Coxeter's Frieze Patterns Arising from Dyck Paths
Agustín Moreno Cañadas, Isaías David Marín Gaviria, Gabriel Bravo Rios, Pedro Fernando Fernández Espinosa
aa r X i v : . [ m a t h . R T ] F e b Coxeter’s Frieze Patterns Arising from Dyck Paths
Agust´ın Moreno Ca˜nadasIsa´ıas David Mar´ın GaviriaGabriel Bravo RiosPedro Fernando Fern´andez Espinosa
Abstract
Frieze patterns are defined by objects of a category of Dyck paths, to do that,it is introduced the notion of diamond of Dynkin type A n . Such diamondsconstitute a tool to build integral frieze patterns. Keywords and phrases : Diamond, Dyck path, Dyck paths category, frieze pattern,seed vector, triangulation.Mathematics Subject Classification 2010 : 16G20; 16G30; 16G60.
Frieze patterns (as shown below) were introduced by Coxeter in the early 70s [9].According to Propp [14], they arose from Coxeter’s study of metric properties of poly-topes, and served as useful scaffolding for various sorts of metric data. . . . . . . . . . m − , − m m m . . .m − , − m − , m , m m . . . . . . . . . . . . . . . . . . . . . . . . Such patterns are defined as grids of numbers bounded from above by an infinite rowof 0s followed by a row of 1s and such that every four adjacent numbers of the followingform ba cd Agust´ın Moreno Ca˜nadas et al satisfy the arithmetic (or frieze) rule ac − bd = 1 . (1)The third line (i.e., the first nontrivial row) can be chosen in an arbitrary way andthen complete using the rule (1), which was named the modular equation by Coxeter[9].A frieze is called closed, if it is also bounded from below by a line of 1s (followed bya line of 0s). A frieze is called integral if it consists of positive integers. The sequenceof integers in the first non-trivial row is called quiddity sequence . This sequence com-pletely determines the frieze pattern. Each frieze pattern is also periodic since it isinvariant under glide reflection. The order of the frieze pattern is defined to be thenumber of rows minus one. It follows that each frieze pattern of order n is n -periodic[3]. Conway and Coxeter classified completely the frieze patterns whose entries arepositive integers, and show that these frieze patterns constitute a manifestation of theCatalan numbers [7, 8] giving a bijection between positive integer frieze patterns andtriangulations of regular polygons with labeled vertices.Frieze patterns appear independently in the 70s in the context of quiver representa-tions, in such a case, the local arithmetic rule is an additive analogue of Coxeter ' sunimodular rule. The generalization of the Coxeter ' s unimodular rule on Auslander-Reiten quivers was found by Caldero and Chapoton [5]. Assem, Reutenauer and Smithintroduced also a generalization of friezes by associating frieze patterns to Cartan ma-trices [1].According to Morier-Genoud [13] there are mainly three approaches for the study offriezes:1. A representation theoretical and categorical approach, in deep connection withthe theory of cluster algebras, where entries in the friezes are rational functions.2. A geometrical approach, in connection with moduli spaces of points in projectivespace and Grassmannians, where entries in the friezes are more often real orcomplex numbers.3. A combinatorial approach, focusing on friezes with positive integer entries.Many authors have studied friezes from the different points of view finding connec-tions with different branches of mathematics [13]. For instance, Baur et al [3] studiedmutation of friezes proving how mutation of a cluster affects the associated frieze. Onthe other hand, Fontaine and Plamondon [11] presented a formula for the number offriezes of type B n , C n , D n , and G . They conjectured that the number of friezes of type E , E , E and F is 868, 4400, 26592 and 112, respectively. In this way, the numberof friezes can be defined as a Dynkin function in the sense of Ringel [15].In this paper, frieze patterns are interpreted as objects of a novel category of Dyckpaths introduced recently by Ca˜nadas and Rios [6].The following is a list of our main results, all of them dealing with integral closedfriezes. oxeter’s Frieze Patterns...
1. It is introduced the notion of diamond of Dynkin type A n , and some of itsproperties are proved (see, Proposition 5, Proposition 6 and Theorem 8). Suchdiamonds are used as a tool to build frieze patterns.2. It is proved that there is a bijective correspondence between the set of all vectorsassociated to positive integral diamonds of Dynkin type A n and triangulationsof a polygon with n + 3 vertices. This result is a consequence of a bijectionbetween such triangulations and Dyck paths of length 2( n + 1) (see Lemma 16and Theorem 17).3. It is proved that if C ( A , t ) = { A t } ≤ t ≤ p − is the minimal p -cycle generated bya diamond A of Dynkin type A n . Then C ( A , t ) is in surjective correspondencewith a direct sum of p indecomposable objects of a Dyck paths category (seeTheorem 21).The paper is distributed as follows. In section 2, we recall main definitions and notationto be used throughout the paper, in particular, it is reminded the notion of a Dyckpaths category. In section 3, we present the main results, where it is introduced thenotion of diamond of Dynkin type A n and some of its properties. In this section, we recall main definitions and notation to be used throughout thepaper [2, 6, 12, 17].
For quivers, cluster algebras are defined as follows:Fix an integer n ≥
1. In this case, a seed (
Q, u ) consists of a finite quiver Q with-out loops or 2-cycles with vertex set { , . . . , n } , whereas u is a free-generating set { u , . . . , u n } of the field Q ( x , . . . , x n ).Let ( Q, u ) be a seed and k a vertex of Q . The mutation µ k ( Q, u ) of (
Q, u ) at k is theseed ( Q ′ , u ′ ), where;(a) Q ′ is obtained from Q as follows;(1) reverse all arrows incident with k ,(2) for all vertices i = j distinct from k , modify the number of arrows between i and j , in such a way that a system of arrows of the form ( i r −→ j, i s −→ k, k t −→ j ) is transformed into the system ( i r + st −→ j, k s −→ i, j t −→ k ). Andthe system ( i r −→ j, j t −→ k, k s −→ i ) is transformed into the system ( i r − st −→ j, i s −→ k, k t −→ j ). Where, r , s and t are non-negative integers, an arrow i l −→ j , with l ≥ l arrows go form i to j and an arrow i l −→ j ,with l ≤ − l arrows go from j to i .(b) u ′ is obtained form u by replacing the element u k with u k = 1 u k Y arrows i → k u i + Y arrows k → j u j . (2) Agust´ın Moreno Ca˜nadas et al
If there are no arrows from i with target k , the product is taken over the empty setand equals 1. It is not hard to see that µ k ( µ k ( Q, u )) = (
Q, u ). In this case the matrixmutation B ′ has the form b ′ ij = ( − b ij , if i = k or j = k,b ij + sgn ( b ik )[ b ik b kj ] + , else , where [ x ] + = max(x , Q is a finite quiver without loops or 2-cycles withvertex set { , . . . , n } , the following interpretations take place:1. the clusters with respect to Q are the sets u appearing in seeds, ( Q, u ) obtainedfrom a initial seed (
Q, x ) by iterated mutation,2. the cluster variables for Q are the elements of all clusters,3. the cluster algebra A ( Q ) is the Q -subalgebra of the field Q ( x , . . . , x n ) generatedby all the cluster variables.As example, the cluster variables associated to the quiver Q = 1 −→ { x , x , x x , x + x x x , x x } .Regarding cluster algebras arising from quivers, we recall that, Fomin and Zelevinsky[10] proved that any cluster algebra A ( Q ) of finite type has a finite set of clustervariables and the following result. Theorem 1.
The cluster algebra A ( Q ) is of finite type if and only if Q is mutation-equivalent to an orientation of a simply-laced Dynkin diagram, A n , n ≥ , D n , n ≥ , E , E and E . An alternative way to define friezes is to say that they are ring homomorphisms froma cluster algebra to the ring of integers such that all cluster variables are sent topositive integers [11]. Let Q be a quiver without loops and 2-cycles and let A ( Q ) bethe corresponding cluster algebra with trivial coefficients [12], then:(i) A frieze of type Q is a ring homomorphism F : A ( Q ) → R from the clusteralgebra to an integral domain R . The frieze is called integral if R = Z .(ii) An integral frieze is said to be positive if every cluster variable in A ( Q ) is mappedby F to a positive integer.Let x = ( x , . . . , x n ) be a cluster of A ( Q ), then:(iii) A vector ( a , . . . , a n ) ∈ R n is called a frieze vector relative to x if the frieze F defined by the assignment F ( x i ) = a i has values in R , F is said to be unitary ifthere exists a cluster x such that F ( x ) is a unit in R , for all x ∈ x . If the frieze F is unitary we say that the frieze vector ( a , . . . , a n ) is unitary .(iv) A vector ( a , . . . , a n ) ∈ Z n> is called a positive frieze vector relative to x if thefrieze F defined by F ( x i ) = a i is positive integral. oxeter’s Frieze Patterns... The following is an example of a frieze pattern. Hereinafter, frieze patterns are as-sumed to be integral closed friezes. . . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .
In this section we recall the definition and main properties of the category of Dyckpaths as Ca˜nadas and Rios describe in [6].A Dyck path is a lattice path in Z from (0 ,
0) to ( n, n ) with steps (1 ,
0) and (0 , y = x . The number of Dyck paths oflength 2 n is equal to C n = n +1 (cid:0) nn (cid:1) , the nth Catalan number [17].The set of Dyck words is the set of words w in the free monoid X ∗ = { U, D } ∗ satisfyingthe following two conditions [2]: • for any left factor u of w ( i.e., w = uv for some suitable word v ), | u | U ≥ | u | D , • | w | U = | w | D ,where | w | a is the number of occurrences of the letter a ∈ X = { U, D } in the word w .Henceforth, Dyck words defined as before are used to denote Dyck paths.Let D n be the set of all Dyck paths of length 2 n , let UW D = Uw . . . w n − D be aDyck path in D n with A = { UD, DU, UU, DD } being the set of all possible choicesin W .The support of UW D (denoted by Supp
UW D ⊆ { , , . . . , n − } = n-1 ) is a set ofindices (of the w i s) such thatSupp UW D = { q ∈ n-1 | w q = UD or w q = UU , 1 ≤ q ≤ n − } . A map f : A −→ A such that for any w ∈ A , it holds that f ( w ) = f ( ab ) = w − = ba , a, b ∈ { U, D } is said to be a shift . An unitary shift is a map f i : D n −→ D n suchthat f i ( Uw . . . w i − w i w i +1 . . . w n − D ) = Uw . . . w i − f ( w i ) w i +1 . . . w n − D. Agust´ın Moreno Ca˜nadas et al
We will denote a unitary shift by a vector of maps from D n to itself of the form(1 , . . . , i − , f i , i +1 , . . . , n − ), where 1 k is the identity map associated to the i -thcoordinate.An elementary shift is a composition of unitary shifts. A shift path of length mUW D −→ UW D −→ · · · −→ UW m D −→ UV D from
UW D to UV D is a composi-tion of m elementary shifts. The set of all Dyck paths in a shift path between UW D and
UV D will be denoted by J ( W, V ). For notation, we introduce the identity shift as the elementary shift (1 , . . . , n − ).Suppose that a map R : D n → D n is defined by the application of successive el-ementary shifts to a given Dyck path. Then R is said to be an irreversible relation over D n if and only if elementary shifts transforming Dyck paths (from one to theother) are not reversible. In other words, if an elementary shift F = f p ◦ · · · ◦ f p q transforms a Dyck path UW D into a Dyck path
UV D then there is not an elementaryshift F ′ = f p ◦ · · · ◦ f p q transforming UV D into
UW D , for some p, q ∈ Z + .If there exist two paths G ◦ F and G ′ ◦ F ′ of irreversible relations (of length 2) trans-forming a Dyck path UW D into the Dyck path
UV D over R in the following form: UW D UW ′ DUW ′′ D UV D , ✏✏✏✏✏✶PPPPPq ✏✏✏✏✏✶PPPPPq FF ′ GG ′ with W ′ = W ′′ . Then G ◦ F is said to be related with G ′ ◦ F ′ (denoted G ◦ F ∼ R G ′ ◦ F ′ )whenever G ′ = F and G = F ′ .As for the case of diagonals [4], Ca˜nadas and Rios [6] defined a F -linear additivecategory ( D n , R ) based on Dyck paths, in this case, objects are F -linear combinationsof Dyck paths in D n with space of morphisms from a Dyck path UW D to a Dyckpath
UV D over F associated to R being the vector spaceHom ( D n ,R ) ( UW D, UV D ) = h{ g | g is a shift path associated to R }i / h∼ R i .Hom ( D n ,R ) ( UW D, UV D ) = 0 if and only if there are shift paths transforming UW D into
UV D and \ UW i D ∈ J ( W,V ) Supp UW i D = ∅ , for each shift path, with UW D and
UV D in D n .Figure 1 shows the elementary shifts over ( D , R ) associated to an irreversible relation R defined over the set of all Dyck paths of length 6. And such that, R ( UW D ) = ( f ( UW D ) , if w = UD,f ( UW D ) , if w = UD. oxeter’s Frieze Patterns... (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)✻ ✻✛✛ Figure 1: Elementary shifts in ( D , R ). Q = ❞ ❞ ❞ ❞ ❞ ✲✛ ✲✛ (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✒ (cid:0)✒ (cid:0)✒(cid:0)✒ (cid:0)✒(cid:0)✒ (cid:0)✒ (cid:0)✒(cid:0)✒ (cid:0)✒❅❅❅❘ ❅❅❅❘❅❅❅❘ ❅❅❅❘ ❅❅❅❘❅❅❅❘ ❅❅❅❘❅❅❅❘ ❅❅❅❘ ❅❅❅❘ Figure 2: Quiver Q and the Dyck path version of the Auslander-Reiten quiverof rep Q (see Corollary 3). If n = { , , . . . , n } is an n -point chain then C (1,n) stands for all admissible sub-chains C of n with min C = 1 and max C = n . Given an admissible subchain C = { j , . . . , j m , i , . . . , i l } then two Dyck paths D and D ′ of length 2 n are said to berelated by a relation of type R i ...i l j ...j m if there is an elementary shift associated to thepoints of the subchain transforming one into the other [6]. Agust´ın Moreno Ca˜nadas et al
We let C n denote the subcategory of ( D n , R i ...i l j ...j m ) whose objects are F -linear combi-nations of Dyck paths with exactly n − R i ...i l j ...j m .The following results describe the structure of the category of Dyck paths [6]. Werecall that a point x where a Dyck path UW D changes from the north to the east issaid to be a peak of the path.In the following theorem the symbol Q is used to denote a quiver of type A n − ( n ≥ { i , . . . , i l } and { j , . . . , j m } are the corresponding sets of sinks and sources.rep Q denotes the corresponding category of representations. Theorem 2 ([6], Theorem 15) . There is a categorical equivalence between categories C n and rep Q . Corollary 3 ([6], Corollary 16) . There exists a bijection between the set of represen-tatives of indecomposable representations
Ind Q of rep Q and the set of Dyck paths oflength n with exactly n − peaks. Corollary 4 ([6], Corollary 17) . The category C n is an abelian category. In this section, we present the main results of the paper. In particular, it is introducedthe notion of integral diamond of Dynkin type A n , which are integer arrays used tobuild frieze patterns associated to triangulations of an ( n + 3)-polygon. A n -Diamonds Let R be an integral domain then a diamond of Dynkin type A n or A n -diamond isan array A = ( a i,j ) with entries in R , such entries satisfy conditions (D1) and (D2)associated to arrays with the following shape: a , a , a , a , ...... a ,n − a ,n a ,n a ,n +1 (D1) a , = a ,n +1 = 1,(D2) a ,j a ,j − a ,j − a ,j +1 = 1 for 1 ≤ j ≤ n ,where, 1 is the identity element of R .If R = Z then A is said to be a positive integral diamond of Dynkin type A n , if it alsosatisfies the following condition (D3) oxeter’s Frieze Patterns... (D3) a , = a (or a , = a + m a ), a , = a + m a (or a , = a ) and a , = a + am a − ≤ a ≤ ⌊ n +22 ⌋ , 1 ≤ m ≤ n and 0 ≤ m a ≤ n + 2(1 − a ) if a > A n -diamonds are assumed to be positiveintegral diamonds of Dynkin type A n .Two A n -diamonds A and B constitute a coupling , denoted A | = B if and only if a ,j = b ,j for 1 ≤ j ≤ n .A set { A t } t ≥ is an A n - sequence of couplings of A n if and only if A r | = A r +1 for r ≥ X = X for any A n -diamond X ). An A n -sequence { A t } t ≥ of couplings is a p - cycle ifthere exists p ∈ N such that A t = A t + p . If the A n -sequence of couplings S t = { A t } t ≥ constitute a frieze pattern F then we will say that S t generates F . In this case, wepoint out that Lemma 7 proves that A = A generates S t , meaning that S t can beobtained from a sequence of couplings starting with A .For example, let R = Z , the sets { A t } t ≥ and { B t } t ≥ ( A and B as shown below)are A -sequences which are 2-cycles with A k = B k +1 = A , A k +1 = B k = B and k ≥
0. 1 1 A = 1 2 B = 2 11 1 (3)In general, it can be written an A n -sequence { A t } t ≥ as an A n -array C A t = ( c i,j ) suchthat c t +1 ,j = a t ,j and c t +1 , = c t +1 ,n +1 = 1, for t ≥
0. For the previous example,1 1 1 . . . . . .C A t = 1 2 1 2 . . . C B t = 2 1 2 1 . . . . . . . . .C A t and C B t are A -arrays associated to { A t } t ≥ and { B t } t ≥ , respectively.If the A n -sequence of couplings is finite of length m , then it can be associated to aninfinity A n -array, C mA t = ( c mi,j ) such that c m ( t +1)+ km,j = a t ,j , c m ( t +1)+ km, = c m ( t +1)+ km,n +1 = 1 , (4)for k ∈ Z . For any A n -sequence of couplings { A t } t ≥ , it is possible to choose an A n - subsequence { B z } z ≥ with B z = A x + z for some x ≥ t . In particular, if { A t } t ≥ is a p -cycle then the subsequence { B s } ≤ s ≤ p − such that B s = A t is called the minimal p - cycle of { A t } t ≥ .The following results give the main properties of diamonds of Dynkin type A n . Proposition 5.
Let { A t } t ≥ be a p -cycle and let B = { B s } ≤ s ≤ p − be its minimal p -cycle. Then, the array C pB is a frieze pattern of order n + 3 . In particular, p divides n + 3 . Agust´ın Moreno Ca˜nadas et al
Proof.
Let C pB = ( c pij ) be the infinity A n -array associated to { B s } ≤ s ≤ p − , identity(4) implies that c p ( s +1)+ kp,j = a s ,j , c p ( s +1)+ kp, = c p ( s +1)+ kp,n +1 = 1 , for k ∈ Z , since that { A t } t ≥ is a p -cycle, then C pB is a frieze pattern. (cid:3) Proposition 6.
Let { A t } t ≥ be a p -cycle of length p , then the subsequences { B s i } with ≤ s i ≤ p − generate the same frieze pattern of order n + 3 , for ≤ i ≤ p − ,and B s i = A i + s i . Proof.
Let { A t } t ≥ be a p -cycle of length 2 p , let C pA = ( c pij ) and C pB = ( c p ′ ij ) be theinfinity arrays of the subsequences A = { B s i } ≤ s i ≤ p − and B = { B s i ′ } ≤ s i ′ ≤ p − of { A t } t ≥ for 0 ≤ i < i ′ ≤ p −
1. The following identities (5) hold by applying thetranslation s i ′ = s i − | i ′ − i | , c ps i ′ +1+ kp,j = a s i ′ + i ′ j = a s i −| i ′ − i | + i ′ j = a s i + iij = c ps i +1+ kp,j . (5)We are done. (cid:3) Lemma 7.
Let { A t } t ≥ be a sequence of couplings. Then { A t } t ≥ is generated by A .In particular, A generates a p -cycle for some p > . Proof.
Let { A t } t ≥ be a sequence, then a x ,j = 1 + ( a x ,j − )( a x − ,j +1 ) a x − ,j , for 1 ≤ j ≤ n , and x ≥ t , a x ,j can be written by using the set { a ,j } ≤ j ≤ n for x > { a , , . . . , a ,n } is a seed of the cluster algebra associated to thelinearly oriented quiver of type A n . Since the cluster variables are finite in the case A n , then there is p = n + 3 (in some cases, it is not minimal) such that A = A p . (cid:3) Theorem 8.
Let A be a diamond of Dynkin type A n then A generates a frieze pattern. Proof.
It is a direct consequence of Lemma 7, and Proposition 5. (cid:3)
For instance, diamonds A and B given in (3) generate the following frieze pattern. . . . . . .. . . . . .. . . . . . In this section, we give an algorithm to build a family of positive integral frieze vectorsassociated to the linearly oriented quiver of type A n . These vectors allow to find out aconnection between the positive integral diamonds of Dynkin type A n , triangulations,and Dyck paths.Let A be a diamond of Dynkin type A n then we can write its first column as a vectorwith the form v A = ( a , . . . , a n ), where a j = a ,j . In such a case, we say that v A isassociated to A and that v A generates A . oxeter’s Frieze Patterns... Proposition 9. If v = ( a , . . . , a n ) is a vector associated to a positive integral diamondof Dynkin type A n with a n = 1 , then the vector v ′ = ( a , . . . , a i , a i + a i +1 , a i +1 , . . . , a n − ) is also associated to a positive integral diamond of Dynkin type A n , for ≤ i < n . Proof.
Let v A = ( a , . . . , a n ) be a vector associated to a positive integral diamond A = ( a j,m ) of Dynkin type A n , then we take the vector v A + i = ( a , . . . , a i , a i + a i +1 , a i +1 , . . . , a n − ) and the array A + i of the following form: b ,m = a ,m , if m ≤ i , a ,i + a ,i +1 , if m = i + 1, a ,m − , if m > i + 1,and b ,m = a ,m , if m ≤ i − a ,i − + a ,i , if m = i , a ,m − , if m ≥ i + 1,then b ,m b ,m − b ,m − b ,m +1 = 1, for 1 ≤ m ≤ n and 1 ≤ i < n . Therefore A + i is apositive integral diamond of Dynkin type A n . (cid:3) Proposition 10.
For each vector v n,z = ( a , . . . , a n ) with a i = ( z + 1 − i, if i < z ,1 , if i ≥ z . (6) there is associated a unique positive integral diamond of Dynkin type A n , for z ∈{ , . . . , n + 1 } . Proof.
Let v n,z be a vector and let z be a natural number between 1 and n + 1, wedefine a positive integral diamond A with a ,i = a i and a ,i = b i where b i = ( , if i < z , i + 2 − z, if i ≥ z , (7)then a ,i a ,i − a ,i − a ,i +1 = 1 for 1 ≤ i ≤ n . (cid:3) Remark 11. v n,z is called a seed vector . The vector v n,z = ( b , . . . , b n ) defines apositive integral diamond B of Dynkin type A n such that b ,i satisfies the followingidentity b ,i = i − , if i < z − b ,i + 1) z − , if z − < i < n , z, if i = n .and b i = b ,i is defined as in (7). Proposition 12.
The positive integral diamonds A and B of Dynkin type A n generatedby vectors v n,z and v n,z respectively constitute a coupling. Proof.
It is a direct consequence of Proposition 10 and Lemma 7. (cid:3)
The number of ways of applying recursively Proposition 9 to a vector w A = ( a , . . . ,a z − , , . . . , ∈ N n is given by the next identity (denoted by f n,z ), Agust´ın Moreno Ca˜nadas et al f n,z = (P ni = z − f n − ,i , if z > P ni =1 f n − ,i , if z = 1,where it is included the trivial move w A +0 = w A , for n >
1, and any z ∈ { , . . . , n +1 } .Actually, these numbers can be arranged as follows: f , f , f , f , f , f , f , f , f , f , f , f , f , f , ... . . . (8)for any of the vectors w A . Since the first choices are v , = (1) and v , = (2), then f , = 1 and f , = 1. The previous triangle (8) appears in the On-Line Encyclopediaof Integer Sequences (OEIS) as A009766 (Catalan triangle [16]). In particular, wegenerate all positive integral diamonds of Dynkin type A n via the seed vectors v n,z .For example, for n = 3, all vectors that generate positive integral diamonds of Dynkintype A are:(1 , ,
1) (1 , ,
2) (1 , ,
1) (1 , ,
3) (1 , ,
2) (2 , ,
1) (2 , , , ,
1) (2 , ,
4) (2 , ,
3) (3 , ,
1) (3 , ,
3) (3 , ,
2) (4 , , G = UD . . . UD . . . be a Dyck path of length 2 n and let m i be the numberof U s before the occurrence of the i -th D in G then G can be written as a vector v G = ( v , . . . , v n − ) where v i = m i − i + 1. As an example, consider the Dyck path G shown in Figure 3, which has associated the vector v G = (5 , , , , , , , (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Figure 3: A Dyck path of length 18 associated to the vector v G =(5 , , , , , , , In what follows, it is defined a map T i between vectors associated to positive integraldiamonds of Dynkin type A n and Dyck paths by using a relation over the coordinatesof a vector u = ( a , . . . , a m ). T i is defined in such a way that, T i : N m → N and: • If a i − a l > l ∈ { , . . . , i } , then we select the maximum of such indexesmax { l } and write r = a i − a l . In the same way, it is chosen max { l } such that r − a l > r = r − a l , this process ends if { l | r t − a l > } = ∅ .Thus, if u ∈ N m then T i ( u ) = r t + t , for some t . oxeter’s Frieze Patterns... • If a i − a l ≤ l ∈ { , . . . , i } , then T i ( u ) = a i .For instance, for the vector u = (14 , , , , , T ( u ) = 14, T ( u ) =13, T ( u ) = 4, T ( u ) = 8, T ( u ) = 3, and T ( u ) = 2. Proposition 13.
Let v n,z be a seed vector, then ( T ( v n,z ) , . . . , T n ( v n,z )) defines aDyck path of length n + 1) . Proof.
For any z ∈ { , . . . , n + 1 } , T i ( v n,z ) = a i with a i given by identity (6), thereis a word G v n,z = w . . . w n +1) in the free monoid { U, D } ∗ such that G v n,z = U . . . U | {z } z − D . . . D | {z } z − UDUD . . . UDUD, for any left factor u s in G v n,z of length s ∈ { , . . . , n +1) } , 0 ≤ | u s | U −| u s | D ≤ z − G v n,z ∈ D n +1) . (cid:3) Proposition 14.
Let v A = ( a , . . . , a n ) be a vector associated to a positive integraldiamond A of Dynkin type A n with a n = 1 , such that ( T ( v A ) , . . . , T n ( v A )) definesa Dyck path in D n +1) . Then ( T ( v A + i ) , . . . , T n ( v A + i )) also defines a Dyck path in D n +1) . Proof.
Let v A = ( a , . . . , a n ) be a vector associated to a positive integral diamond A with a n = 1, then there exists a Dyck path G v A ∈ D n +1) such that any left factor u s of length s satisfies | u s | U ≥ | u s | D for 1 ≤ s ≤ n + 1). Let v A + i be a vector associatedto the positive integral diamond A + i with T m ( v A + i ) = T m ( v A ) , if 1 ≤ m ≤ i , T m ( v A ) + 1 , if m = i + 1, T m − ( v A ) , if m ≥ i + 1,then there is a word G A + i = w ′ , . . . , w ′ n +1) in { U, D } ∗ , where we take the index m of the i -th D in G A + i , any left factor u ′ s in G A + i satisfies the identities | u ′ s | U = | u s | U , if 1 ≤ s ≤ m , | u m | U + 1 , if s = m + 1, | u s − | U + 1 , if s ≥ m + 2,and | u ′ s | D = | u s | D , if 1 ≤ s ≤ m , | u m | D , if s = m + 1, | u s − | U + 1 , if s ≥ m + 2,then, we have the following possibilities: • If 1 ≤ s ≤ m , | u ′ s | U = | u s | U ≥ | u s | D = | u ′ s | D . • If s = m + 1, | u ′ m +1 | U = | u m | U + 1 > | u m | D = | u ′ m +1 | D . • If m + 2 ≤ s ≤ n + 1), | u ′ s | U = | u s − | U + 1 ≥ | u s − | D + 1 = | u ′ s | D .Therefore, G A + i ∈ D n +1) . (cid:3) Agust´ın Moreno Ca˜nadas et al
Lemma 15.
There is a bijective correspondence between the set of all vectors associ-ated to positive integral diamonds of Dynkin type A n and the set of all Dyck paths oflength n + 1) . Proof.
Let D A n be the set of all vectors associated to positive integral diamonds ofDynkin type A n and let D n +1) be the set of all Dyck paths of length 2( n + 1) thenwe define a map f : D A n → D n +1) with f ( u A ) = (cid:0) T ( u A ) , . . . , T n ( u A ) (cid:1) , Propositions13 and 14 allow us to establish that f is well defined. In order to prove that the map f is injective, suppose that u A = v B , we take the minimum l such that u l = v l . If l = 1then T ( u A ) = T ( v B ). If l > u l = m ( u l − ) + a and v l = m ′ ( u l − ) + a with m = m ′ is a consequence of Proposition 9 then r t ul = r t vl , therefore T l ( u A ) = T k ( v B ). (cid:3) An alternative way of writing a Dyck path G ∈ D n +1) can be defined by using avector λ G = ( λ , . . . , λ n ) where λ i is the number of D s before the occurrence of the( n + 2 − i )-th U in G . For example, Dyck path in Figure 3 has associated the followingvector λ G = (4 , , , , , , , λ be a vector associated to a Dyck path of length 2( n + 1) then a triangulation ofan ( n + 3)-polygon can be realized by λ as follows: • Fix a labeling for the vertices of polygon K n +30 = ( v n +30 , . . . , v n +3 n +2 ) with v n +3 i = i , for 0 ≤ i ≤ n + 2. • For λ i , we draw a diagonal l λ i i between λ i and λ i + 2. Afterwards, we label thelast polygon with n + 3 − i vertices K n +3 − ii = ( v n +3 − i , . . . , v n +3 − in +2 − i ), and v n +3 − ij = ( v n +3 − ( i − j , if j ≤ λ i , v n +3 − ( i − j +1 − , if j > λ i ,for i = 1 , . . . , n . l l l l l l −→ −→ Figure 4: Example of a triangulation realized by a Dyck path of length 8.
The previous algorithm describes that if l λ i i is a diagonal then it does not cross thediagonals l λ , . . . , l λ i − i − for 1 ≤ i ≤ n . For instance, let λ G = (2 , ,
1) be the vectorassociated to G = UDUDUUDD , then the triangulation of λ G is shown in Figure 4.If we fix a labeling K over all vertices of a polygon with n + 3 vertices, a triangulation T is written as a sequence T = ( l v , . . . , l v n n ), where v i belongs to the set of vertices. Lemma 16.
There is a bijective correspondence between the set of all triangulationsof a polygon with n + 3 vertices and the set of all Dyck paths of length n + 1) . oxeter’s Frieze Patterns... Proof.
Let T n be the set of all triangulations of a polygon with n + 3 verticesthen we can define a map g : D n +1) → T n with g ( λ ) = T λ . In order to provethat g is one to one, we fix a labeling K and suppose that g ( λ G ) = g ( σ G ′ ), then( l λ , . . . , l λ n n ) = ( l σ , . . . , l σ n n ), provided that l λ j j = l σ j j . Since by definition there arediagonals connecting vertices λ j with ( λ j + 2) and σ j with ( σ j + 2), therefore λ j = σ j for j = 1 , . . . , n . We are done. (cid:3) The next theorem presents the main result regarding relationships between positiveintegral diamonds of Dynkin type A n and triangulations of an ( n + 3)-polygon. Theorem 17.
There is a bijective correspondence between the set of all vectors asso-ciated to positive integral diamonds of Dynkin type A n and triangulations of a polygonwith n + 3 vertices. Proof.
We fix a labeling K in a polygon with n + 3 vertices, the map F : D A n → T n defined by the formula F ( v A ) = ( g ◦ f )( v A ) (9)is a bijection (see Lemmas 15 and 16). (cid:3) Figure 5 presents an example of the bijective correspondence between a positive inte-gral diamond of Dynkin type A , a Dyck path of length 10, and a triangulation of apolygon with 7 vertices. In this section, we describe an algebraic interpretation of frieze patterns as a directsum of indecomposable objects of Dyck paths categories.
Lemma 18.
Vectors v n,z and v n,z realize the same triangulation except for one anti-clockwise rotation. Proof.
Let v n,z and v n,z be frieze vectors, fixed a labeling K in an ( n + 3)-polygon,then (cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) zn +1 f ( v n,z ) f ( v n,z ) Agust´ın Moreno Ca˜nadas et al
12 23 34 11 21 ✲(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✠❅❅❅❅❅❅■ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)
Figure 5: Map F defined in the proof of Theorem 17 allows to establish identi-fications between an A -diamond, a Dyck path of length 10 and a triangulationof a heptagon. the following identities hold by applying map F (see (9)) as follows: F ( v n,z ) = ( n |{z} , . . . , z |{z} n − z , |{z} n − z +1 , . . . , |{z} n ) ,F ( v n,z ) = ( n − | {z } , . . . , z − | {z } n − z , z − | {z } n − z +1 , . . . , |{z} n ) , if we change K for K by making the following replacements: • Vertex l ∈ K is changed for l − ∈ K , 1 ≤ l ≤ n + 2, • Vertex 0 ∈ K is changed for n + 2 ∈ K ,then the diagonals from 0 to r in K are diagonals from r − n + 2 in K , andthe diagonals from r in K are diagonals from r − K , for 0 ≤ r ≤ z ≤ r ≤ n .Therefore F ( v n,z ) ∈ K coincides with F ( v n,z ) ∈ K . (cid:3) Note that, there exists a permutation σ = (cid:18) . . . n − z − n − z n − z + 1 n − z + 2 . . . n − n . . . n − z − n − z n n − . . . n − z + 2 n − z + 1 (cid:19) in S n that describes a bijection between the coordinates of the vector F ( v n,z ) =( u , . . . , u n ) and the vector F ( v n,z ) = ( u ′ , . . . , u ′ n ) such that σ ( F ( v n,z )) = ( u σ (1) , . . . ,u σ ( n ) ) = ( u ′ , . . . , u ′ n ) = F ( v n,z ) in K . In general, if v and w realize the same oxeter’s Frieze Patterns... triangulation except for one anti-clockwise rotation, then there exists a permutation σ ′ ∈ S n such that σ ′ ( F ( v )) = F ( w ) in K . Lemma 19.
Let A and B be a coupling of positive integral diamonds of Dynkin type A n , and v A = ( a , . . . , a z , . . . , a n ) a corresponding associated vector with a t = 1 for z ≤ t ≤ n . If v A and v B realize the same triangulation except for one anti-clockwiserotation. Then vectors v A + i = ( a , . . . , a i − , a i − + a i , a i +1 , . . . , a n − ) and v B + i − = ( b , . . . , b i − , b i − + b i − , b i − , . . . , b n − ) , realize the same triangulation except for one anti-clockwise rotation for z − ≤ i ≤ n ,and i ≥ . Proof.
Let v A and v B be associated vectors to the A n -diamonds A and B , respectively.Since v A and v B realize the same triangulation except for one anti-clockwise rotationthen there exists a permutation σ ∈ S n such that σ ( F ( v A )) = F ( v B ) in K . Thefollowing options arise from the map f , such that:(1) If i > z ≥
1, then f ( v A ) = ( . . . , |{z} i − , |{z} i , . . . ) ,f ( v B ) = ( . . . , d |{z} i − , |{z} i − , |{z} i , . . . ) , (see Figure 6)(1.1) If d = 1, then F ( v A ) = ( . . . , i |{z} n − i , i − | {z } n +1 − i , . . . ) ,F ( v B ) = ( . . . , i − | {z } n − i , i − | {z } n +1 − i , . . . ) , and σ satisfies the expression, σ ( r ) = ( r, if r ≤ n + 1 − i , m, otherwise,for some m > n + 1 − i . Applying F to v A + i and v B + i − , it holds that F ( v A + i ) = ( . . . , i − | {z } n − i , i − | {z } n +1 − i , . . . ) ,F ( v B + i − ) = ( . . . , i − | {z } n − i , i − | {z } n +1 − i , . . . ) , then there exits σ ′ ∈ S n such that σ ′ = σ and σ ′ ( F ( v A + i )) = F ( v B + i − ) in K (see Figure 6).(1.2) The case for d = 2 is the same as the previous case.(1.3) If d = 3, then A and B do not realize the same triangulation.Note that, if z = 1, this case satisfies the condition (1.1) and (1.2) without d . Agust´ın Moreno Ca˜nadas et al n +1 − i i i i (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) d =1 d =2 i i i (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) d =1 d =2 Figure 6: Dyck paths associated to vectors v A , v B (left), v A + i and v B + i − (right) for i > z . (2) If i = z ≥
2, then f ( v A ) = ( . . . , |{z} i − , |{z} i , . . . ) ,f ( v B ) = ( . . . , b |{z} i − , a |{z} i − , |{z} i , . . . ) , (see Figure 7) . (2.1) If a = 1 and b = 1, then F ( v A ) = ( . . . , i |{z} n − i , . . . ) ,F ( v B ) = ( . . . , i − | {z } n − i , i − | {z } n +1 − i , . . . ) , and σ is defined by the following cases: σ ( r ) = r, if r ≤ n − i , n + 1 − i, if r = n , m, otherwise,for some m > n + 1 − i . Applying F , we obtain F ( v A + i ) = ( . . . , i − | {z } n − i , . . . ) ,F ( v B + i − ) = ( . . . , i |{z} n − i , i − | {z } n +1 − i , . . . ) , then there exits σ ′ ∈ S n satisfying the following cases: σ ′ ( r ) = n − i, if r = n , n + 1 − i, if r = n − i , σ ( r ) , otherwise,therefore σ ′ ( F ( v A + i )) = F ( v B + i − ) in K (see Figure 8).(2.2) If a = 1 and b = 2, then conditions defined in the case (2.1) hold.(2.3) For a = 2 and b = 1 or b = 2, we have only contradictions. oxeter’s Frieze Patterns... n +1 − i i i i i (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) a =1 a =1 a =2 b =1 b =2 b =3 Figure 7: Dyck paths associated to vectors v A and v B for i = z . (2.4) If a = 2 and b = 3, F ( v B ) = ( . . . , i − | {z } n − i , . . . ) and σ = σ . Applying F to v B + i − , it holds that F ( v B + i − ) = ( . . . , i − | {z } n − i , . . . ) then there exits σ ′ ∈ S n such that σ ′ = σ and σ ′ ( F ( v A + i )) = F ( v B + i − ) in K (see Figure 8).(2.5) Case (2.3) holds for a = 3 and b = 1 , , n +1 − i i i i i (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) a =1 a =1 a =2 b =1 b =2 b =3 Figure 8: Dyck paths associated to vectors v A + i and v B + i − for i = z . Note that, if z = 2 then conditions for a = 1 , b hold.(3) If i = z − ≥
3, then f ( v A ) = ( . . . , |{z} i , |{z} i +1 , . . . ) ,f ( v B ) = ( . . . , b |{z} i − , a |{z} i , |{z} i +1 , . . . ) (see Figure 9) . (3.1) If a = 1 and b = 1, then F ( v A ) = ( . . . , i + 1 | {z } n − i − , . . . ) ,F ( v B ) = ( . . . , i |{z} n − i − , i |{z} n − i , i − | {z } n +1 − i , . . . ) , and σ ( r ) = r, if r ≤ n − i − n − i, if r = n , n + 1 − i, if r = n − m, otherwise, Agust´ın Moreno Ca˜nadas et al for some m > n + 1 − i . Provided that F ( v A + i ) = ( . . . , i − | {z } n − i − , . . . ) , and F ( v B + i − ) = ( . . . , i + 1 | {z } n − i − , i |{z} n − i , i − | {z } n +1 − i , . . . ) , then, there exits σ ′ ∈ S n such that σ ′ ( r ) = n − i − , if r = n , n − i, if r = n − n + 1 − i, if r = n − i − σ ( r ) , otherwise,then σ ′ ( F ( v A + i )) = F ( v B + i − ) in K (see Figure 10). n +1 − i i i i i (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) a =1 a =1 a =2 b =1 b =2 b =3 Figure 9: Dyck paths associated to vectors v A and v B for i − z . (3.2) If a = 1 and b = 2. It holds that, F ( v B ) = ( . . . , i |{z} n − i − , i |{z} n − i , . . . ) , whereas, σ is given by the identities σ ( r ) = r, if r ≤ n − i − n − i, if r = n , m, otherwise,for some m > n − i . Applying F to v B + i − , we get F ( v B + i − ) = ( . . . , i + 1 | {z } n − i − , i − | {z } n − i , . . . ) , then there exists σ ′ with σ ′ ( r ) = n − i − , if r ≤ n , n − i, if r = n − i − σ ( r ) , otherwise,therefore σ ′ ( F ( v A + i )) = F ( v B + i − ) in K (see Figure 10). oxeter’s Frieze Patterns... (3.3) If a = 2 and b = 3. F ( v B ) = ( . . . , i |{z} n − i − , . . . ), provided that σ ( r ) = ( r, if r ≤ n − i − m, otherwise,for some m > n − i . In this case, F ( v B + i − ) = ( . . . , i − | {z } n − i − , . . . ), and there is σ ′ = σ such that σ ′ ( F ( v A + i )) = F ( v B + i − ) in K (see Figure 10). n +1 − i i i i i (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) a =1 a =1 a =2 b =1 b =2 b =3 Figure 10: Dyck paths associated to vectors v A + i and v B + i − for i − z . Same arguments are used for the remaining cases (see item (2) of this proof). (cid:3)
Proposition 20.
Two positive integral diamonds of Dynkin type A n are in the sameminimal p -cycle if their triangulations are in the same mutation class. Proof.
It is a direct consequence of Proposition 9, Lemmas 18 and 19 (cid:3)
The following result gives a way to build frieze patterns.
Theorem 21.
Let A be a positive integral diamond of Dynkin type A n and let { A t } ≤ t ≤ p − be the minimal p -cycle generated by A . Then: (i) A and F ( v A ) generate the same frieze pattern (see (9) ). (ii) { A t } ≤ t ≤ p − is in surjective correspondence with a direct sum of p indecompos-able objects of a Dyck paths category. Proof.
Let D A n be the set of all vectors associated to positive integral diamonds ofDynkin type A n , A a positive integral diamond of Dynkin type A n , and { A t } ≤ t ≤ p − the minimal p -cycle generated by A .(i) Let K be a labeling of an ( n + 3)-polygon, Theorem 17 implies that F ( v A ) = g (( a , T ( v A ) , . . . , T n ( v A )))= g ( λ ( a ,T ( v A ) ,...,T n ( v A )) )= g (( λ , . . . , λ n +1 − a , , . . . , | {z } a ))= ( l v , . . . , l v n +1 − a n +1 − a , l n − a , . . . , l n ) , then, there are a − a triangles incident with vertex 0. Proposition 20 allows us to establishthat a i is the number of triangles incident with the vertex i , for 1 ≤ i ≤ n + 3, i = pm and 1 ≤ m ≤ p | ( n + 3). Therefore A and F ( v A ) generate the samefrieze pattern. Agust´ın Moreno Ca˜nadas et al (ii) Let ( D n +1) , R ) be any Dyck paths category, we take objects of ( D n +1) , R )defined by the following identity Ob ( D n , R ) = (cid:26) M G i ∈ D n G i (cid:12)(cid:12)(cid:12)(cid:12) g ( λ G i ) and g ( λ G j ) are in the same mutation class (cid:27) , the map ϕ : D A n → Ob ( D n , R ), such that ϕ ( v A ) = f ( v A ) ⊕ · · · ⊕ f ( v A p − ) , with { A t } ≤ t ≤ p − is surjective as a consequence of Theorem 17 and Proposition20. (cid:3) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ⊕ ⊕ Figure 11: Examples of objects of a Dyck paths category.
As an example of the use of Theorem 21, we choose the object D of a Dyck pathscategory ( D n +1) , R ) shown in Figure 11. Then D has associated the following friezepattern where red numbers is a positive integral diamond of Dynkin type A n . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . . References [1] I. Assem, C. Reutenauer, and D. Smith,
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