´ α w T K ˜ c w T . (7.5) Pr o of. By ( 2.22 ), this can b e sho wn b y a direct calculation. □ R YOT A AKAGI AND ZHICHAO CHEN 36 Based on these facts, w e deﬁne the following v ectors and sets. Deﬁnition 7.3. F or eac h branc h ∆ ě w p B q , let ¯ g w “ ˜ g w K ´ p α w S K q ´ 1 ˜ g w S ´ p α w T K q ´ 1 ˜ g w T , ¯ c w “ α w S K ˜ c w S ´ α w T K ˜ c w T . (7.6) W e also deﬁne the following t wo subsets of R 3 . V w ˝ “ C ˝ p ˜ g w S , ˜ g w T , ¯ g w q , V w “ C p ˜ g w S , ˜ g w T q Y V w ˝ . (7.7) The set V w is called the lo c al upp er b ound of the branch ∆ ě w p B q . Note that V w ˝ ma y b e expressed as V w ˝ “ H ` D ´ Ă lim n Ñ8 ˜ c w S n S ¯ X H ` D ´ Ă lim n Ñ8 ˜ c w T S n S ¯ X H ` D p ˜ c w K q . (7.8) By deﬁnition, these tw o sets V w ˝ and V w are conv ex cones. Since ˜ g w K “ ¯ g w ` p α w S K q ´ 1 ˜ g w S ` p α w T K q ´ 1 ˜ g w T P V w ˝ , (7.9) w e obtain C p G w q Ă V w . Hence, these vectors and sets can b e illustrated as in Figure 10 and Figure 11 . w K S T w S w T ¯ g w ¯ g w S ¯ g w T H D ´ Ă lim n Ñ8 ˜ c w S n S ¯ H D ´ Ă lim n Ñ8 ˜ c w T S n S ¯ H D p ˜ c w K q Figure 11. Lo cal upp er b ounds. The thick blue lines form V w , and the thin blue lines form V w S and V w T . The cen tral red line is H D p ¯ c w q . The following theorem gives a strong restriction for the structure of branc hes in the G -fan. Theorem 7.4 (Lo cal upp er b ound) . F or any br anch ∆ ě w p B q , the fol lowing inclusions hold. | ∆ ě w p B q| Ă C p G w q Y V w S ˝ Y V w T ˝ Ă V w . (7.10) Mor e over, the fol lowing statements hold. ( a ) Two sets V w S ˝ and V w T ˝ ar e sep ar ate d by the plane H D p ¯ c w q , that is, V w S ˝ Ă H ´ D p ¯ c w q and V w T ˝ Ă H ` D p ¯ c w q hold. Mor e over, the union C p G w q Y V w S ˝ Y V w T ˝ is disjoint. GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 37 ( b ) The lo c al upp er b ounds ar e monotonic al ly de cr e asing. Namely, V w Ą V w S and V w Ą V w T hold. This theorem can b e illustrated in Figure 11 . T o proceed, w e ﬁrst pro v e the follo wing relations, whic h form a key p oint in the pro of of Theorem 7.4 . Lemma 7.5. L et ∆ ě w p B q b e a br anch. Then, we have ¯ g w S „ ¯ g w ` “ p α w T K q ´ 1 ´ p α w T S α w S K q ´ 1 ‰ ˜ g w T , ¯ g w T „ ¯ g w ` “ p α w S K q ´ 1 ´ p α w T S α w T K q ´ 1 ‰ g w S . (7.11) In p articular, ¯ g w S P C p ¯ g w , ˜ g w T q and ¯ g w T P C p ¯ g w , ˜ g w S q hold. Pr o of. W e aim to pro v e the ﬁrst relation for w S . (The second one can b e sho wn by a sim ilar argumen t.) By Deﬁnition 7.3 , we hav e ¯ g w S “ ˜ g w S K ´ p α w S S K q ´ 1 ˜ g w S S ´ p α w S T K q ´ 1 ˜ g w S T . (7.12) By Lemma 3.15 , we may obtain α w S S K “ α w S K and α w S T K “ α w T S . Moreov er, by substituting the equalities in Lemma 3.15 , w e obtain ¯ g w S “ p´ ˜ g w S ` p w S K ˜ g w K q ´ p α w S K q ´ 1 ˜ g w K ´ p α w T S q ´ 1 ˜ g w T “ “ p w S K ´ p α w S K q ´ 1 ‰ ˜ g w K ´ ˜ g w S ´ p α w T S q ´ 1 ˜ g w T . (7.13) By ( 3.32 ), w e ha v e p w S K ´ p α w S K q ´ 1 “ α w S K . Thus, it holds that ¯ g w S “ α w S K ˜ g w K ´ ˜ g w S ´ p α w T S q ´ 1 ˜ g w T „ ˜ g w K ´ p α w S K q ´ 1 ˜ g w S ´ p α w S K α w T S q ´ 1 ˜ g w T “ ¯ g w ` “ p α w T K q ´ 1 ´ p α w T S α w S K q ´ 1 ‰ ˜ g w T , (7.14) where „ on the second ro w is obtained by multiplying p α w S K q ´ 1 to the previous one. Moreo v er, b y ( 3.9 ), the co eﬃcien t of ˜ g w T is non-negative. Thus, ¯ g w S P C p ¯ g , ˜ g w T q holds. □ Pr o of of The or em 7.4 . Firstly , w e show ( a ). The inclusion V w S ˝ Ă H ` D p ¯ c w q can b e prov ed by: x ˜ g w S S , ¯ c w y D “ x ˜ g w K , α w S K ˜ c w S ´ α w T K ˜ c w T y D “ 0 , x ˜ g w S T , ¯ c w y D “ x ˜ g w T , α w S K ˜ c w S ´ α w T K ˜ c w T y D “ ´ α w T K ă 0 , x ¯ g w S , ¯ c w y D „ x ¯ g w ` β ˜ g w T , α w S K ˜ c w S ´ α w T K ˜ c w T y D “ ´ β α w T K ď 0 , (7.15) where we set β “ p α w T K q ´ 1 ´ p α w T S α w S K q ´ 1 ě 0. Note that the intersection C p ¯ g w S , ˜ g w S S , ˜ g w S T q X H D p ¯ c w q is a subset of C p ¯ g w S , ˜ g w S S q , which does not in tersect with V w S ˝ . By the same argument, w e may obtain V w T ˝ Ă H ´ D p ¯ c w q . The remaining problem is to show C p G w q X V w S ˝ “ C p G w q X V w T ˝ “ H . This is sho wn b y C p G w q Ă H ` D p ˜ c w S q X H ` D p ˜ c w T q , V w S ˝ Ă H ` D ` ˜ c w S K ˘ “ H ´ D p ˜ c w S q , and V w T ˝ Ă H ` D ` ˜ c w S T ˘ “ H ´ D p ˜ c w T q . R YOT A AKAGI AND ZHICHAO CHEN 38 Next, we show ( b ). By deﬁnition and ( 7.9 ), w e ha v e ˜ g w S S “ ˜ g w K P V w and ˜ g w S T “ ˜ g w T P V w . By Lemma 7.5 , w e hav e ¯ g w S P C p g w S T , ¯ g w q , which is a subset of the b oundary of V w . Th us, we ha v e V w S Ă V w . Lastly , we sho w the inclusions ( 7.10 ). The second one has already b een shown b y p b q . W e aim to sho w the ﬁrst one. F or an y cone C p G w X q with X “ M 1 M 2 ¨ ¨ ¨ M r P M ( r ě 1, M i P t S, T u ), the following inclusions hold. V w M 1 Ą V w M 1 M 2 Ą ¨ ¨ ¨ Ą V w X Ą C p G w X q . (7.16) Th us, w e hav e C p G w X q Ă V w M 1 Ă C p G w q Y V w M 1 ˝ . □ Example 7.6. W e give t wo examples in Figure 12 . The left one is the case of ν 1 “ ν 2 “ 0 and the right one is the case of ν 1 , ν 2 ă 0. There is remark able, sp ecial phenomenon when ν 1 “ ν 2 “ 0, whic h is equiv alen t to p 12 “ p 23 “ p 31 “ 2 as in Remark 4.7 . In this case, for any w P T , p w 12 “ p w 23 “ p w 31 “ 2 and α w 12 “ α w 23 “ α w 31 “ 1 hold. By considering the equality in Lemma 7.5 , this condition implies that ¯ g w „ ¯ g w S „ ¯ g w T . Th us, if ν 1 “ ν 2 “ 0, the direction of ¯ g w dep ends on its maximal branc h ∆ ěr i s S n T p B q . In fact, more strongly , this vector ¯ g w dep ends on the initial mutation i “ 1 , 2 , 3 only , which is given by ¯ g w “ ˜ e k 0 ´ ˜ e s 0 ´ ˜ e t 0 . (7.17) Figure 12. Pictures of G -fans. The left one corresp onds to the case ν 1 “ ν 2 “ 0 (e.g. the Mark o v quiv er), and the righ t one corresp onds to the case ν 1 , ν 2 ă 0 (general cluster-cyclic type). The thick blue lines indicate the global upp er b ounds. The dashed blue lines indicate the local upp er b ounds of maximal branc hes. GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 39 8. Sep ara teness among local upper bounds Recall that Theorem 7.4 guaran tees that every G -cone except the initial one C p G H q is a subset of either V r i s S n T or | ∆ ăr i s S 8 p B q| , where i “ 1 , 2 , 3 and n P Z ě 0 . Let T i b e the in terior of | ∆ ăr i s S 8 | . By ( 6.7 ), it is given by T i “ C ˝ p ˜ e t 0 , ˜ e s 0 , α k 0 s 0 ˜ e s 0 ´ ˜ e k 0 q “ H ´ D p ˜ e k 0 q X H ` D p ˜ e t 0 q X H ` D p α s 0 k 0 ˜ e k 0 ` ˜ e s 0 q . (8.1) In this section, we establish the follo wing prop osition concerning the separateness among the lo cal upp er b ounds and the in teriors of trunks. Prop osition 8.1. F or any W, U P t T 1 , T 2 , T 3 u Y t V r i s S n T ˝ | i “ 1 , 2 , 3; n P Z ě 0 u , if W ‰ U , we have W X U “ H . (8.2) 8.1. Relations among maximal branches. In this subsection, we ﬁx an initial mutation direction i P t 1 , 2 , 3 u , and set k 0 “ K pr i sq , s 0 “ S pr i sq , and t 0 “ T pr i sq as in T able 1 . Firstly , w e giv e the expressions of some v ectors for maximal branches V r i s S n T ˝ . Lemma 8.2. L et i “ 1 , 2 , 3 , and set k 0 , s 0 , t 0 P t 1 , 2 , 3 u as in T able 1 . Then, for any n P Z ě 0 , we have ˜ g r i s S n T M “ $ ’ ’ & ’ ’ % ´ ˜ e t 0 ` p r i s S n S T p u n p p k 0 s 0 q ˜ e s 0 ´ u n ´ 1 p p k 0 s 0 q ˜ e k 0 q M “ K , u n p p k 0 s 0 q ˜ e s 0 ´ u n ´ 1 p p k 0 s 0 q ˜ e k 0 M “ S, u n ` 1 p p k 0 s 0 q ˜ e s 0 ´ u n p p k 0 s 0 q ˜ e k 0 M “ T , (8.3) ¯ g r i s S n T “ ´ ˜ e t 0 ´ ” p α r i s S n T K q ´ 1 u n ` 1 p p k 0 s 0 q ´ α r i s S n S T u n p p k 0 s 0 q ı ˜ e s 0 ` ” p α r i s S n T K q ´ 1 u n p p k 0 s 0 q ´ α r i s S n S T u n ´ 1 p p k 0 s 0 q ı ˜ e k 0 . (8.4) Ă lim m Ñ8 ˜ c r i s S n T S m S „ α r i s S n S T ˜ e t 0 ` u n p p k 0 s 0 q ˜ e s 0 ` u n ` 1 p p k 0 s 0 q ˜ e k 0 , (8.5) Ă lim m Ñ8 ˜ c r i s S n T 2 S m S „ ´p α r i s S n T K q ´ 1 ˜ e t 0 ´ u n ´ 1 p p k 0 s 0 q ˜ e s 0 ´ u n p p k 0 s 0 q ˜ e k 0 . (8.6) Pr o of. They can b e sho wn b y Lemma 6.1 , Lemma 3.16 , and Lemma 5.3 . □ Motiv ated by the equalities ( 8.5 ) and ( 8.6 ), we deﬁne c ` i p n q “ α r i s S n S T ˜ e t 0 ` u n p p k 0 s 0 q ˜ e s 0 ` u n ` 1 p p k 0 s 0 q ˜ e k 0 , (8.7) c ´ i p n q “ ´p α r i s S n T K q ´ 1 ˜ e t 0 ´ u n ´ 1 p p k 0 s 0 q ˜ e s 0 ´ u n p p k 0 s 0 q ˜ e k 0 . (8.8) By ( 7.8 ) and ˜ c r i s S n T K “ ´ ˜ e t 0 , we hav e V r i s S n T ˝ “ H ` D ` c ` i p n q ˘ X H ` D ` c ´ i p n q ˘ X H ´ D p ˜ e t 0 q . (8.9) Then, the relation among maximal branches can b e giv en as follows: R YOT A AKAGI AND ZHICHAO CHEN 40 Lemma 8.3. Fix an initial mutation dir e ction i “ 1 , 2 , 3 . Then, for any m, n P Z ě 0 , the fol lowing statements hold. ( a ) If m ă n , we have V r i s S m T ˝ Ă H ` D ` c ` i p n q ˘ , V r i s S m T ˝ Ă H ´ D ` c ´ i p n q ˘ . (8.10) ( b ) If m ą n , we have V r i s S m T ˝ Ă H ´ D ` c ` i p n q ˘ , V r i s S m T ˝ Ă H ` D ` c ´ i p n q ˘ . (8.11) ( c ) If m “ n , we have V r i s S n T ˝ Ă H ` D ` c ` i p n q ˘ , V r i s S n T ˝ Ă H ` D ` c ´ i p n q ˘ . (8.12) In this pro of, we often use the following fact. Lemma 8.4. Fix one w P T ztHu , and set p “ p w S K and α “ α w S K . F or e ach l P Z ě 0 , let F p l q “ α w S l T K α l and G p l q “ α w S l T K α ´ l . Then, F p l q is non-de cr e asing and G p l q is non-incr e asing. Pr o of. Note that α “ α w S l S K and α w S l ` 1 T K “ α w S l S T hold by ( 3.33 ). Thus, w e hav e F p l ` 1 q ´ F p l q “ α l ´ α w S l S T α w S l S K ´ α w S l T K ¯ ě 0 , G p l ` 1 q ´ G p l q “ α ´ l ´ 1 ´ α w S l S T ´ α w S l T K α w S l S K ¯ ď 0 , (8.13) where the last inequalities are obtained by ( 3.9 ). □ Pr o of of L emma 8.3 . F or simplicity , w e write p “ p k 0 s 0 and u n “ u n p p q for any n P Z ě´ 2 . Firstly , w e pro ve the claim for c ` i p n q . By Lemma 8.2 and ( 8.7 ), w e ha v e x ˜ g r i s S m T S , c ` i p n qy D “ u m u n ´ u m ´ 1 u n ` 1 , x ˜ g r i s S m T T , c ` i p n qy D “ u m ` 1 u n ´ u m u n ` 1 , x ¯ g r i s S m T , c ` i p n qy D “ ´ α r i s S n S T ´ p α r i s S m T K q ´ 1 p u m ` 1 u n ´ u m u n ` 1 q ` α r i s S m S T p u m u n ´ u m ´ 1 u n ` 1 q . (8.14) Set f p m, n q “ u m ` 1 u n ´ u m u n ` 1 . Then, the ab o v e equalities can b e expressed as x ˜ g r i s S m T S , c ` i p n qy D “ f p m ´ 1 , n q , x ˜ g r i s S m T T , c ` i p n qy D “ f p m, n q , x ¯ g r i s S m T , c ` i p n qy D “ ´ α r i s S n S T ´ p α r i s S m T K q ´ 1 f p m, n q ` α r i s S m S T f p m ´ 1 , n q . (8.15) If m, n ě ´ 1, w e ha ve f p m, n q “ p pu m ´ u m ´ 1 q u n ´ u m p pu n ´ u n ´ 1 q “ u m u n ´ 1 ´ u m ´ 1 u n “ f p m ´ 1 , n ´ 1 q . (8.16) Let m ą n . Then, by applying the ab ov e equality rep eatedly , we hav e f p m, n q “ f p m ´ n ´ 2 , ´ 2 q “ ´ u m ´ n ´ 1 , f p m ´ 1 , n q “ f p m ´ n ´ 3 , ´ 2 q “ ´ u m ´ n ´ 2 . (8.17) Since m ´ n ě 1, w e obtain x ˜ g r i s S m T S , c ` i p n qy D “ ´ u m ´ n ´ 2 ď 0 , x ˜ g r i s S m T T , c ` i p n qy D “ ´ u m ´ n ´ 1 ă 0 . (8.18) GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 41 Moreo v er, w e obtain x ¯ g r i s S m T , c ` i p n qy D “ ´ α r i s S n S T ` p α r i s S m T K q ´ 1 u m ´ n ´ 1 ´ α r i s S m S T u m ´ n ´ 2 “ ´ α r i s S n S T ` p α r i s S m T K q ´ 1 ´ u m ´ n ´ 1 ´ α r i s S m T K α r i s S m S T u m ´ n ´ 2 ¯ . (8.19) By ( 3.9 ), we ha v e α r i s S m T K α r i s S m S T ě α r i s S m S K . Moreo ver, by ( 5.5 ), p r i s S m S K “ p r i s S n S K “ p holds. So, let us write α “ α p p q “ α r i s S m S K “ α r i s S n S K . Then, w e ha v e x ¯ g r i s S m T , c ` i p n qy D ď ´ α r i s S n S T ` p α r i s S m T K q ´ 1 p u m ´ n ´ 1 ´ αu m ´ n ´ 2 q (8.20) Set l “ m ´ n ě 1. Then, we hav e u l ´ 1 ´ αu l ´ 2 “ p p ´ α q u l ´ 2 ´ u l ´ 3 “ α ´ 1 u l ´ 2 ´ u l ´ 3 “ α ´ 1 p u l ´ 2 ´ αu l ´ 3 q . (8.21) By applying this equality rep eatedly , we obtain u l ´ 1 ´ αu l ´ 2 “ α 1 ´ l (8.22) and x ¯ g r i s S m T , c ` i p n qy D ď ´ α w S T ` p α w S l T K q ´ 1 α 1 ´ l , (8.23) where we set w “ r i s S n . If l “ 1, the righ t hand side is non-p ositiv e by α w S K , α w T K ě 1. Moreov er, since p α w S l T K q ´ 1 α 1 ´ l “ α ´ α w S l T K α l ¯ ´ 1 , the right hand side is non-increasing b y Lemma 8.4 . Th us, x ¯ g r i s S m T , c ` i p n qy D ď 0 holds. Therefore, we obtain V r i s S m T ˝ Ă H ´ D ` c ` i p n q ˘ when m ą n . Let m ď n . Then, by applying ( 8.16 ) rep eatedly , we ha ve f p m, n q “ f p´ 2 , n ´ m ´ 2 q “ u n ´ m ´ 1 , f p m ´ 1 , n q “ f p´ 2 , n ´ m ´ 1 q “ u n ´ m . (8.24) Th us, b y ( 8.14 ), w e ha v e x ˜ g r i s S m T S , c ` i p n qy D “ u n ´ m ą 0 , x ˜ g r i s S m T T , c ` i p n qy D “ u n ´ m ´ 1 ą 0 , (8.25) and x ¯ g r i s S m T , c ` i p n qy D “ ´ α r i s S n S T ´ p α r i s S m T K q ´ 1 u n ´ m ´ 1 ` α r i s S m S T u n ´ m “ ´ α r i s S n S T ` α r i s S m S T ´ u n ´ m ´ p α r i s S m S T α r i s S m T K q ´ 1 u n ´ m ´ 1 ¯ ( 3.9 ) ě ´ α r i s S n S T ` α r i s S m S T ` u n ´ m ´ α ´ 1 u n ´ m ´ 1 ˘ , (8.26) where α “ α r i s S m S K . Set w “ r i s S m and l “ n ´ m ě 0. Then, by the same argument in ( 8.22 ), w e ha v e u l ´ α ´ 1 u l ´ 1 “ α l . (8.27) Th us, w e obtain x ¯ g r i s S m T , c ` i p n qy D ě ´ α w S l S T ` α w S T α l “ α l ´ ´ α w S l S T α ´ l ` α w S T ¯ . (8.28) R YOT A AKAGI AND ZHICHAO CHEN 42 When l “ 0, ´ α w S l S T α ´ l ` α w S T “ 0 holds. Moreo v er, b y Lemma 8.4 , α w S l S T α ´ l “ α α w S l ` 1 T K α ´p l ` 1 q is non-increasing. Thus, ´ α w S l S T α ´ l ` α w S T is non-decreasing, and we hav e x ¯ g r i s S m T , c ` i p n qy D ě 0. Therefore, we obtain V r i s S m T ˝ Ă H ` D ` c ` i p n q ˘ when m ď n . F or c ´ i p n q , we hav e x ˜ g r i s S m T S , c ´ i p n qy D “ ´ f p m ´ 1 , n ´ 1 q , x ˜ g r i s S m T T , c ´ i p n qy D “ ´ f p m, n ´ 1 q , x ¯ g r i s S m T , c ´ i p n qy D “ p α r i s S n T K q ´ 1 ` p α r i s S m T K q ´ 1 f p m, n ´ 1 q ´ α r i s S m S T f p m ´ 1 , n ´ 1 q . (8.29) If m ă n , b y ( 8.24 ), we ha ve x ˜ g r i s S m T S , c ´ i p n qy D “ ´ u n ´ m ´ 1 ă 0 , x ˜ g r i s S m T T , c ´ i p n qy D “ ´ u n ´ m ´ 2 ď 0 , (8.30) and x ¯ g r i s S m T , c ´ i p n qy D “ p α r i s S n T K q ´ 1 ` p α r i s S m T K q ´ 1 u n ´ m ´ 2 ´ α r i s S m S T u n ´ m ´ 1 “ p α r i s S n T K q ´ 1 ´ α r i s S m S T ´ u n ´ m ´ 1 ´ p α r i s S m T K α r i s S m S T q ´ 1 u n ´ m ´ 2 ¯ ( 3.9 ) ď p α r i s S n T K q ´ 1 ´ α r i s S m S T ` u n ´ m ´ 1 ´ α ´ 1 u n ´ m ´ 2 ˘ . (8.31) Set l “ n ´ m ě 1 and w “ r i s S m . By ( 8.27 ), we ha v e x ¯ g r i s S m T , c i p n qy D ď p α w S l T K q ´ 1 ´ α w S T α l ´ 1 “ p α w S l T K q ´ 1 ´ 1 ´ α w S T α w S l T K α l ´ 1 ¯ . (8.32) If l “ 1, 1 ´ α w S T α w S l T K α l ´ 1 is non-p ositive b ecause α w S T , α w S l T K ě 1. Moreov er, by Lemma 8.4 , the factor 1 ´ α w S T α w S l T K α l ´ 1 is non-increasing. Th us, x ¯ g r i s S m T , c ´ i p n qy D ď 0 holds. Therefore, w e obtain V r i s S m T ˝ Ă H ´ D ` c ´ i p n q ˘ when m ă n . If m ě n , b y ( 8.17 ), we ha ve x ˜ g r i s S m T S , c ´ i p n qy D “ u m ´ n ´ 1 ą 0 , x ˜ g r i s S m T T , c ´ i p n qy D “ u m ´ n ą 0 , (8.33) and x ¯ g r i s S m T , c ´ i p n qy D “ p α r i s S n T K q ´ 1 ´ p α r i s S m T K q ´ 1 u m ´ n ` α r i s S m S T u m ´ n ´ 1 “ p α r i s S n T K q ´ 1 ´ p α r i s S m T K q ´ 1 ´ u m ´ n ´ α r i s S m T K α r i s S m S T u m ´ n ´ 1 ¯ ( 3.9 ) ě p α r i s S n T K q ´ 1 ´ p α r i s S m T K q ´ 1 p u m ´ n ´ αu m ´ n ´ 1 q . (8.34) Set l “ m ´ n ě 0 and w “ r i s S n . Then, b y ( 8.22 ), w e ha v e x ¯ g r i s S m T , c ´ i p n qy D ě p α w T K q ´ 1 ´ p α w S l T K q ´ 1 α ´ l ě p α w T K q ´ 1 ´ p α w S 0 T K q ´ 1 α 0 “ 0 (8.35) b y Lemma 8.4 . Therefore, we obtain V r i s S m T ˝ Ă H ` D ` c ´ i p n q ˘ when m ě n . □ Thanks to Lemma 8.3 , ev ery lo cal upper b ound V r i s S m T ˝ is even tually included in H ` D ` c ` i p n q ˘ for enough large n . The limit of c ` i p n q is given as follo ws. GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 43 Lemma 8.5. F or e ach i “ 1 , 2 , 3 , we have Ă lim n Ñ8 c ` i p n q „ p´ p s 0 t 0 ` α s 0 k 0 p k 0 t 0 q ˜ e t 0 ` ˜ e s 0 ` α s 0 k 0 ˜ e k 0 (8.36) Pr o of. Set p “ p s 0 k 0 and α “ α s 0 k 0 . Supp ose p ą 2. By ( 5.4 ) and ( 6.3 ), w e ha v e lim n Ñ8 a p 2 ´ 4 α n ` 1 p r i s S n S T “ ´ p s 0 t 0 ` αp k 0 t 0 . (8.37) Since α r i s S n S T “ 1 2 ˆ p r i s S n S T ` b p p r i s S n S T q 2 ´ 4 ˙ , we also hav e lim n Ñ8 a p 2 ´ 4 α n ` 1 α r i s S n S T “ ´ p s 0 t 0 ` αp k 0 t 0 (8.38) Therefore, we obtain lim n Ñ8 a p 2 ´ 4 α n ` 1 c ` i p n q “ p´ p s 0 t 0 ` αp k 0 t 0 q ˜ e t 0 ` ˜ e s 0 ` α ˜ e k 0 . (8.39) When p “ 2, the claim is shown b y replacing ? p 2 ´ 4 α n ` 1 with 1 n in the ab ov e argument. □ Based on the observ ations in this subsection, let us deﬁne c ` i “ p´ p s 0 t 0 ` α s 0 k 0 p k 0 t 0 q ˜ e t 0 ` ˜ e s 0 ` α s 0 k 0 ˜ e k 0 , c ´ i “ c ´ i p 0 q “ ´ ˜ e k 0 ´ α ´ 1 t 0 k 0 ˜ e t 0 . (8.40) Then, we obtain the follo wing upp er b ound. Lemma 8.6. F or e ach i “ 1 , 2 , 3 , we have ď m P Z ě 0 V r i s S m T ˝ Ă H ` D ` c ` i ˘ X H ` D ` c ´ i ˘ X H ´ D p ˜ e t 0 q . (8.41) Pr o of. The inclusions V r i s S m T ˝ Ă H ´ D p ˜ e t 0 q and V r i s S m T ˝ Ă H ` D ` c ´ i ˘ are sho wn b y ( 8.9 ) and Lemma 8.3 . W e sho w V r i s S m T ˝ Ă H ` D ` c ` i ˘ . Let x P V r i s S m T ˝ . Then, by Lemma 8.3 , w e hav e x x , c ` i p n qy D ą 0 for an y n ą m . Since c ` i „ Ă lim n Ñ8 c ` i p n q , b y taking its limit with some normaliza- tion, we hav e x x , c ` i y D ě 0 . (8.42) Th us, w e hav e V r i s S m T ˝ Ă H ` D ` c ` i ˘ . Since V r i s S m T ˝ is an op en set, this implies V r i s S m T ˝ Ă ´ H ` D ` c ` i ˘ ¯ ˝ “ H ` D ` c ` i ˘ as desired. □ F rom Lemma 8.6 , we ma y give other rough but simple upp er b ounds via some planes. F or example, since ´ ˜ e k 0 ´ ˜ e t 0 “ c ´ i ´ p 1 ´ α ´ 1 t 0 k 0 q ˜ e t 0 (8.43) and 1 ´ α ´ 1 t 0 k 0 ě 0, we ha ve ď m P Z ě 0 V r i s S m T ˝ Ă H ` D p´ ˜ e k 0 ´ ˜ e t 0 q . (8.44) R YOT A AKAGI AND ZHICHAO CHEN 44 This is b ecause, if x b elongs to the set in the right hand size of ( 8.41 ), then we hav e x x , ´ ˜ e t 0 ´ ˜ e k 0 y D “ x x , c ´ i y D ´ p 1 ´ α ´ 1 s 0 k 0 qx x , ˜ e t 0 y D ą 0 . (8.45) Similarly , since ˜ e s 0 ` α s 0 k 0 ˜ e k 0 “ c ` i ´ p´ p s 0 t 0 ` α s 0 k 0 p k 0 t 0 q ˜ e t 0 (8.46) and ´ p s 0 t 0 ` α s 0 k 0 p k 0 t 0 ě 0 by ( 3.8 ), we ha ve ď m P Z ě 0 V r i s S m T ˝ Ă H ` D p ˜ e s 0 ` α s 0 k 0 ˜ e k 0 q . (8.47) Moreo v er, b y ( 6.7 ), this region H ` D p ˜ e s 0 ` α s 0 k 0 ˜ e k 0 q also includes the trunk ∆ ăr i s S 8 p B q except for one line C p ˜ e t 0 q . So, w e obtain the following prop osition. Prop osition 8.7. F or e ach i “ 1 , 2 , 3 , we have | ∆ ěr i s p B q| Ă H ` D p ˜ e s 0 ` α s 0 k 0 ˜ e k 0 q Y C p ˜ e t 0 q . (8.48) Pr o of. This follows from ( 6.7 ) and ( 8.47 ). □ 8.2. Pro of of Prop osition 8.1 . In this subsection, we fo cus on pro ving Prop osition 8.1 . Pr o of of Pr op osition 8.1 . W e pro v e the claim by case-by-case calculation in the follo wing list: (1) W “ T i and U “ T j with i ‰ j . (2) W “ V r i s S n T ˝ and U “ V r i s S m T ˝ with n ‰ m . (3) W “ V r i s S n T ˝ and U “ T j . (4) W “ V r i s S n T ˝ and U “ V r j s S m T ˝ with i ‰ j . In this pro of, we set k 0 “ K pr i sq , s 0 “ S pr i sq , and t 0 “ T pr i sq . ( 1 ) This follows from T i Ă H ´ D p ˜ e i q X H ` D p ˜ e j q and T j Ă H ` D p ˜ e i q X H ´ D p ˜ e j q by Figure 9 . ( 2 ) This follows from Lemma 8.3 . ( 3 ) If j “ i , this follows from V r i s S n T ˝ Ă H ´ D p ˜ e t 0 q (Lemma 8.6 ) and T i Ă H ` D p ˜ e t 0 q . Suppose i ‰ j . If j “ s 0 , we hav e K pr j sq “ s 0 , S pr j sq “ t 0 , and T pr j sq “ k 0 b y ( 3.42 ). By ( 8.1 ), we ha v e T j “ H ´ D p ˜ e s 0 q X H ` D p ˜ e k 0 q X H ` D p α t 0 s 0 ˜ e s 0 ` ˜ e t 0 q . (8.49) Note that H ´ D p ˜ e s 0 q X H ` D p α t 0 s 0 ˜ e s 0 ` ˜ e t 0 q Ă H ` D p ˜ e t 0 q since for any x in the left hand side, x x , ˜ e t 0 y D “ x x , α t 0 s 0 ˜ e s 0 ` ˜ e t 0 y D ´ α t 0 s 0 x x , ˜ e s 0 y D ą 0 . (8.50) Hence, T j Ă H ` D p ˜ e t 0 q holds. On the other hand, by ( 8.41 ), we hav e V r i s S n T ˝ Ă H ´ D p ˜ e t 0 q . Thus, they are disjoin t. If j “ t 0 , w e hav e K pr j sq “ t 0 , S pr j sq “ k 0 , and T pr j sq “ s 0 b y ( 3.43 ). Then, w e ha v e T j “ H ´ D p ˜ e t 0 q X H ` D p ˜ e s 0 q X H ` D p α k 0 t 0 ˜ e t 0 ` ˜ e k 0 q . (8.51) Note that H ´ D p ˜ e t 0 q X H ` D p α k 0 t 0 ˜ e t 0 ` ˜ e k 0 q Ă H ` D p ˜ e t 0 ` ˜ e k 0 q holds b ecause, for an y x in the left hand side, x x , ˜ e t 0 ` ˜ e k 0 y D “ x x , α k 0 t 0 ˜ e t 0 ` ˜ e k 0 y D ´ p α k 0 t 0 ´ 1 qx x , ˜ e t 0 y D ą 0 , (8.52) GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 45 where we use α k 0 t 0 ě 1. On the other hand, b y ( 8.44 ), we hav e V r i s S n T ˝ Ă H ´ D p ˜ e t 0 ` ˜ e k 0 q holds. Th us, the claim holds. ( 4 ) In this case, w e may additionally assume j “ t 0 without loss of generality . (If j “ s 0 , then i “ T pr j sq holds.) Then, we ha ve K pr j sq “ t 0 , S pr j sq “ k 0 , and T pr j sq “ s 0 b y ( 3.43 ). By ( 8.47 ), we ha v e V r j s S m T ˝ Ă H ` D p ˜ e k 0 ` α k 0 t 0 ˜ e t 0 q . On the other hand, H ´ D p ˜ e t 0 ` ˜ e k 0 q X H ´ D p e t 0 q Ă H ´ D p ˜ e k 0 ` α k 0 t 0 ˜ e t 0 q holds since for an y x in the left hand side, x x , ˜ e k 0 ` α k 0 t 0 ˜ e t 0 y D “ x x , ˜ e k 0 ` ˜ e t 0 y D ` p α s 0 k 0 ´ 1 qx x , ˜ e t 0 y D ă 0 . (8.53) Th us, by ( 8.41 ) and ( 8.44 ), we hav e V r i s S n T ˝ Ă H ´ D p ˜ e k 0 ` α k 0 t 0 ˜ e t 0 q , which completes the pro of. □ 9. Applica tions to g -vectors In this section, w e presen t an application of Theorem 4.12 , Theorem 7.4 , and Prop osition 8.1 to the g -v ectors, esp ecially ab out the non-p erio dicity and the sign structure. 9.1. Non-p erio dicity of g -v ectors. In [ AC25b , Thm. 5.2], we show ed that there is no p eri- o dicit y among G -matrices, nor among G -cones. Here, w e pro v e a stronger result on g -vectors. W e fo cus on an equality of g -vectors g w l “ g u m for w , u P T and l, m “ 1 , 2 , 3. An equalit y g w l “ g u m is said to b e trivial if l “ m and u “ w r k 1 , . . . , k r s with k i ‰ l . (In this case, this equalit y alw a ys holds by the m utation rules ( 2.5 ) of g -vectors.) Theorem 9.1. Supp ose that an initial exchange matrix is cluster-cyclic of r ank 3 . Then, al l e qualities g w l “ g u m ( l, m “ 1 , 2 , 3 , w , u P T ) of g -ve ctors ar e trivial. Pr o of. More strongly , we claim that there are no nontrivial cone equalities C p g w l q “ C p g u m q . F or this purp ose, we can use the mo diﬁed g -vectors instead of ordinary ones because C p g w l q “ C p ˜ g w l q holds. Note that by Lemma 3.15 and Lemma 3.16 , for any M “ S, T , ˜ g w M S and ˜ g w M T resp ectiv ely coincide with one of t ˜ g w K , ˜ g w S , ˜ g w T u , and the induced equalities are trivial. Moreov er, all mo diﬁed g -vectors except the initial ones ˜ e 1 , ˜ e 2 , ˜ e 3 can b e expressed as the form ˜ g w K with w ‰ H . Th us, if a nontrivial equalit y exists, one of them should b e expressed as C p ˜ g w K q “ C p ˜ g u K q with w ‰ u or C p ˜ g w K q “ C p ˜ e m q , where m P t 1 , 2 , 3 u and u , w ‰ H . W e will prov e that this phenomenon never o ccurs. If both w and u are in trunks, this fact can be shown by Lemma 5.1 . Supp ose that w is in a branc h, that is, w “ r i s S n T X for some i “ 1 , 2 , 3, n P Z ě 0 , and X P M . Then, b y ( 7.9 ) and Theorem 7.4 , w e ha v e ˜ g w K P V w ˝ Ă V r i s S n T ˝ . If u is in a trunk, then ˜ g u K is on the b oundary of some T j ( j “ 1 , 2 , 3), which is the closure of T j . Since V w ˝ is an op en set, V r i s S n T ˝ X T j “ H holds by Prop osition 8.1 . Th us, C p ˜ g w K q ‰ C p ˜ g u K q holds. The same argument w orks if we replace ˜ g u K with ˜ e m b ecause ˜ e m also b elongs to T j . Supp ose that u is also in a branc h, that is u “ r j s S m T Y for some j “ 1 , 2 , 3, m P Z ě 0 , and Y P M . If p i, n q ‰ p j, m q , then the claim follows from Prop osition 8.1 . (Note that ˜ g w K P V r i s S n T ˝ and ˜ g u K P V r j s S m T ˝ .) W e R YOT A AKAGI AND ZHICHAO CHEN 46 no w assume that p i, n q “ p j, m q . Let w 0 b e the maxim um common preﬁx of w and u . Since r i s S n T ď w 0 , this preﬁx w 0 also b elongs to a branch. Then, b y Theorem 7.4 , each of ˜ g w K and ˜ g u K b elongs to one of C p G w 0 q , V w 0 S ˝ , or V w 0 T ˝ . F urthermore, since w 0 is the maximum common preﬁx of w and u , these tw o vectors cannot b e con tained in the same set. Note that these three sets are pairwise disjoint. Hence, w e obtain C p ˜ g w K q ‰ C p ˜ g u K q . □ Thanks to the results in [ CIKLFP13 , CL20 , Nak23 ], we can restate this theorem via cluster v ariables. Let us consider an inte ger cluster-cyclic initial exchange matrix. In this case, cluster v ariables x w i ( i “ 1 , 2 , 3, w P T ) are deﬁned as in [ FZ02 ]. By [ Nak23 , Thm. I I.7.2], the equality g w l “ g u m of g -vectors holds if and only if the equality x w l “ x u m of cluster v ariables holds for an y l , m “ 1 , 2 , 3 and w , u P T . Thus, Theorem 9.1 implies the following corollary . (Here, we deﬁne the triviality of equalities of cluster v ariables as the same wa y in g -vectors.) Corollary 9.2. Supp ose that an initial exchange matrix is integer cluster-cyclic of r ank 3 . Then, al l the e qualities of cluster variables x w l “ x u m ar e trivial. 9.2. Sign structure of g -vectors. It is known that every row vector of eac h G -matrix is sign- coheren t [ GHKK18 ]. Let τ w i P t˘ 1 u b e the sign of i th row vector of G w . As exp ected from Figure 1 , these signs are given as follows. Theorem 9.3. Consider the G -matric es asso ciate d with a cluster-cyclic initial exchange matrix B of r ank 3 . Fix an initial mutation dir e ction i “ 1 , 2 , 3 , and set k 0 “ K pr i sq , s 0 “ S pr i sq , t 0 “ T pr i sq as in T able 1 . Then, for e ach n P Z ě 0 and X P M , the signs of G -matric es ar e given in T able 2 . w r i s S n r i s S n ` 1 T X r i s T S n r i s T S n T X ¨ ˝ τ w k 0 τ w s 0 τ w t 0 ˛ ‚ ¨ ˝ ´ ` ` ˛ ‚ ¨ ˝ ´ ` ´ ˛ ‚ ¨ ˝ ´ ` ´ ˛ ‚ ¨ ˝ ` ` ´ ˛ ‚ T able 2. The list of signs of G -matrices. Example 9.4. F or a given B and i , w e can easily obtain the characterization of w in T able 2 . F or example, consider the case of ( 3.23 ). Then, b y T able 1 , we ha ve k 0 “ 1, s 0 “ 3, and t 0 “ 2. By following the rule in Lemma 3.10 , each w ě r 1 s can b e characterized as follo ws: ‚ w “ r i s S n : A ﬁnite sequence of the form w “ r 1 , 3 , 1 , 3 , . . . s . ‚ w “ r i s S n ` 1 T X : A ﬁnite sequence starting with r 1 , 3 s and containing at least one term 2. (Namely , w “ r 1 , 3 , . . . , 2 , . . . s .) ‚ w “ r i s T S n : A ﬁnite sequence of the form w “ r 1 , 2 , 1 , 2 , . . . s . ‚ w “ r i s T S n T X : A ﬁnite sequence starting with r 1 , 2 s and containing at least one term 3. (Namely , w “ r 1 , 2 , . . . , 3 , . . . s .) GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 47 Remark 9.5. F or the later purpose, w e giv e one corollary of Theorem 9.3 . Consider a g -v ector g w j “ p x 1 , x 2 , x 3 q J with w ě r i s . Then, by Theorem 9.3 , we directly obtain x s 0 ě 0 . (9.1) Consider the sign of x t 0 . Then, b y Theorem 9.3 , if w R T ăr i s S 8 , we hav e x t 0 ď 0. In fact, b y ( 6.4 ), the unique p ositive vector is e t 0 . Thus, w e can conclude that, except for one vector g r i s S n t 0 “ e t 0 , we hav e x t 0 ď 0 . (9.2) The claim for the ﬁrst case w “ r i s S n in Theorem 9.3 can b e obtained directly by Lemma 6.1 . F or the third case w “ r i s T S n , this follows from the following formulas. Lemma 9.6. Set p 1 “ p r i s T S K . F or any n P Z ě 0 , we have ˜ g r i s T S n K “ u n ` 1 p p 1 q ˜ e s 0 ´ u n p p 1 q ˜ e t 0 , ˜ g r i s T S n S “ u n p p 1 q ˜ e s 0 ´ u n ´ 1 p p 1 q ˜ e t 0 , ˜ g r i s T S n T “ ´ ˜ e k 0 ` p k 0 s 0 ˜ e s 0 . (9.3) Pr o of. When n “ 0, the claim holds by a direct calculation based on Lemma 3.18 and Lemma 3.16 . Moreo v er, b y taking w “ r i s T in ( 5.6 ) , we obtain the formula. □ In the following pro of, w e alwa ys assume that B is skew-symmetric. By Prop osition 2.17 , it is enough to sho w this claim. In this case, by taking D “ diag p 1 , 1 , 1 q , each mo diﬁed g -vector coincides with the corresp onding ordinary g -vector, and the inner pro duct x , y D coincides with the usual Euclidean inner pro duct x , y . 9.2.1. Key lemma. F or the pro of of Theorem 9.3 , we ﬁx i , k 0 , s 0 and t 0 as stated b efore. Let v b e the vector given b y ( 4.11 ) with resp ect to B r i s , and write v “ p v 1 , v 2 , v 3 q J . T o simplify the notation, we write q lm “ p r i s lm , β lm “ α r i s lm . (9.4) Note that q lm and β lm corresp ond to not the p l , m q th entry of the initial exc hange matrix B but the one of B r i s (although they coincide if l “ i or m “ i ). The following inequality pla ys an essential role in the forthcoming pro of. Lemma 9.7. F or any l, m P t 1 , 2 , 3 u with l ‰ m , we have β lm v l ´ v m ě 0 . (9.5) Pr o of. Let n P t 1 , 2 , 3 uzt l , m u . T o simplify the notation, we write p “ q lm , q “ q ln , r “ q mn , and α p “ β lm . Then, b y ( 4.15 ), w e ha v e v l “ p λ ´ 2 q 2 ` p p ` q qp λ ´ 2 q ` pr ` q r ´ r 2 , v m “ p λ ´ 2 q 2 ` p p ` r qp λ ´ 2 q ` pq ` r q ´ q 2 . (9.6) R YOT A AKAGI AND ZHICHAO CHEN 48 By a direct calculation, w e ha v e β lm v l ´ v m “ p α p ´ 1 qp λ ´ 2 q 2 ` α p q r ´ α p r 2 ´ q r ` q 2 ` rp α p ´ 1 q p ` p α p q ´ r qsp λ ´ 2 q ` p p α p r ´ q q ě p α p ´ 1 qp λ ´ 2 q 2 ` α p q r ´ α p r 2 ´ q r ` q 2 , (9.7) where the last inequality follo ws from ( 3.8 ) and α p ´ 1 ě 0. Since p λ ´ 2 q 2 ą p 2 ` q 2 ` r 2 b y ( 4.10 ), we hav e β lm v l ´ v m ě p α p ´ 1 qp p 2 ` q 2 ` r 2 q ` α p q r ´ q r ´ α p r 2 ` q 2 “ p α p ´ 1 q p 2 ` q p α p q ´ r q ` r p α p q ´ r q ě 0 . (9.8) □ 9.2.2. Case of r i s T S n T X . Let D b e the union of C p G r i s T S n T X q with n P Z ě 0 and X P M . Let D ˝ b e its interior. W e giv e the following upp er b ound. Lemma 9.8. Supp ose that B is skew-symmetric. Then, we have D ˝ Ă H ` p ´ e t 0 ´ β k 0 t 0 e k 0 q X H ` p e k 0 q X H ` p v q . (9.9) Pr o of. The inclusion D ˝ Ă H ` p v q follows from Theorem 4.12 . By Theorem 7.4 , w e ha ve D ˝ Ă V r i s T ˝ “ H ` ´ Ă lim n Ñ8 c r i s T S n S ¯ X H ` ´ Ă lim n Ñ8 c r i s T 2 S n S ¯ X H ´ p e t 0 q . By ( 3.34 ), ( 3.36 ), and ( 3.38 ), for an y m P Z ě 0 , we hav e c r i s T S m T K “ ´ c r i s T S m T “ ´ c r i s T T “ ´ c r i s K “ e k 0 . (9.10) In particular, by ( 7.8 ), we hav e V r i s T S m ˝ Ă H ` p e k 0 q . Since D ˝ is a subset of the union of V r i s T S m T ˝ , we hav e D ˝ Ă H ` p e k 0 q . Moreov er, by Lemma 3.16 and Lemma 5.3 , w e ha ve Ă lim n Ñ8 c r i s T 2 S n S „ ´ e t 0 ´ β k 0 t 0 e k 0 . Thus, w e also obtain D ˝ Ă H ` p´ e t 0 ´ β k 0 t 0 e k 0 q . □ In what follows, let us prov e this case in Theorem 9.3 . Pr o of. Let x “ p x 1 , x 2 , x 3 q J P H ` p´ e t 0 ´ β k 0 t 0 e k 0 q X H ` p e k 0 q X H ` p v q . By x P H ` p e k 0 q , x k 0 ą 0 holds. By x P H ` p´ e t 0 ´ β k 0 t 0 e k 0 q , we hav e ´ x t 0 ´ β k 0 t 0 x k 0 ą 0. This implies x t 0 ă ´ β k 0 t 0 x k 0 ă 0 . (9.11) Since x P H ` p v q , we ha ve x k 0 v k 0 ` x s 0 v s 0 ` x t 0 v t 0 ą 0. Since x t 0 ă ´ β k 0 t 0 x k 0 , we hav e x s 0 v s 0 ą ´ x k 0 v k 0 ´ x t 0 v t 0 ą x k 0 p β k 0 t 0 v t 0 ´ v k 0 q ě 0 , (9.12) where the last inequality follo ws from ( 9.5 ). This completes the pro of. □ 9.2.3. Case of r i s S n ` 1 T X . In parallel to the pro of of the case of r i s S n T X , we giv e the following upp er b ound. GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 49 Lemma 9.9. Supp ose that B is skew-symmetric. Then, we have ď n P Z ě 0 V r i s S n ` 1 T ˝ Ă H ` ` c ´ i p 1 q ˘ X H ´ p e t 0 q X H ` p v q . (9.13) Pr o of. This can b e sho wn b y Theorem 4.12 , Lemma 8.3 , and ( 8.41 ). □ As a consequence, we can completely ﬁnish the pro of of Theorem 9.3 . Pr o of. Let x “ p x 1 , x 2 , x 3 q J P H ` ` c ´ i p 1 q ˘ X H ´ p e t 0 q X H ` p v i q . By x P H ´ p e t 0 q , w e hav e x t 0 ă 0. By x P H ` ` c ´ i p 1 q ˘ and p k 0 s 0 “ p r i s k 0 s 0 “ q k 0 s 0 , we hav e x x , c ´ i p 1 qy “ ´p α r i s S T K q ´ 1 x t 0 ´ x s 0 ´ q k 0 s 0 x k 0 ą 0 . (9.14) Note that, by ( 3.33 ), we ha ve α r i s S T K “ α r i s S T “ β s 0 t 0 . Thus, w e obtain ´ β ´ 1 s 0 t 0 x t 0 ´ x s 0 ´ q k 0 s 0 x k 0 ą 0 . (9.15) By x P H ` p v q , we ha ve v t 0 x t 0 ` v s 0 x s 0 ` v k 0 x k 0 ą 0 . (9.16) By ( 9.15 ) and ( 9.16 ), we obtain p v t 0 ´ β ´ 1 s 0 t 0 v s 0 q x t 0 ` p v k 0 ´ q k 0 s 0 v s 0 q x k 0 ą 0 , (9.17) and it implies p q k 0 s 0 v s 0 ´ v k 0 q x k 0 ă β ´ 1 s 0 t 0 p β s 0 t 0 v t 0 ´ v s 0 q x t 0 . (9.18) Then, by x t 0 ă 0 and ( 9.5 ), the righ t hand side is non-p ositive. On the other hand, since q k 0 s 0 ą β k 0 s 0 , we hav e q k 0 s 0 v s 0 ´ v k 0 ą β k 0 s 0 v s 0 ´ v k 0 ě 0 . (9.19) Th us, the inequality ( 9.18 ) implies x k 0 ă 0. By x t 0 , x k 0 ă 0 and ( 9.16 ), w e obtain x s 0 ą 0. □ 10. Under the minimum assumption F or most rank 3 cluster-cyclic matrices, it is known that their m utation equiv alence classes con tain the minimum element. By c ho osing the initial exchange matrix to b e such a minim um one, several additional structural properties emerge. In this section, w e establish the monotonic- it y of g -v ectors and introduce a simple global upp er b ound under this condition. 10.1. Minim um assumption. T o state the minim um assumption, we deﬁne the preorder ď on M 3 p R q as A “ p a ij q ď A 1 “ p a 1 ij q ð ñ | a ij | ď | a 1 ij | p@ i, j “ 1 , 2 , 3 q . (10.1) Note that by ( 2.1 ) and ( 2.2 ), for an y j P t 1 , 2 , 3 u , either µ j p B q ě B or B ě µ j p B q (or b oth) holds. Moreov er, the following fact is known. R YOT A AKAGI AND ZHICHAO CHEN 50 Lemma 10.1 ([ BBH11 , Lem. 2.1 (a)], [ Ak a24 , Lem. 7.5]) . F or any cluster-cyclic skew-symmetrizable matrix B , ther e ar e at le ast two indic es j P t 1 , 2 , 3 u satisfying µ j p B q ě B . Mor e over, we set t j, l, m u “ t 1 , 2 , 3 u . If µ j p B q ě B , the lar gest entry in t p r j s lm , p r j s j l , p r j s j m u is p r j s lm . Deﬁnition 10.2. F or any cluster-cyclic sk ew-symmetrizable matrix B P M 3 p R q , w e call the sequence (admitting empty , ﬁnite, and inﬁnite) δ p B q “ r k 1 , k 2 , . . . s a de cr e asing se quenc e for B if the following statements hold. ‚ F or each l “ 1 , 2 , . . . , the inequality µ k l p B q ě B do es not hold. (Note that the statements exclude the p ossibility that µ k l p B q “ ´ B .) ‚ If δ p B q is a ﬁnite sequence, then it holds that µ j p B δ p B q q ě B δ p B q for any j “ 1 , 2 , 3. If the decreasing sequence for B is empt y ( δ p B q “ H ), we sa y that B is minimum in this B -pattern B p B q . By Lemma 10.1 , this decreasing sequence δ p B q is uniquely determined by B . Moreov er, since µ j is an inv olution, δ p B q should b e reduced. By the follo wing prop erty , most B -patterns corresp onding to cluster-cyclic matrices hav e the minim um elemen t. Lemma 10.3 ([ Ak a24 , Lem. 7.5, Thm. 10.3]) . L et B P M 3 p R q b e a cluster-cyclic skew-symmetrizable matrix. ( a ) The de cr e asing se quenc e δ p B q is ﬁnite if and only if either of the fol lowing holds. ‚ C p B q ă 4 . ‚ Sk p B q is mutation e quivalent to the tr ansp osition or p ermutation of the fol lowing matrix. ¨ ˚ ˝ 0 ´ p p p 0 ´ 2 ´ p 2 0 ˛ ‹ ‚ p p P R ě 2 q . (10.2) ( b ) Supp ose that δ p B q is ﬁnite. L et B 1 b e mutation e quivalent to B . Then, δ p B 1 q is also ﬁnite, and B δ p B q “ ˘p B 1 q δ p B 1 q holds. In other wor ds, if the minimum element B δ p B q exists, it is unique up to the diﬀer enc e of the sign. F or any inte ger cluster-cyclic exchange matrix, its decreasing sequence is known to b e ﬁnite. Th us, w e can take the minim um elemen t in its mutation equiv alence class. Example 10.4. W e giv e three types of examples as follows. ( a ) Consider the case of B “ ¨ ˚ ˝ 0 ´ 228 1795 228 0 ´ 409252 ´ 1795 409252 0 ˛ ‹ ‚ . (10.3) GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 51 The Marko v constant is C p B q “ ´ 7. Then, we obtain δ p B q “ r 1 , 2 , 3 , 2 , 1 s as follows. ´ 0 ´ 228 1795 228 0 ´ 409252 ´ 1795 409252 0 ¯ ´ 0 228 ´ 1795 ´ 228 0 8 1795 ´ 8 0 ¯ ´ 0 ´ 228 29 228 0 ´ 8 ´ 29 8 0 ¯ ´ 0 4 ´ 29 ´ 4 0 8 29 ´ 8 0 ¯ ´ 0 ´ 4 3 4 0 ´ 8 ´ 3 8 0 ¯ ´ 0 4 ´ 3 ´ 4 0 4 3 ´ 4 0 ¯ ´ 0 ´ 4 13 4 0 ´ 4 ´ 13 4 0 ¯ ´ 0 ´ 8 3 8 0 ´ 4 ´ 3 4 0 ¯ ď 2 1 ě 2 ě 3 ě 1 ě 2 ď 3 ď (10.4) In this B -pattern, the minim um element is uniquely determined. ( b ) Consider that B “ ¨ ˚ ˝ 0 ´ 3 18 3 0 ´ 7 ´ 18 7 0 ˛ ‹ ‚ , B 1 “ ¨ ˚ ˝ 0 3 ´ 18 ´ 3 0 7 18 ´ 7 0 ˛ ‹ ‚ . (10.5) The Marko v constant is C p B q “ 4. Then, we obtain the following decreasing sequence: ´ 0 ´ 3 18 3 0 ´ 7 ´ 18 7 0 ¯ ´ 0 3 ´ 3 ´ 3 0 7 3 ´ 7 0 ¯ ´ 0 ´ 3 3 3 0 ´ 2 ´ 3 2 0 ¯ ´ 0 3 ´ 18 ´ 3 0 7 18 ´ 7 0 ¯ ´ 0 ´ 3 3 3 0 ´ 7 ´ 3 7 0 ¯ ´ 0 3 ´ 3 ´ 3 0 2 3 ´ 2 0 ¯ 2 ě 1 ě 2 ě 1 ě 2 , 3 ď ě (10.6) In this B -pattern, there are tw o p ossible minimum elemen ts (up to sign) as follo ws B δ p B q “ ¨ ˚ ˝ 0 ´ 3 3 3 0 ´ 2 ´ 3 2 0 ˛ ‹ ‚ , p B 1 q δ p B 1 q “ ¨ ˚ ˝ 0 3 ´ 3 ´ 3 0 2 3 ´ 2 0 ˛ ‹ ‚ . (10.7) ( c ) Lastly , w e consider the exchange matrix B “ ¨ ˚ ˝ 0 ´ 14 . 5 4 . 75 14 . 5 0 ´ 3 . 5 ´ 4 . 75 3 . 5 0 ˛ ‹ ‚ . (10.8) The Mark o v constan t is C p B q “ 4. This is an example where δ p B q is inﬁnite. (The classiﬁcation of this case is given by [ FT19 , Rem. 4.6].) F or example, w e obtain the following decreasing sequence δ p B q “ r 1 , 2 , 3 , . . . s . ´ 0 ´ 14 . 5 4 . 75 14 . 5 0 ´ 3 . 5 ´ 4 . 75 3 . 5 0 ¯ ´ 0 2 . 125 ´ 4 . 75 ´ 2 . 125 0 3 . 5 4 . 75 ´ 3 . 5 0 ¯ ´ 0 ´ 2 . 125 2 . 6875 2 . 125 0 ´ 3 . 5 ´ 2 . 6875 3 . 5 0 ¯ 1 ě 2 ě 3 ě ¨ ¨ ¨ (10.9) As a conclusion of this subsection, we introduce the following assumption. R YOT A AKAGI AND ZHICHAO CHEN 52 Deﬁnition 10.5. F or a B -pattern B asso ciated with a cluster-cyclic exc hange matrix, if w e tak e an initial exchange matrix B as the minim um element in B “ B p B q , we call this choice the minimum assumption . Lemma 10.6 ([ Ak a24 , Lem. 7.5]) . L et B P M 3 p R q b e a cluster-cyclic initial exchange matrix and i P t 1 , 2 , 3 u satisfying µ i p B q ě B . F or any w , u ě r i s , if u ď w , we have B u ď B w . In p articular, under the minimum assumption, we have B u ď B w for any u ď w . 10.2. Monotonicit y of g -vectors. F or an y vector x “ p x 1 , x 2 , x 3 q J P R 3 , w e deﬁne its absolute v alue vector as abs p x q “ p| x 1 | , | x 2 | , | x 3 |q J . (10.10) Under the minimum assumption, w e can show the follo wing monotonicit y of g -v ectors. Theorem 10.7. L et B P M 3 p R q b e a cluster-cyclic initial exchange matrix, and let i P t 1 , 2 , 3 u satisfy µ i p B q ě B . ( a ) F or any w , u ě r i s , if w ď u , we have abs p ˜ g w K q ď abs p ˜ g u K q . (10.11) ( b ) F or any w , u ě r i s and j “ 1 , 2 , 3 , if w ď u , we have abs p g w j q ď abs p g u j q , (10.12) wher e g w j and g u j ar e the or dinary g -ve ctors, not the mo diﬁe d ones. In p articular, under the minimum assumption, the ab ove ine qualities ( 10.11 ) and ( 10.12 ) always hold for any w ď u and j “ 1 , 2 , 3 . Unfortunately , the assumption µ i p B q ě B cannot b e eliminated, see Example 10.10 . Note that the claim ( a ) can directly imply ( b ) due to ( 2.19 ). And, the k ey point for the pro of of Theorem 10.7 is the following lemma. Lemma 10.8. L et B P M 3 p R q b e a cluster-cyclic initial exchange matrix. L et w P T b e in a br anch. Supp ose that abs p ˜ g w K q ě abs p ˜ g w S q , abs p ˜ g w T q . (10.13) Then, for e ach M “ S, T , we have abs p ˜ g w M K q ě abs p ˜ g w M S q , abs p ˜ g w M T q , abs p ˜ g w K q . (10.14) In p articular, if ( 10.13 ) holds, then we have ˜ g u 1 K ě ˜ g u K for any u 1 ě u ě w . Pr o of. W e show the case of M “ S . (The case of M “ T is similar.) Since ˜ g w S S “ ˜ g w K and ˜ g w S T “ ˜ g w T , it suﬃces to sho w abs p ˜ g w S K q ´ abs p ˜ g w M 1 q ě 0 for an y M 1 “ K , S, T . First, we obtain abs p ˜ g w S K q “ p w S K abs p ˜ g w K q ´ abs p ˜ g w S q . (10.15) GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 53 Let g w K “ p x 1 , x 2 , x 3 q J and g w S “ p y 1 , y 2 , y 3 q J . Then, by the sign-coherence of G -matrices, the sign of x i and y i is the same (admitting to combine 0 and another sign). Let τ i P t˘ 1 u b e the sign of x i and y i . Then, we may express abs p ˜ g w K q “ p τ 1 x 1 , τ 2 x 2 , τ 3 x 3 q J and abs p g w S q “ p τ 1 y 1 , τ 2 y 2 , τ 3 y 3 q J . Thus, w e ha v e p w S K abs p ˜ g w K q ´ abs p ˜ g w S q “ ¨ ˚ ˝ τ 1 p p w S K x 1 ´ y 1 q τ 2 p p w S K x 2 ´ y 2 q τ 3 p p w S K x 3 ´ y 3 q ˛ ‹ ‚ . (10.16) Note that the sign of p w S K x i ´ y i should b e the same as τ i b ecause p w S K ě 2 and | x i | ě | y i | . Thus, w e obtain ( 10.15 ). By using this equality , we decomp ose abs p ˜ g w S K q ´ abs p ˜ g w M 1 q into the sum of the following three terms. p p w S K ´ 2 q abs p ˜ g w K q ` p abs p ˜ g w K q ´ abs p ˜ g w S qq ` p abs p ˜ g w K q ´ abs p ˜ g w M 1 qq . (10.17) W e can show that eac h term is larger or equal to 0 b y using p w S K ě 2 and abs p ˜ g w K q ě abs p ˜ g w S q , abs p ˜ g w T q . Thus, w e obtain the claim. □ Thanks to this claim, once the inequalit y ( 10.13 ) holds, the monotonicit y ( 10.11 ) alw ays holds after that. Unfortunately , this condition do es not alwa ys hold, so we need to consider more carefully to prov e Theorem 10.7 . The strategy of the follo wing pro of is that w e prov e ( 10.13 ) when w “ r i s S n T ( n ě 1) and w “ r i s T S n T ( n ě 0), and w e directly chec k the monotonicity for the other cases, w “ r i s S n and w “ r i s T S n . Pr o of of The or em 10.7 . By Lemma 10.8 , it suﬃces to show the following four conditions. (In the pro of of ( 2 ) and ( 4 ), the assumption µ i p B q ě B is needed.) (1) abs p ˜ g r i s S n K q ď abs p ˜ g r i s S m K q for 0 ď n ď m . (2) Let w “ r i s S n T with n “ 1 , 2 , . . . . Then, ( 10.13 ) holds. (3) abs p ˜ g r i s T S n K q ď abs p ˜ g r i s T S m K q for 0 ď n ď m . (4) Let w “ r i s T S n T with n “ 0 , 1 , 2 , . . . . Then, ( 10.13 ) holds. Due to Lemma 10.8 , for each w “ r i s S n T and w “ r i s T S n T , ( 2 ) and ( 4 ) imply the monotonicity after that. Moreo v er, since ˜ g r i s S n K “ ˜ g r i s S n T T and ˜ g r i s T S n K “ ˜ g r i s T S n T S b y ( 3.36 ) and ( 3.38 ), the inequalities abs p ˜ g r i s S n K q ď abs p ˜ g r i s S n T K q and abs p ˜ g r i s T S n K q ď abs p ˜ g r i s T S n T K q can b e shown by ( 2 ) and ( 4 ). ( 1 ) W e can chec k it by Lemma 5.1 . (Note that u n ` 1 p p q ě u n p p q .) ( 2 ) By ( 8.3 ), it suﬃces to sho w p r i s S n S T u n ´ 1 p p k 0 s 0 q ě u n p p k 0 s 0 q (10.18) for any n P Z ě 1 . Since u n ´ 2 p p k 0 s 0 q ě u ´ 1 p p k 0 s 0 q “ 0, we hav e u n p p k 0 s 0 q “ p k 0 s 0 u n ´ 1 p p k 0 s 0 q ´ u n ´ 2 p p k 0 s 0 q ď p k 0 s 0 u n ´ 1 p p k 0 s 0 q . Moreov er, according to Lemma 10.1 , the largest element in t p r i s S n 12 , p r i s S n 23 , p r i s S n 31 u should b e expressed as p r i s S n S T . In particular, we obtain p r i s S n S T ě p r i s S n k 0 s 0 . By Lemma 10.6 , w e hav e p r i s S n k 0 s 0 ě p k 0 s 0 . Th us, w e hav e u n p p k 0 s 0 q ď p k 0 s 0 u n ´ 1 p p k 0 s 0 q ď R YOT A AKAGI AND ZHICHAO CHEN 54 p r i s S n S T u n ´ 1 p p k 0 s 0 q as we desired. ( 3 ) This can b e v eriﬁed b y ( 9.3 ). ( 4 ) Set p 1 “ p r i s T S K . By Lemma 3.16 and ( 9.3 ), w e ha v e ˜ g r i s T S n T S “ ˜ g r i s T S n K “ u n ` 1 p p 1 q ˜ e s 0 ´ u n p p 1 q ˜ e t 0 , ˜ g r i s T S n T T “ ˜ g r i s T S n S “ u n p p 1 q ˜ e s 0 ´ u n ´ 1 p p 1 q ˜ e t 0 , ˜ g r i s T S n T K “ ˜ e k 0 ` p p r i s T S n K T u n ` 1 p p 1 q ´ p k 0 s 0 q ˜ e s 0 ´ p r i s T S n K T u n p p 1 q ˜ e t 0 . (10.19) Th us, it suﬃces to sho w the following inequality for an y n P Z ě 0 : p r i s T S n K T u n ` 1 p p 1 q ´ p k 0 s 0 ě u n ` 1 p p 1 q . (10.20) W e aim to show p p r i s T S n K T ´ 1 q u n ` 1 p p 1 q ě p k 0 s 0 . By p r i s T S n K T ě 2, we ha ve p p r i s T S n K T ´ 1 q u n ` 1 p p 1 q ě u n ` 1 p p 1 q ě u 1 p p 1 q “ p 1 . (10.21) By ( 3.35 ), w e ha v e p 1 “ p r i s T S K “ p r i s S T “ p r i s s 0 t 0 . (10.22) Note that i ‰ s 0 and i ‰ t 0 . Th us, by Lemma 10.1 , the largest num b er in t p r i s k 0 s 0 , p r i s s 0 t 0 , p r i s t 0 k 0 u is p 1 “ p r i s s 0 t 0 . Namely , we ha ve p 1 ě p r i s k 0 s 0 ( 3.44 ) “ p k 0 s 0 . (10.23) Th us, w e obtain p p r i s T S n K T ´ 1 q u n ` 1 p p 1 q ě p k 0 s 0 as desired. This completes the pro of. □ Remark 10.9. Due to the ab ov e pro of, we can replace the assumption with a more restrictive one. Namely , if the conditions ( 2 ) and ( 4 ) are satisﬁed, that is, if p r i s S n ` 1 S T u n p p k 0 s 0 q ě u n ` 1 p p k 0 s 0 q , p r i s T S n K T u n ` 1 p p r i s T S K q ´ p k 0 s 0 ě u n ` 1 p p r i s T S K q (10.24) for any n P Z ě 0 , then the monotonicity holds in T ěr i s . Example 10.10. W e give one coun terexample when we lose the assumption µ i p B q ě B . Let an initial exchange matrix b e the one in ( 10.3 ). By a direct calculation, we may obtain the follo wing G -matrices asso ciated with the m utation sequence w “ r 1 , 2 , 1 s . ´ 1 0 0 0 1 0 0 0 1 ¯ 1 ÞÑ ´ ´ 1 0 0 0 1 0 1795 0 1 ¯ 2 ÞÑ ´ ´ 1 0 0 0 ´ 1 0 1795 8 1 ¯ 1 ÞÑ ´ 1 0 0 ´ 228 ´ 1 0 29 8 1 ¯ . (10.25) In particular, the p 3 , 1 q -th en try decreases by the last mutation. Th us, in this case, the mono- tonicit y do es not hold in general. (If we restrict the mutations in T ěr 2 s and T ěr 3 s , the mono- tonicit y holds.) GEOMETRIC STRUCTURES OF G -F ANS ASSOCIA TED WITH RANK 3 CLUSTER-CYCLIC EXCHANGE MA TRICES 55 On the other hand, ev en if the condition µ i p B q ě B do es not hold, the monotonicit y sometimes still holds. F or example, let an initial exchange matrix b e B “ ¨ ˚ ˝ 0 ´ 4 3 4 0 ´ 8 ´ 3 8 0 ˛ ‹ ‚ . (10.26) It holds that µ 1 p B q “ ´ 0 4 ´ 3 ´ 4 0 4 3 ´ 4 0 ¯ ğ B . (10.27) Ho w ever, b y a direct calculation, we may chec k that the inequalities in ( 10.24 ) hold, whic h implies that the monotonicity holds in this case. 10.3. Simpliﬁcation of global upp er b ounds. In Section 4 , we in tro duced the global upp er b ound, whic h is an upp er b ound only dep ending on the initial mutation i “ 1 , 2 , 3. In this subsection, we simplify this upp er b ound under the minim um assumption. Let Q initial , Q ` initial and Q ´ initial b e the sets deﬁned b y Q , Q ` and Q ´ in ( 4.18 ) with resp ect to the initial exchange matrix B together with ˜ A . Theorem 10.11. L et B P M 3 p R q b e a cluster-cyclic initial exchange matrix, and let i P t 1 , 2 , 3 u satisfy µ i p B q ě B . Then, we have | ∆ ěr i s p B q| Ă Q ` initial . (10.28) In p articular, under the minimum assumption, we have | ∆ p B q| Ă Q ` initial . (10.29) Fix one initial mutation i “ 1 , 2 , 3, and set k 0 , s 0 , and t 0 as in T able 1 . Note that b y Theorem 4.12 , ( 9.1 ) and ( 9.2 ), all the g -vectors except for e t 0 b elong Q ` i X H ` D p e s 0 q X H ´ D p e t 0 q . Before pro ceeding with the pro of, w e establish the following lemma. Lemma 10.12. Fix i “ 1 , 2 , 3 satisfying µ i p B q ě B . Then, we have Q ` i X H ` D p e s 0 q X H ´ D p e t 0 q Ă Q ` initial , Q ´ i X H ` D p e s 0 q X H ´ D p e t 0 q Ă Q ´ initial . (10.30) Pr o of. Firstly , w e pro ve Q i X H ` D p e s 0 q X H ´ D p e t 0 q Ă Q initial . Let x “ p x 1 , x 2 , x 3 q P Q i X H ` D p e s 0 q X H ´ D p e t 0 q . If x “ 0 , w e ha v e 0 P Q initial . Assume that x ‰ 0 . Then, we hav e x s 0 ě 0, x t 0 ď 0, and d k 0 x 2 k 0 ` d s 0 x 2 s 0 ` d t 0 x 2 t 0 ` a d k 0 d s 0 p r i s k 0 s 0 x k 0 x s 0 ` a d k 0 d t 0 p r i s k 0 t 0 x k 0 x t 0 ` a d s 0 d t 0 p r i s s 0 t 0 x s 0 x t 0 ą 0 . (10.31) R YOT A AKAGI AND ZHICHAO CHEN 56 By ( 3.44 ), w e ha ve p r i s k 0 s 0 “ p k 0 s 0 and p r i s k 0 t 0 “ p k 0 s 0 . By the assumption µ i p B q ě B , we hav e p r i s s 0 t 0 ě p s 0 t 0 . Since x s 0 ě 0 and x t 0 ď 0, we ha ve p r i s s 0 t 0 x s 0 x t 0 ď p s 0 t 0 x s 0 x t 0 , and it implies d k 0 x 2 k 0 ` d s 0 x 2 s 0 ` d t 0 x 2 t 0 ` a d k 0 d s 0 p k 0 s 0 x k 0 x s 0 ` a d k 0 d t 0 p k 0 t 0 x k 0 x t 0 ` a d s 0 d t 0 p s 0 t 0 x s 0 x t 0 ě d k 0 x 2 k 0 ` d s 0 x 2 s 0 ` d t 0 x 2 t 0 ` a d k 0 d s 0 p r i s k 0 s 0 x k 0 x s 0 ` a d k 0 d t 0 p r i s k 0 t 0 x k 0 x t 0 ` a d s 0 d t 0 p r i s s 0 t 0 x s 0 x t 0 ą 0 . (10.32) This implies x P Q initial . No w, we assume that the inclusions ( 10.30 ) do not hold. By the symmetry , we might assume that Q ` i X H ` D p e s 0 q X H ´ D p e t 0 q Ć Q ` initial without loss of generality . Since Q initial is decomp osed in to Q ` initial and Q ´ initial zt 0 u , the set Q ` i X H ` D p e s 0 q X H ´ D p e t 0 q intersects with Q ´ initial zt 0 u . Note that Q ` initial zt 0 u also in tersects with Q ` i X H ` D p e s 0 q X H ´ D p e t 0 q . F or example, e s 0 is an in tersection of these t w o sets. T ake an element y P Q ` i X H ` D p e s 0 q X H ´ D p e t 0 q so that y P Q ´ initial zt 0 u . Then, since Q ` initial zt 0 u and Q ´ initial zt 0 u are disconnected b y ( 4.27 ), there exist p ositive num b ers a, b P R ą 0 suc h that a e s 0 ` b y R Q initial . (Note that a e s 0 ` b y ‰ 0 holds because y P H ` D p e s 0 q zt 0 u .) On the other hand, since Q ` i X H ` D p e s 0 q X H ´ D p e t 0 q is a conv ex cone by Lemma 4.9 , w e ha v e a e s 0 ` b y P Q ` i X H ` D p e s 0 q X H ´ D p e t 0 q . This con tradicts with the inclusion Q i X H ` D p e s 0 q X H ´ D p e t 0 q Ă Q initial . Hence, we complete the pro of. □ Pr o of of The or em 10.11 . Consider a g -v ector g w j with w ě r i s . If this g -vector is e t 0 , the claim is sho wn directly . F or another g -vector, b y Theorem 4.12 , ( 9.1 ) and ( 9.2 ), it b elongs to Q ` i X H ` D p e s 0 q X H ´ D p e t 0 q . By Lemma 10.12 , we obtain the claim. □ Remark 10.13. Note that for the exc hange matrix corresp onding to the Mark o v quiv er, it is minim um. Hence, we can give the upp er b ound describ ed in Theorem 10.11 , whic h is also the same as the one giv en in Theorem 4.12 . Ac kno wledgemen ts. The authors would lik e to express their sincere gratitude to T omoki Nak anishi for his though tful guidance. The authors also wish to thank Peigen Cao, Y asuaki Gy o da, Salv atore Stella and T oshiya Y urikusa for their v aluable discussions and insightful sug- gestions. In addition, Z. Chen wan ts to thank Xiao wu Chen, Zhe Sun and Y u Y e for their help and supp ort. R. Ak agi is supp orted by JSPS KAKENHI Gran t Num b er JP25KJ1438 and Ch ub ei Itoh F oundation. Z. 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Email addr ess : ryota.akagi.e6@math.nagoya-u.ac.jp School of Ma thema tical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China. Email addr ess : czc98@mail.ustc.edu.cn

Geometric structures of $G$-fans associated with rank $3$ cluster-cyclic exchange matrices

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