A GRAPH THEORETIC EXPANSION FORMULA FOR CLUSTER ALGEBRAS ...musiker/Finite.pdf · The ﬁnite type case is deﬁned as the case where the cluster variable generation procedure only

A GRAPH THEORETIC EXPANSION FORMULA FOR CLUSTER

ALGEBRAS OF CLASSICAL TYPE

GREGG MUSIKER

Abstract. In this paper we give a graph theoretic combinatorial interpreta-tion for the cluster variables that arise in most cluster algebras of finite typewith bipartite seed. In particular, we provide a family of graphs such that aweighted enumeration of their perfect matchings encodes the numerator of theassociated Laurent polynomial while decompositions of the graphs correspondto the denominator. This complements recent work by Schiffler and Carroll-Price for a cluster expansion formula for the An case while providing a novelinterpretation for the Bn, Cn, and Dn cases.

Contents

1. Introduction 12. An 43. Cn 164. Bn and Dn 185. G2 316. Future Directions 31References 34

1. Introduction

Several years ago, Sergey Fomin and Andrei Zelevinsky introduced a new mathe-matical object known as a cluster algebra which is related to a host of other combi-natorial and geometric topics. Some of these include canonical bases of semisimplealgebraic groups, generalized associahedra, quiver representations, tilting theory,and Teichmuller theory. In the proceeding we will use the definitions and conven-tions used in Fomin and Zelevinsky’s initial papers, [9, 11]. Starting with a subset{x1, x2, . . . , xn} of cluster algebra A, one applies binomial exchange relations toobtain additional generators of A, called cluster variables. The (possibly infinite)set of cluster variables obtained this way generate A as an algebra. It was provenin [9] and [10] that any cluster variable is a Laurent polynomial in {x1, x2, . . . , xn},i.e. of the form

P (x1, . . . , xn)

xa11 x

a22 · · ·x

ann

(Note that xi =1

x−1i

is also allowed)

Date: June 19, 2008.

2000 Mathematics Subject Classification. 05E15, 16S99.Key words and phrases. cluster algebras, classical type, perfect matchings, laurentness.This work was done with support of an NSF Mathematical Sciences Postdoctoral Fellowship.

1

2 GREGG MUSIKER

where P (x1, . . . , xn) is a primitive polynomial (not divisible by any monomial)with integer coefficients and the exponents ai are (possibly negative) integers. Itis further conjectured that the polynomials P (x1, . . . , xn) have nonnegative integercoefficients for any cluster algebra. However, this conjecture has been proved in alimited number of cases, and excluding the Ptolemy algebra described below, havemostly been for coefficient-free cluster algebras. These cases include the bipartitefinite type case, as proved in [11], rank two affine cluster algebras as demonstratedin [4], [19], [23], and [25], and cluster algebras arising from acyclic quivers [6].

The finite type case is defined as the case where the cluster variable generationprocedure only yields a finite set of cluster variables associated to A. By a com-binatorial and geometric miracle, one which has sparked much interest in thesealgebras, the cluster algebras of finite type exactly correspond to the Lie algebrasof finite type. Furthermore, in these cases, the cluster variables (except for thexi’s) have denominators with nonnegative exponents and can be put in a 1-to-1correspondence with the positive roots of the associated root system.

Study of a particular finite type cluster algebra of type An with coefficients, alsoknown as the Ptolemy algebra, has been especially fruitful as it can be realized interms of the Grassmannian and Plucker embedding. In 2003, as part of the REACHresearch group under Jim Propp’s direction, Gabriel Carroll and Gregory Price [7]described two combinatorial interpretations of the associated cluster variables, onein terms of paths and one in terms of perfect matchings. Further, Ralf Schifflerrecently independently discovered and extended the paths interpretation [21].

In the present paper, we go beyond An, and describe a combinatorial interpreta-tion for the cluster variables in all four families of finite type, namely An, Bn, Cn,and Dn, for the coefficient-free case. Our combinatorial model will involve perfectmatchings, in the spirit of [19], and agrees with Carroll and Price’s interpretationin the An case. Unlike the aforementioned work we do not attempt to give theLaurent expansion of cluster variables in terms of an arbitrary seed but only interms of the initial bipartite seed, whose definition we remind the reader of below.By restricting ourselves to expansions in this initial seed, we are able to explic-itly write down families of graphs which encode the cluster algebra using weightedperfect matchings.

We shall use the following notation throughout this paper. Let G = (V,E) be afinite graph with vertex set V = {v1, . . . , vm} and edge set E ⊆ {{u, v} : u, v ∈ V }.For each edge e ∈ E, we set we to be the weight of e, where we is allowed to be 1or xi for i ∈ {1, 2, . . . , n}. A perfect matching M of graph G is a subset of E suchthat for every vertex v ∈ V , there is exactly one edge e ∈ M containing v. Theweight of a perfect matching is defined to be the product w(M) =

∏

e∈M we, andwe let P (G) denote the matching polynomial, or matching enumerator, of graph G,defined as

P (G) =∑

M is a perfect matching of G

w(M).

Let Φ be a root system of classical type and denote its positive roots as Φ+. Weshall let {α1, α2, . . . , αn} denote the simple roots of Φ and let [α : αi] denote themultiplicity of αi in root α. The result of this paper is the following theorem.

Theorem 1. For each such classical root system Φ, we explicitly construct a familyof graphs, GΦ, with the following three properties.

(1) |GΦ| = |Φ+|.

CLUSTER ALGEBRAS OF CLASSICAL TYPE 3

(2) For each α ∈ Φ+, there exists a unique GΦα ∈ GΦ associated to α.

(3) We have the cluster expansion formula

x[α]Φ =P (GΦ

α)

x[α:α1]1 · · ·x

[α:αn]n

,

where x[α]Φ denotes the cluster variable corresponding to positive root α(in type Φ) under Fomin and Zelevinsky’s bijection.

We will describe the explicit bijection between positive roots of Φ+ and graphsof GΦ in more detail in the ensuing sections. In brief, for each root system Φ,we shall define a family of tiles {T1, T2, . . . , Tn}, which is a finite set of graphswith weighted edges such that each Ti is isomorphic to a cycle graph with an evennumber of vertices. For each type of classical root system Φ, i.e. An, Bn, Cn, andDn, we define gluing rules so that for all nonnegative integer vectors of length n,(t1, t2, . . . , tn), there exists at most one graph in GΦ which can be constructed fromt1 copies of T1, t2 copies of T2, . . . , tn copies of Tn. To be more precise, given graphG constructed by gluing together tiles, we define

LP (G) to be the Laurent polynomialP (G)

xt11 · · ·x

tnn

where ti is the number of tiles in graph G labeled as Ti. Further, there exists sucha graph exactly when t1α1 + t2α2 + · · ·+ tnαn is a positive root of Φ+.

We shall use the convention from [11], so that the initial exchange matrix B =||bij ||

ni,j=1 contains columns of like sign. Any rank n cluster algebra of finite type

has such a seed consisting of a cluster of initial variables {x1, . . . , xn} and a set ofn binomial exchange relations of the form

xjx′j =

n∏

i=1

x|bij |i + 1.

After mutating in the kth direction, i.e. applying an exchange relation of the formxkx

′k = binomial, we obtain a new seed with cluster {x1, x2, . . . , xn} ∪ {x

′k} \ {xk}

and exchange matrix B′ = ||b′ij ||ni,j=1 such that the b′ij ’s satisfy

b′ij =

{

−bij if i = k or j = k,

bij + max(−bik, 0) · bkj + bik ·max(bkj , 0) otherwise.

As we mention below in Remark 2, we shall use an ordering of mutations in this pa-per so that we need only work with binomial exchanges of the form xkx

′k = (Monomial

+1). Note that we shall use the notation Pα(x1, x2, . . . , xn) to denote the numer-

ator of the cluster variable with denominator x[α:α1]1 x

[α:α2]2 · · ·x

[α:αn]n despite its

similarity with the notation of P (G) for the matching polynomial of graph G.

The outline of the paper is as follows. We proceed to prove Theorem 1 separatelyfor the four families of non-exceptional type, starting with the well-studied case ofAn. We will use different language than in [7], [11], or [21], and we include ourown proof of this case to familiarize the reader with the techniques which we willutilize later in the paper. Since the type of the cluster algebra will frequently beclear from context, we will simply denote tiles as Ti or graphs as Gα (instead of as

GαΦ). We end with some comments and directions for further research.

4 GREGG MUSIKER

1 3

12

3

1

4

3

2

3

1

4

12

1

3

1

3

1

3

2 2

4

21 1

21

3

4

1

3

12 2

1

3

12 2

4

2

3

21

2

3

4

24

Figure 1. The collection GB4 (edge weights described in Section 4).

Remark 1. In [12], Fomin and Zelevinsky explicitly constructed Fibonacci poly-nomials for types An and Dn and bipartite seeds using combinatorial formulas. Aslight variant, and generalization of these polynomials to other types, are defined in[13], where they are referred to as F-polynomials. See Definition 11.5 and Theorem11.6 of [13] for the connection between F-polynomials and Fibonacci polynomialsin the case of bipartite seeds. Fibonacci polynomials give expansions of Yα’s, whichare algebraically related to the cluster variables, as described by Corollary 6.3 of[13].

2. An

The work in this section was done independently of the work of Carroll-Price[7] and the work of Schiffler [21] mentioned in the introduction. We will use thenotation and the techniques of this section later in the paper for the Bn, Cn, andDn

cases. Thus we include this section even though the combinatorial interpretationgiven by Proposition 1 is not new in this case, although we believe our proof is new.

We begin by reviewing the necessary characteristics of the cluster algebra oftype An. Recall that Lie algebra An has a Dynkin diagram consisting of a line ofn vertices connected by edges of weight one.


54321

5

432

4

xx

x

x

x

1

3

x

xx2

Figure 2. The tiles for cluster algebra of type A5.

2 3 41 n

Thus the associated Cartan matrix has the form

2 −1 0 0 . . . 0 0−1 2 −1 0 . . . 0 00 −1 2 −1 . . . 0 00 0 −1 2 . . . 0 0. . . . . . . . . . . . . . . . . . . . .0 0 0 0 . . . −1 2

,

and thus using the convention given in [11] the associated exchange matrix is

BAn = ||bij || =

0 1 0 0 . . . 0 0−1 0 −1 0 . . . 0 00 1 0 1 . . . 0 00 0 −1 0 . . . 0 0. . . . . . . . . . . . . . . . . . . . .0 0 0 0 . . . (−1)n+1 0

.

Notice that every column has like sign and that the matrix is skew-symmetrizable(and in fact skew-symmetric in this case). The bipartite seed for a cluster algebraof type An therefore consists of an initial cluster of variables {x1, x2, . . . , xn} andexchange matrix BAn which encodes the following exchange binomials, readingdown the columns,

x1x′1 = x2 + 1

x2x′2 = x1x3 + 1

x3x′3 = x2x4 + 1

. . .

xn−1x′n−1 = xn−2xn + 1

xnx′n = xn−1 + 1.

We describe a set of tiles from which we will build our family of graphs. In the caseof An, let tiles T1, . . . , Tn be squares defined as follows:

Definition 1. Tile T1’s northern edge is given weight x2 while the other three aregiven weight 1. Tile Tn’s southern edge is weighted with value xn−1 and the restare weighted with value 1. Finally all other Ti have a weight of xi+1 given to theirnorthern edge, xi−1 for their southern edge while the eastern and western edges aregiven weight 1.

Let GAnbe the set of graphs that can be built from these n tiles given the following

gluing rule.

6 GREGG MUSIKER

4

4321543

3

24

432 215

2

4321

32

1 5

3

3 454321

5

Figure 3. The collection GA5 .

Rule 1. Without allowing reflections or rotations of the tiles, tile Ti can be gluedto tile Tj if and only if the identified edge (as an edge of Ti) lies clockwise from anedge weighted xj and clockwise from an edge weighted xi (as an edge of Tj).

Since tile Ti only contains edges of weight xi+1 and xi−1, and these weights appearacross from each other, this rule uniquely describes how the blocks can connect.

Lemma 1. Given the above tiles, TAn, and the above gluing rule, the collection of

possible graphs is enumerated by the set of subsets

{Ti, Ti+1, . . . , Tj−1, Tj}

for 1 ≤ i < j ≤ n.

This collection GAnhas the same cardinality as the set of positive roots of the Lie

algebra of type An using the bijection

ψAn: Ti ∪ Ti+1 ∪ Ti+2 ∪ · · · ∪ Tj−1 ∪ Tj → αi + αi+1 + · · ·+ αj .

As shown in [11], this implies that the cardinality is also the same as the numberof non-initial cluster variables for the bipartite cluster algebra of type An.

Proposition 1. The set of graphs GAnis in bijection with the set of non-initial

cluster variables for a coefficient-free cluster algebra of type An and satisfy the state-ment of Theorem 1. Under this bijection, the graph Ti ∪Ti+1 ∪ · · · ∪Tj correspondsto the unique cluster variable with denominator xixi+1 · · ·xj.

To prove this proposition we will take a detour through a case we refer to asA∞, i.e. n is arbitrarily large. In this case, our set of tiles is in bijection with theintegers, and we define Ti to have yi+1 on its northern edge, yi−1 on its southernedge for all i ∈ Z. Without the issue of boundaries, it is easier to show that a certainarray of graphs corresponds to the non-initial cluster variables. After doing so, we


x1 x3 x5 . . . xn−2 xn

x2 x4 x6 . . . xn−1

x(1)1 x

(1)3 x

(1)5 . . . x

(1)n−2 x

(1)n

x(2)2 x

(2)4 x

(2)6 . . . x

(2)n−1

x(3)1 x

(3)3 x

(3)5 . . . x

(3)n−2 x

(3)n

x(4)2 x

(4)4 x

(4)6 . . . x

(4)n−1

Figure 4. Top six rows of the array constructed via Remark 2(assuming n odd).

choose a periodic specialization for the initial variables to recover a correspondingregion of this array for any specific An. We start with the following observation.

Remark 2. For all n, if we start with the above exchange matrix BAn and applythe binomial exchanges corresponding to relations 1, 3, 5, . . . n (resp. n − 1) if nis odd (resp. even) the resulting exchange matrix is −B. Afterward, applying therelations 2, 4, 6, . . . n−1 (resp. n) if n is odd (resp. even) to exchange matrix −BAn

results in the initial exchange matrix BAn . In fact, in both of these cases, the orderof the exchanges does not matter, and the intermediate exchange matrices will havecolumns of like sign for all relevant xk not already exchanged. By the definition ofmatrix mutation, this procedure will in fact work for any cluster algebra where theseed has an exchange matrix that is tri-diagonal (bij = 0 if |i − j| 6= 1). Thus wecan calculate a row of cluster variables at a time by applying the exchange relationsrelative to the two previous rows. The tri-diagonal condition includes the cases An,Bn, Cn, and G2 and minor modifications to the procedure will allow it to work forDn.

Returning to the An case, after applying exchange 1, 3, 5, . . . , n (resp. n− 1) wehave cluster

{x(1)1 , x2, x

(1)3 , x4, x

(1)5 , . . . , xn−1, x

(1)n }

(resp. {x(1)1 , x2, x

(1)3 , x4, x

(1)5 , . . . , xn−2, x

(1)n−1, xn} )

where x(1)i = xi−1xi+1+1

xiusing the convention x0 = xn+1 = 1. Analogously, apply-

ing exchanges 2, 4, 6, . . . , n− 1 (resp. n) we obtain the cluster

{x(1)1 , x

(2)2 , x

(1)3 , x

(2)4 , x

(1)5 , . . . , x

(2)n−1, x

(1)n }

(resp. {x(1)1 , x

(2)2 , x

(1)3 , x

(2)4 , x

(1)5 , . . . , x

(2)n−2, x

(1)n−1, x

(2)n } )

where x(2)i =

xi−2x2i xi+2+xi−2xi+xi−1xi+1+xixi+2+1

xi−1xixi+1for 1 ≤ i ≤ n, if we set x−1 =

xn+2 = 0. By Remark 2 we can make an array of cluster variables by applyingexchanges iteratively (row-by-row) in this order, see Figure 4.

By the binomial exchange relations, this array satisfies the diamond conditionwhich states that the relation ad = bc+ 1 holds for any four elements arranged asa diamond.

ab c

d

8 GREGG MUSIKER

. . . −x3 −x1 0 x1 . . . xn 0 −xn . . .

. . . −x2 −1 1 x2 . . . 1 −1 . . .

. . . −x(1)3 −x

(1)1 0 x

(1)1 . . . x

(1)n 0 −x

(1)n . . .

. . . −x(2)2 −1 1 x

(2)2 . . . 1 −1 . . .

. . . −x(3)3 −x

(3)1 0 x

(3)1 . . . x

(3)n 0 −x

(3)n . . .

. . . −x(4)2 −1 1 x

(4)2 . . . 1 −1 . . .

Figure 5. First six rows of extended frieze pattern (assuming n odd).

Remark 3. Arrays of integers satisfying the diamond condition are known as friezepatterns, and were studied by Conway and Coxeter [8] in the 1970’s. They havealso been studied in connection with cluster algebras elsewhere in work of Caldero[5] and work of Propp [20]. When we use cluster variables rather than integers asentries, these arrays are also special cases of the bipartite belt described in [13];each pair of consecutive rows of the frieze pattern corresponds to a seed of the belt.

In our example, xix(1)i = xi−1xi+1 +1 for i ∈ {2, 3, . . . , n−1}. Furthermore, this

frieze pattern can be extended periodically by using the conventions x−1 = xn+2 =0, x0 = xn+1 = 1, and extending further using negatively weighted variables. Notethat the negatives are necessary since we wish the configurations

0 0b 1 1 c

0 0

to satisfy the diamond condition.This pattern can also continue infinitely in the vertical direction, as well as

horizontally, extending vertically in the unique way that preserves the diamondcondition throughout the entire frieze pattern, see Figure 5. Consequently, all An

can be treated simultaneously by considering the infinite diamond pattern (A∞)which starts with sequence {. . . , y−2, y−1, y0, y1, y2, . . . } zig-zagging to create theinitial two rows. To obtain the extended An-frieze pattern for a specific n wefirst set the initial variables y−1, y0, y1, . . . , yn+2 to be certain values in terms ofthe xi’s, extend to yn+3, yn+4, . . . , y2n+4 by a reflective symmetry about yn+2, andthen periodically extend to all yk for k∈ Z:

y−1 = yn+2 = x0(1)

y0 = yn+1 = 1(2)

yi = xi for i ∈ {1, 2, . . . , n}(3)

if yn+2−k = zk ∈ {1, x0, x1, x2, . . . , xn}, then

yn+2+k = −zk for k ∈ {1, 2, . . . , n+ 2}(4)

if yk = zk ∈ {1, x0, x1, x2, . . . , xn}, then

y2n+6+k = zk for all k ∈ Z.(5)

If P is a Laurent polynomial in the variables {yi : i ∈ Z}, we henceforth let P denotethe Laurent polynomial in {x0, x1, x2, . . . , xn} obtained after applying substitutions(1)-(5). As a last step, we take the limit as x0 goes to zero, which is well-defined byProposition 2 appearing below. Note that we do not set y−1 and yn+2 to be zerodirectly since this would sometimes result in indeterminate expressions of the form“0/0”. Nonetheless, since the fourth identity is invalid for k = 0 or k = n+ 3, our


values for each yi are unambiguously defined. For example, y2n+5 = y−1 = x0 asopposed to −x0 by periodically extending.

As a consequence of these substitutions, it suffices to start the proof of Proposi-tion 1 by proving the combinatorial interpretation for the infinite diamond patterncorresponding to A∞, which we write in terms of yi’s and tiles Ti for i ∈ Z. Even

though we now have a boundary-less frieze pattern, every given y(j)i can be com-

puted locally by considering the necessary mutations stemming from a finite halfdiamond extending back to the initial two rows of yi’s. More precisely, to compute

y(j)i requires the initial values of yi−j , yi−j+1, . . . , yi+j .

We now proceed to prove the combinatorial interpretation in this boundary-lessversion. We start with the base case where we can easily see that the combinato-rial interpretation works for cluster variables with denominator yi. To see this, we

observe that yiy(1)i = yi−1yi+1 + 1 corresponds to the two perfect matchings of the

graph consisting of tile Ti by itself. Similarly, we observe the bijection works for

the second row of non-initial cluster variables by the definition of y(2)i . Recall our

definition of LP (G) of a graph G to be the Laurent polynomial whose numeratoris the matching enumerator of G and whose denominator vector encodes the occur-

rences of tiles Ti in G. By comparing with the formula for y(2)i given after Remark

2, we directly verify that y(2)i = LP (Ti−1 ∪ Ti ∪ Ti+1), where Ti−1 ∪ Ti ∪ Ti+1 is

shorthand for the graph consisting of tiles Ti−1, Ti, Ti+1 glued together in thatorder. By a technique of graphical condensation developed by Eric Kuo [17], weobtain the following combinatorial interpretation for the rest of the rows.

Lemma 2. Cluster variable y(j)i is equal to

LP (Ti−j+1 ∪ · · · ∪ Ti+j−1)

where Ti−j+1 ∪ · · · ∪ Ti+j−1 denotes the graph containing exactly 2j − 1 tiles, con-nected in order.

Proof. The proof follows from a slight variant of the argument given in [19]. Herewe need to be more careful with the labeling scheme, but the same pairings will yieldthe desired result. In fact if one lets yi = x if i even and yi = y if i odd, one recoversthe A(2, 2) case analyzed in [19]. In particular, we inductively assume for all i ∈ Z

that y(j−1)i = LP (Gi

1) where Gi1 = Ti−j+2∪· · ·∪Ti+j−2 and y

(j−2)i = LP (Gi

2) where

Gi2 = Ti−j+3 ∪ · · · ∪ Ti+j−3, i.e. y

(j−1)i =

P (Gi1)

yi−j+2···yi+j−2and y

(j−2)i =

P (Gi2)

yi−j+3···yi+j−3.

It thus suffices to show, for all i ∈ Z, that the Laurent polynomials y(j)i , defined as

y(j−1)i−1 y

(j−1)i+1 +1

y(j−2)i

, equal LP (Gi0) =

P (Gi0)

yi−j+1···yi+j−1with Gi

0 = Ti−j+1 ∪ · · · ∪ Ti+j−1. We

use our recursive definition and clear denominators to rewrite our desired equationas

P (Gi0)P (Gi

2) = P (Gi−11 )P (Gi+1

1 ) + yi−j+1yi−j+2y2i−j+3 · · · y

2i+j−3yi+j−2yi+j−1.(6)

One can decompose graph Gi0 into a superposition of graphs Gi−1

1 ∪ Gi+11 so that

Gi2 is the intersection of overlap. Out of the two subgraphs, only Gi−1

1 contains

tiles Ti−j+1, Ti−j+2 and only Gi+11 contains Ti+j−1, Ti+j−2. Let M(G) denote

the set of perfect matchings of graph G, m0′ denote the matching of Gi

0 using

the horizontal edges of Ti−j+2, Ti−j+4, . . . , Ti+j−4, Ti+j−2, and m2′ denote the

matching of Gi2 using the horizontal edges of Ti−j+3, Ti−j+5, . . . , Ti+j−5, Ti+j−3.

10 GREGG MUSIKER

The pair of matchings (m′0,m

′2) has exactly the weight of the excess monomial

yi−j+1yi−j+2y2i−j+3 · · · y

2i+j−3yi+j−2yi+j−1.

We finish the proof of Lemma 2 by exhibiting a weight-preserving bijection betweenM(Gi

0)×M(Gi2) \ {(m0

′,m2′)} and M(Gi−1

1 )×M(Gi+11 ), thus showing (6).

We define our bijection piece-meal on M(Gi0)×M(Gi

2) \ {(m0′,m2

′)}, first con-

sidering the case where the horizontal edges of penultimate tile Ti+j−2 in Gi0 are

not used. In this case, the pair of matchings from M(Gi0) × M(Gi

2) reduces to

a pair from M(Gi−11 ∪ Ti+j−1) ×M(Gi

2). We define φ(m0,m2) = (m−1,m1) for

such matchings by letting m−1 be the corresponding matching of Gi−11 and build

matching m1 by adjoining the matching of Gi2 to the matching of Ti+j−1. In other

words, map φ takes tiles Ti+j−1, Ti+j−2 and slides them down from Gi0 onto Gi

2 to

obtain Gi+11 with the matching included. This also leaves Gi−1

1 in place of Gi0.

If on the other hand, the horizontals of Ti+j−2 are used, then the situation ismore complicated. If we restrict further to the case where the rightmost verticaledge is used in Gi

2, we can define φ analogously by sliding down tiles Ti+j−2 and

Ti+j−1, followed by swapping tiles Ti+j−3 of graphs Gi0 and Gi

2. We are also forced

to use the rightmost vertical edge of Gi−11 in this case.

We can continue defining φ iteratively, defining it for classes characterized bythe length of the pattern of horizontals on the right-hand sides of Gi

0 and Gi2.

If the horizontals of Ti+j−2, Ti+j−4, . . . , Ti+j−2ℓ in Gi0, and the horizontals of

Ti+j−3, Ti+j−5, . . . , Ti+j−2ℓ−1 (resp. Ti+j−2ℓ+1) in Gi2 are used, accompanied by

a vertical edge between tiles Ti+j−2ℓ−2 and Ti+j−2ℓ−1 of Gi0 (resp. Ti+j−2ℓ−1 and

Ti+j−2ℓ of Gi2), then φ swaps the right-hand sides of these two graphs, leaving the

left-hand sides alone up until tile Ti+j−2ℓ−3 (resp. Ti+j−2ℓ−2). This constructionmakes sense as long as we eventually encounter a vertical edge as we move leftward;in this case, a vertical edge and the horizontal edges appearing in such patternsensure that neither the matching of Gi

0 nor of Gi2, will use the horizontal edges of

tile Ti+j−2ℓ−2 (resp. Ti+j−2ℓ−1).Map φ is injective since the inverse map just swaps back the right-hand sides

as dictated by the alternating pattern of horizontals. Since we have exhaustivelyenumerated the pairs of matchings (m−1,m1) by splitting into classes according tothe longest alternating pattern of horizontals, we also have surjectivity. Lastly, it iseasy to verify that (m′

0,m′2) is the unique matching which cannot be decomposed

into a pair (m−1,m1). �

Analogous pairings will also appear below in the arguments for the case of Bn.Notice that by Lemma 2, the diagonals of the frieze pattern satisfy the followingtwo properties:

• On any of the diagonals travelling from SW to NE, all graphs end with thesame tile.• On any of the diagonals travelling from NW to SE, all graphs start with the

same tile.

We now wish to show how to specialize to the case of a specific An by imposingperiodicity and boundary conditions on the initial two rows of variables. We shallshow that after specialization (1)-(5) and the limits, the An-frieze pattern sits inside


. . . y(1)1 y

(1)3 y

(1)5 y

(1)7 y

(1)9 y

(1)11 . . .

. . . y(2)2 y

(2)4 y

(2)6 y

(2)8 y

(2)10 . . .

. . . y(3)1 y

(3)3 y

(3)5 y

(3)7 y

(3)9 y

(3)11 . . .

. . . y(4)2 y

(4)4 y

(4)6 y

(4)8 y

(4)10 . . .

. . . y(5)1 y

(5)3 y

(5)5 y

(5)7 y

(5)9 y

(5)11 . . .

. . . y(6)2 y

(6)4 y

(6)6 y

(6)8 y

(6)10 . . .

. . . y(7)1 y

(7)3 y

(7)5 y

(7)7 y

(7)9 y

(7)11 . . .

. . . y(8)2 y

(8)4 y

(8)6 y

(8)8 y

(8)10 . . .

. . . y(9)1 y

(9)3 y

(9)5 y

(9)7 y

(9)9 y

(9)11 . . .

. . . y(10)2 y

(10)4 y

(10)6 y

(10)8 y

(10)10 . . .

. . . y(11)1 y

(11)3 y

(11)5 y

(11)7 y

(11)9 y

(11)11 . . .

. . . y(12)2 y

(12)4 y

(12)6 y

(12)8 y

(6)12 . . .

Figure 6. The first twelve rows of the A∞-frieze pattern.

the extended An-frieze pattern as columns 1 through n of the first (n + 3) rows,including two rows of initial cluster variables. To accomplish this goal, we must

(a) Verify that each row of the frieze pattern with entries given by limx0→0 y(j)i ,

for i ∈ Z, 1 ≤ j ≤ n+ 1, has the same horizontal pattern as the extended-An-frieze

pattern. Recall that for each i, j ∈ Z, the element limx0→0 y(j)i denotes the Laurent

polynomial that we obtain from applying relations (1)-(5) to the yi’s of Laurent

polynomial y(j)i .

(b) Verify that this frieze pattern is vertically periodic. Thus the set of Laurentpolynomial entries are associated to a finite set of graphs, which we readily identifyas the set GAn

.To show (a), it suffices to show that after applying relations (1) - (5), we have

the correct behavior on the boundaries, i.e. limx0→0 y(j)−1 = 0 = limx0→0 y

(j)n+2 and

limx0→0 y(j)0 = 1 = limx0→0 y

(j)n+1 for 1 ≤ j ≤ n + 1. The diamond condition will

then induce the analogous horizontal reflective symmetry (4) and periodicity (5)for non-initial cluster variables.

Notice that (4) implies that for k ∈ {1, 2, . . . , n + 2}, the variables yn+2+k andyn+2−k specialize to values which are negatives of each other. By periodicity, (5),we have that yk−1 and y−k−1 also specialize to negatives of each other, for allk 6≡ −1 mod (n+ 3). Note that yk−1 = x0 otherwise.

Lemma 3. The Laurent polynomial y(j)i is divisible by x0 whenever i ≡ −1 mod (n+

3) and 1 ≤ j ≤ n + 1. Consequently, the limit of such y(j)i ’s as we take x0 → 0

exists, and is zero for such graphs.

Proof. We shall prove the result for y(j)−1. The case y

(j)n+2 is analogous by symmetry

from the identity yn+2−k = −yn+2+k for k ∈ {0, 1, . . . , n + 2}, and by periodicityy2n+6+k = yk, we obtain the result for the remaining i such that i ≡ −1 mod (n+

3). By Lemma 2, y(j)i = LP (Ti−j+1 ∪ · · · ∪ Ti+j−1) for all i ∈ Z, j ≥ 1. Returning

to the i = −1 case, since j ≤ n + 1, it follows that graph T−j ∪ · · · ∪ Tj−2, which

we denote as G−10 , contains tile T−1 but not tile T−n−4 nor Tn+2. Consequently,

12 GREGG MUSIKER

LP (G−10 ) equals the Laurent polynomial

y(j)−1 =

P (G−10 )

y−1(y−2y0)(y−3y1)(y−4y2) · · · (y−jyj−2)

which specializes to

y(j)−1 =

P (G−10 )

x0(−1)(1)(−x1)(x1)(−x2)(x2) · · · (−xj−2)(xj−2).

Any matching of graph G−10 that uses neither the horizontals of tile T−2 nor of tile

T0 come in two types: those that use the horizontals of tile T−1, and those that use

both verticals of T−1. Since the two horizontals of T−1 are weighted with y0 = 1and y−2 = −1, the weights of these matchings will be identical except for their sign.

Consequently, in the sum P (G−10 ), all such matchings of this form cancel out each

other.We next consider the signed sum of matchings which contain either the horizon-

tals of tile T0 or the horizontals of T−2, but not both. Again, these two cases havethe same weighted number of matchings except for a difference in sign, and thustheir contribution to the associated Laurent polynomial is zero as well. We are leftenumerating matchings which contain the horizontals of both T0 and T−2. Thesematchings contribute a term with a factor of x2

0 in the numerator, but there is only

a single x0 in the denominator since j ≤ n + 1. Thus LP (G−10 ) is divisible by x0,

and we obtain limx0→0 LP (G−10 ) = 0. �

Lemma 4. Let 1 ≤ j ≤ n + 1 and Hj signify the graph T−j ∪ · · · ∪ Tj−1, i.e.

a graph with an even number of tiles, with leftmost tile T−j, and centered around

subgraph T−1 ∪ T0. Let Hj denote the graph T−j+1 ∪ · · · ∪ Tj−1, which contains an

odd number of tiles centered around T0. Then we have the following identities asLaurent polynomials:

limx0→0

LP (Hj) = yj and limx0→0

LP (Hj) = 1.

(We use the notation yj instead of xi since in the case j = n+ 1, yn+1 = 1 ratherthan xn+1.)

Proof. We start by proving the result for graphs with an even number of tiles,beginning with two base cases. The graph H1 = T−1 ∪ T0 has three perfect match-ings, but due to signs, two of them cancel with one another. We thus obtainP (T−1 ∪ T0) = y−1y1. After dividing through by y−1y0 = x0(1) we get Laurentpolynomial y1.

Secondly, the graph H2 = T−2 ∪ T−1 ∪ T0 ∪ T1 has eight perfect matchings, butone of them has a weight containing submonomial x2

0, which maps to zero in thelimit x0 → 0 since the denominator contains only a single factor of x0. Six of theother seven cancel with each other after using y−i−1 = −yi−1 and y0 = 1. We areleft with one perfect matching, which has weight y−3y−1y0y2 = −y1x0(1)y2. Thistime the denominator is y−2y−1y0y1 = (−1)x0(1)y1 and we obtain y2 after division.

We now assume that 3 ≤ j ≤ n+ 1 and wish to show that

P (Hj) = y−jy−j+1 · · · y−1(−1)x0(1)y1 · · · yj−1yj +O(x20)


for all such j. This approximation is sufficient since the denominator of the Laurentpolynomial associated to graphHj contains x0 to the first power, and terms divisibleby x2

0 vanish in the limit x0 → 0.Notice that if the rightmost vertical edge of graph Hj is used in a matching,

our computation of P (Hj) reduces to the computation of P (H ′j), where H ′

j =

T−j ∪ · · · ∪ T−1 ∪ · · · ∪ Tj−2, a graph centered around tile T−1. However, we knowfrom the proof of Lemma 3 that such a graph has a matching enumerator divisibleby x2

0. Thus any matching with the rightmost vertical edge of graph H ′j does not

contribute to the lowest term of the enumerator P (Hj). Thus we must use thetwo rightmost horizontal edges, which have weight yj−2yj = −(y−j)yj , and then

compute P (Hj−1) where Hj−1 = T−j∪· · ·∪Tj−3. However, such a graph is centered

around T−2 ∪ T−1, and is the horizontal reflection of graph Hj−1 about tile T−1.Thus by analogous logic, we are now forced to use the two leftmost horizontaledges in our matchings to get a nontrivial contribution. Such edges have weighty−j−3y−j+1 = (−yj−1)y−j+1. Consequently, after two iterations, and two helpfullyplaced negative signs, we have the identity

P (Hj) = (−y−j)(y−j+1)P (Hj−2)(−yj−1)yj +O(x20).

Induction thus yields the result for the case of Hj , i.e. a graph with an even number

of tiles centered around T−1 ∪ T0.We now prove this result for the corresponding graphs with an odd number

of tiles. Let Hj = T−j+1 ∪ · · · ∪ Tj−1. Since 1 ≤ j ≤ n + 1, the monomialy−j+1y−j+2 · · · yj−1 only contains one occurrence of x0. We will use this fact whentaking the limit of

P (Hj)

y−j+1 · · · y−3y−2y−1y0y1 · · · yj−1=

P (Hj)

y−j+1 · · · y−3(−1)x0(1)y1 · · · yj−1

as x0 → 0.We look at possible perfect matchings of graph Hj , and note that if both hor-

izontal edges or both vertical edges of tile Tj−1 are used, then we are reduced

to computing P (H ′j−1 ⊔ Tj−1) = P (H ′

j−1)P (Tj−1) with H ′j−1 = T−j+1 ∪ · · · ∪

Tj−3. However, since H ′j−1 is centered around tile T−1, the Laurent polyno-

mial P (H ′j−1)/y−1 = P (H ′

j−1)/x0 tends to zero as x0 → 0. We conclude that

any matching of Hj resulting in a nontrivial contribution to limx0→0 LP (Hj) =

limx0→0P (Hj)

y−j+1···y−3y−2y−1y0y1···yj−1= limx0→0

P (Hj)y−j+1···y−3(−1)x0(1)y1···yj−1

must uti-

lize the two horizontal edges of tile Tj−2 with weight yj−3yj−1.However, this step reduces our calculation to that of the matching polynomial

of Hj−1 = T−j+1 ∪ · · · ∪ Tj−4, which has an even number of tiles and is centered

around T−2 ∪ T−1. By our earlier logic, we therefore can only have a nontrivial

contribution to P (Hj−1) in the case where we use the two leftmost horizontal edges,which have weight y−jy−j+2. This reduction results in subgraph Hj−3, which is

14 GREGG MUSIKER

centered around T−1 ∪ T0, and so by induction

P (Hj) =

(

yj−3yj−1

)(

y−jy−j+2

)(

y−j+3 · · · yj−3

)

+O(x20)

= (−y−j+1)(yj−1)(−yj−2)(y−j+2)

(

y−j+3 · · · yj−3

)

+O(x20)

whose lowest term with respect to x0 is the same as the denominator correspondingto Hj = T−j+1∪· · ·∪Tj−1. Thus after division and the limit, the Laurent polynomial

corresponding to P (Hj) is one. �

By analogous logic, but messier notation, we also find that limx0→0 P (H) = 1 if

H has an odd number of tiles and is centered around Tn+1.We thus have shown (a), i.e. that after applying substitutions and taking the

limit x0 → 0 to the initial row of the A∞-frieze pattern, we get the extendedAn-frieze pattern with the proper horizontal boundary, symmetry, and periodicity.

Since the y(j)i ’s satisfy the same recurrence relations as the cluster variables x

(j)i ,

subject to specializations (1) - (5), LP (Ti−j+1 ∪ · · · ∪ Ti+j−1) = limx0→0 y(j)i = x

(j)i

for 1 ≤ j ≤ n+1 as desired. We now need to show the vertical finiteness of the friezepattern indicated by property (b), thus obtaining the combinatorial interpretationstated in Proposition 1.

At this point, we switch back to our original definition of tiles T1 through Tn,with edge weights from the set {1, x1, x2, . . . , xn}, as defined in the statement of

Proposition 1. In particular, for i ∈ {1, 2, . . . , n}, tile Ti becomes Ti after thespecializations (2) and (3). We must now also extend our collection of tiles toinclude tiles which we denote as T0, T−1, T−2, . . . and Tn+1, Tn+2, Tn+3, . . .

where we assign Ti to be Ti after specializations (2) and (3) for all i ∈ Z. Thissequence of tiles is periodic of length 2n + 6, where a set of representative tiles isgiven by

{T−n−2, T−n−3, . . . , Tn+3}.

Comparing the horizontal edges of Ti and T−i−2 for i ∈ {1, 2, . . . , n}, we see thatthey are identical except that the value of the northern edge in T−i+2 is the negativeof the weight of the southern edge in Ti, and vice-versa. Also tile Tn+2 has −1 asthe weight of its northern edge and 1 as the weight of its southern edge, the reverseof the weights of tile T−1. Note that for the purposes of computing the matchingenumerator, the tiles Ti and T−i+2 are identical, as are tiles T−1 and Tn+2. Thereason is that a particular matching either contains or does not contain the twohorizontal edges as a pair, hence swapping their placement on the tile or negatingthe weight of both of them will not effect the matching enumerator.

The remaining tiles are T0 (resp. T−2) which has weight x1 (resp. −x1) acrossfrom an edge weighted by x0, and Tn+1 (resp. Tn+3) which has weight xn (resp.−xn) across from an edge weighted by x0. Our frieze pattern of graphs correspondsto an array where after the first two initial rows of variables, the jth row correspondsto graphs consisting of 2j−1 consecutive tiles glued together horizontally. We wishto finish the proof of Proposition 1 by showing that for 1 ≤ j ≤ n+ 1, if graph Gγ

is constructed by gluing tiles Ti together and LP (Gγ) equals the cluster variable

xγ = x(j)i , then there is a connected subgraph Hγ of Gγ , only containing tiles T1

through Tn such that LP (Hγ) also equals x(j)i .


Proposition 2. After specializations (1)-(5), the limits limx0→0 y(j)i are well-defined

for all 1 ≤ i ≤ n and 1 ≤ j ≤ n+ 1. Furthermore, whenever a graph contains thetile T0 (resp. Tn+1), we may shorten the graph by removing the largest connectedsubgraph centered around T0 (resp. Tn+1) without changing the associated Laurentpolynomial. This transformation will be henceforth referred to as excision.

Proof. Since we assume 1 ≤ i ≤ n, the graph G(j)i = Ti−j∪· · ·∪Ti+j corresponding

to limx0→0 y(j)i only contains tile T0 (resp. Tn+1) if it also contains tile T1 (resp.

Tn). We first consider the case where G(j)i contains T0 or Tn+1 but not both.

Without loss of generality, assume that graph G(j)i contains tiles T0 and T1, and

has the form G(j)i = T−k+1 ∪ · · · ∪ T0 ∪ T1 ∪ · · · ∪ Tk−1 ∪ Tk ∪ · · · ∪ Tk+ℓ−1 for

1 ≤ k, ℓ ≤ n + 1. We must show that the limit limx0→0P (G

(j)i

)

y−k+1···yk+ℓ−1is well-

defined, and that graphs G(j)i and G

(j)i

′= Tk ∪ · · · ∪ Tk+ℓ−1 associate to the same

cluster variable, i.e. LP (G(j)i ) = LP (G

(j)i

′) = limx0→0 y

(j)i .

We categorize perfect matchings of G(j)i based on whether or not two horizontal

edges appear on the tile Tk. If they do not, then that matching of G(j)i decomposes

into a matching of subgraph T−k−1∪· · ·∪Tk−1, and a matching of Tk+1∪· · ·∪Tk+ℓ−1.Thus we obtain

P (T−k−1 ∪ · · · ∪ Tk−1) · P (Tk+1 ∪ · · · ∪ Tk+ℓ−1)

y−k+1 · · · yk+ℓ−1

=P (T

−k+1 ∪ · · · ∪ Tk−1)

y−k+1 · · · yk−1

·P (Tk+1 ∪ · · · ∪ Tk+ℓ−1)

yk · · · yk+ℓ−1

= 1 ·P (Tk+1 ∪ · · · ∪ Tk+ℓ−1)

yk · · · yk+ℓ−1

as a contribution to LP (G(j)i ), where the second equality follows from Lemma 4.

Notice that the second fraction does not contain an x0 in the denominator since

G(j)i does not contain Tn+1 in this case, thus the limit is well-defined.If on the other hand, the horizontal edges of Tk are used, the matching decom-

poses differently and we get a contribution of

P (T−k+1 ∪ · · · ∪ Tk−2)

y−k+1 · · · yk

·yk−1yk+1

yk−1ykyk+1·P (Tk+2 ∪ · · · ∪ Tk+ℓ−1)

yk+2 · · · yk+ℓ−1,

which specializes to

yk−1 ·yk+1

ykyk+1·P (Tk+2 ∪ · · · ∪ Tk+ℓ−1)

yk+2 · · · yk+ℓ−1

by Lemma 4, and the limit x0 → 0 is again well-defined. Comparing the sum of

these two Laurent polynomials to the cluster variable corresponding to graph G(j)i

′

finishes the proof in the case that G(j)i contains T0 or Tn+1 but not the other.

We now consider the case where G(j)i contains both T0 and Tn+1. Since 1 ≤ i ≤ n

and 1 ≤ j ≤ n + 1, the largest subgraphs centered at T0 and Tn+1, respectively,

do not overlap. (If G(j)i = T−a ∪ · · · ∪ T0 ∪ · · · ∪ Tn+1 ∪ · · · ∪ Tn+1+b, then the

largest subgraph centered at T0 is T−a∪· · · ∪T0 ∪· · · ∪Ta and the largest subgraphcentered at Tn+1 is Tn+1−b ∪ · · · ∪Tn+1 ∪ · · · ∪Tn+1+b. These do not overlap unless

a > n+1− b, but G(j)i contains a+ b+(n+2) tiles, and for j ≤ n+1, this number

must be ≤ 2n+1.) We therefore excise subgraph T−a∪· · ·∪T0∪· · ·∪Ta by the aboveargument, using that y−2 = −1, y−1 = x0, y0 = 1, but not specializing any other

16 GREGG MUSIKER

yi’s for the moment. We then excise subgraph Tn+1−b ∪ · · · ∪ Tn+1 ∪ · · · ∪ Tn+1+b,yielding graph Ta+1 ∪ · · · ∪ Tn−b. �

Using Proposition 2, we can replace all graphs in our frieze pattern with sub-graphs containing only tiles between T1 and Tn. Additionally, the graph corre-

sponding to limx0→0 y(n+1)i = x

(n+1)i ,

Ti−n ∪ · · · ∪T0 ∪ · · · ∪Tn−i ∪ Tn+1−i ∪ Tn+1−(i−1) ∪ · · · ∪Tn+1 ∪ · · · ∪Tn+1+(i−1),

associates to the same cluster variable as the graph of the single tile Tn+1−i for ieven (resp. odd) if n odd (resp. n even). Thus the first row of our frieze patterncorresponds to tiles Ti for i odd while the (n + 1)st row of our graphical friezepattern corresponds to tiles Ti for i even, but written backwards. Propagating thediamond pattern, we obtain the first two rows of initial cluster variables again;however in reverse order and with the even indexed initial variables appearing first.The next (n+1) rows are an exact mirror image of the first (n+1) rows, and henceafter this second iteration we obtain vertical periodicity, and thus (b).

We refer to the region of the extended An-frieze pattern that lies between thetwo columns of positive ones and in the first (n + 1) rows underneath the initialvariables as the An-frieze pattern. The diagonals of the An-frieze pattern inherit theproperties of the A∞-frieze pattern and this implies that all the cluster variables doin fact appear in our frieze pattern. In particular, their denominators are in bijectionwith positive roots of An’s root system and we have the desired corresponding graphfor each of them. Thus Proposition 1 is proven. In particular, the extended An

frieze pattern reduces to doubly-periodic copies of the An frieze pattern containinggraphs involving only tiles T1, . . . , Tn.

Remark 4. We can build a graphical frieze pattern for the An case directly byplacing graph Gα in the location of cluster variable xα in the An-frieze pattern,and adding columns consisting of empty sets on the left-hand and right-hand sidesof this pattern. This pattern satisfies a tropical-like diamond condition where oneof the four following hold.

a = b ∪ c and d = b ∩ c

a = b ∩ c and d = b ∪ c

b = a ∪ d and c = a ∩ d

b = a ∩ d and c = a ∪ d

Remark 5. Hugh Thomas [24] brought it to the author’s attention that onecan also derive the above frieze patterns via the algorithm for constructing theAuslander-Reiten quiver [1] starting from projective representations of the pathalgebra corresponding to the alternating An Dynkin diagram; in particular thepattern of denominator vectors agrees with the dimension vectors of the indecom-posables in the AR quiver.

3. Cn

The Lie algebra Cn has the following Dynkin diagram

2 3 41 n


54321

5

432

4

xx

x

x

2x

2x

x

x

3

1

x

Figure 7. Tiles for C5.

and thus the bipartite exchange matrix is:

0 1 0 0 . . . 0 0−2 0 −1 0 . . . 0 00 1 0 1 . . . 0 00 0 −1 0 . . . 0 0. . . . . . . . . . . . . . . . . . . . .0 0 0 0 . . . (−1)n+1 0

.

To build the corresponding graphs we let TCnbe identical to TAn

except that tileT1 now has weights of x2 and x2 opposite each other instead of a lone weightededge. This change to T1 corresponds to the change to the exchange polynomialassociated to label 1 in the seed of this cluster algebra.

We use gluing rule 1 again which leads us to a collection similar to GAnexcept

now tile T1 can connect to tile T2 on either side, and we allow rotations of tiles by180◦. Thus the collection of possible graphs, GCn

corresponds to the sets of theform

{Ti, Ti+1, Ti+2, . . . , Tj−1, Tj}

for 1 ≤ i < j ≤ n or multisets of the form

{Ti, Ti−1, Ti−2, . . . , T3, T2, T1, T2, T3, Tj−1, Tj}

for 2 ≤ i ≤ j ≤ n. This collection GCnhas the same cardinality as the collection of

non-initial cluster variables for a cluster algebra of type Cn and thus the collectionof positive roots for a root system of type Cn, as in the last case [11, 16].

Proposition 3. The set of graphs GCnis in bijection with the set of non-initial

cluster variables for a coefficient-free cluster algebra of type Cn such that the state-ment of Theorem 1 holds.

This can be proved quickly by using the folding procedure as in [11, 12]. Weidentify A2n−1 with Cn by letting xk = xn+1−k for k ∈ {1, . . . , n − 1}. We letxn = x1 and let xk = xk−n+1 for k ∈ {n + 1, . . . , 2n − 1}. Our frieze patternwill contain repeats but we can restrict our list to the right half, including thecentral axis, to obtain the correct number of graphs. Thus Proposition 1 impliesProposition 3.

Remark 6. Though our convention for Bn and Cn agrees with the convention of[11], up to re-ordering of initial variables, it differs with comment (2.20) of [12],where Fomin and Zelevinsky describe that A2n−1 folds onto Bn and Dn+1 foldsonto Cn. As indicated above, we instead find from the graph theoretic side of thestory that A2n−1 folds onto Cn, and in the next section we fold Dn+1 onto Bn.

18 GREGG MUSIKER

1

3213 1

313

2

22

2 2

22 1

3

3

3

3

3

32

2 2

22

2

2

1

11

1

2

Figure 8. The collection GC3 with duplicates.

4. Bn and Dn

In the previous two cases, all of the exchange polynomials had degree two or less.For the cases of Bn and Dn, exactly one of the exchange polynomials has degreethree. We will deal with such exchanges by including hexagons as potential tiles.We start with the case of Bn, which is a folded version of the simply-laced Dn case.By folding, our proofs will require less notation and as we will see, the Dn case hasa symmetry such that we can easily derive this case from the results for Bn.

In the case of Bn, the Dynkin diagram is

2 3 41 n

and thus the bipartite exchange matrix is:

0 2 0 0 . . . 0 0−1 0 −1 0 . . . 0 00 1 0 1 . . . 0 00 0 −1 0 . . . 0 0. . . . . . . . . . . . . . . . . . . . .0 0 0 0 . . . (−1)n+1 0

.

We will now use the notation T1 through Tn to refer to a collection of tiles, TBn,

related to Bn. We construct TBnfrom TAn

by first replacing T2 with a hexagonhaving weights 1, x1, 1, x1, 1, and x3 in clockwise order starting from the top. Welet T1 be a trapezoid with a single weighted edge of x2 on its northern side. Notethat T1 is homeomorphic to its previous definition. Then for all i > 2 we defineTi, for type B, as a 90◦ counter-clockwise rotation of the An-tile Ti, including theboundary tile Tn which has a single weighted edge of xn−1 on its eastern side.

The gluing rule will be more complicated now that hexagons are involved. Asin the An case, we do not allow rotations or reflections of the tiles T1, T3, T4, . . . ,


x2

x3 x1

x1

2x2

1

5

4

3

x

x

x

x5

4

3

4

Figure 9. Tiles for B5.

Tn; those tiles must be connected in the orientations as described above. However,we do allow a 120◦ clockwise rotation of tile T2. As a first approximation, theset of graphs GBn

will include any graphs that can be constructed from TBnwhile

conforming to Rule 1. Any such graph will resemble either a tower of tiles Ta

through Tb for 3 ≤ a ≤ b ≤ n, a base involving hexagon T2 with or withouttrapezoid T1 on its western side, or may be a complex of a tower beginning withT3 on top of a base.

In addition, we enlarge the set of GBnby allowing any graphs that obey the

following second rule, whose description requires the following total order ω on thepositive integers. We define ω : Z≥1 → Z≥1 so that m′ < m implies

ω(2m+ 1) < ω(2m′′)

ω(2m) < ω(2m′)

ω(2m′ + 1) < ω(2m+ 1)

for all integers m,m′,m′′ ≥ 1. For example, the integers {1, 2, 3, 4, 5, 6, 7, 8, 9}would be re-ordered as {1, 3, 5, 7, 9, 8, 6, 4, 2} under ω.

Rule 2. The trapezoidal tile T1 may appear twice if and only if the graph has oneof the following two forms:

12

1

3

m1

1 1

33

2 2

4

m

m

2

1

where 2 ≤ m1,m2 ≤ n, ω(m2) > ω(m1), and in the second case, the leftmost tileT2 has the usual orientation and the rightmost tile T2 is rotated clockwise 120◦.

20 GREGG MUSIKER

x1 x3 x5 . . .x2 x4 x6 . . .

x(1)1 x

(1)3 x

(1)5 . . .

x(2)2 x

(2)4 x

(2)6 . . .

x(3)1 x

(3)3 x

(3)5 . . .

x(4)2 x

(4)4 x

(4)6 . . .

. . . . . . . . . . . .

Figure 10. The frieze pattern for B∞.

Note that the case m1 or m2 = 2 corresponds to a tower consisting of the base T2

with no tiles above.

Remark 7. Notice that Rule 1 is now broken when we connect trapezoid T1 to ahexagon T2 on its left. Furthermore, in the last of these cases, we have adjoinedan additional arc which had not been allowed or required in previous examples.However, there is precedent for using such additional arcs, see Section 3 of [19].

One can check that the collection of graphs GBnobeying Rule 1 or Rule 2 has

the cardinality equal to the number of positive roots for Bn, and that Theorem 1is satisfied by these definitions.

Proposition 4. The set of graphs GBnis in bijection with the set of non-initial

cluster variables for a coefficient-free cluster algebra of type Bn such that the state-ment of Theorem 1 holds.

Analogous to the An case, we will first prove the result for the B∞ case, thatis we assume that n is arbitrarily large so that for i ≥ 3 tile Ti always has exactlytwo weighted edges (xi−1 on its east and xi+1 on its west). This greatly simplifiesthe proofs by allowing easier notation and bypassing case-by-case analysis. We willlater discuss how to obtain the result for a specific Bn from such graphs. In the B∞

case, we amend Rule 2 by restricting m1 and m2 to both be odd. Thus total orderω reduces to the usual order on the integers in this case. We create a semi-infinitearray whose entries satisfy a deformed diamond condition.

Without the boundary on the right, any collection of four variables

ab c

d

such that b 6= x(j)1 and c 6= x

(j)2 will satisfy ad − bc = 1. A diamond such that

c = x(j)2 will satisfy the truncated condition ad − c = 1, and a diamond which

contains b = x(j)1 will satisfy the relation ad− b2c = 1. For notational convenience,

we will sometimes denote the identity x(j)i = LP (G) as x

(j)i ↔ G.

Given this setup along with the initial assignments of x(0)i = xi for i ≥ 1, we

directly verify that

x(1)1 ←→

x2

1, x

(2)2 ←→

12

3

1,


i−j = 0 i−j = 2

Lemma 7 Lemma 6 Lemma 5

i−j = −2

Lemma 8

i−j = −4

��

��

Figure 11. A model of how these Lemmas fit together and relateto the B∞-frieze pattern.

x(3)1 ←→

2

3

1, x

(4)2 ←→

1

3

1

3

2 2

4

5

,

where the weights of the edges are as dictated by the definitions of tiles T1 throughTn. (The rightmost tile T2 in G

x(4)2

has weights rotated 120◦.) Additionally for

i− j ≥ 2, the only initial variables used to determine x(j)i are {x3, x4, x5, . . . } and

thus we recover the regular diamond pattern used in the A∞ case. Consequently,we immediately obtain

Lemma 5. x(j)i ←→

a a+1 a+2 bfor a = i− j + 1, b = i+ j − 1

when i− j ≥ 2. We also find that 3 ≤ a ≤ b.

We proceed with the rest of the proof in three steps described below as Lemmas6, 7, and 8. The first two steps are proved inductively by using Lemma 5 as well

as {x(1)1 , x

(2)2 , x

(3)1 , x

(4)2 } as a base case. We will prove the inductive step via the

usual diamond condition ad − bc = 1 which will hold for the diagonals i − j = 0and i− j = −2 while i ≥ 2.

22 GREGG MUSIKER

Lemma 6. x(i)i ←→

2i−1

1

3

21

for i ≥ 2.

Lemma 7. x(i+2)i ←→ 1 1

33

2 2

4

2i+1

for i ≥ 2.

The proof of these two Lemmas will prove Proposition 4 for all x(j)i such that

i − j ≥ −2. We then use variants of the diamond conditions (ad − c = 1 and

ad − b2c = 1) to extend down columns x(j)1 and x

(j)2 respectively. Interlaced with

this induction, we must also extend down the diagonals, as illustrated in Figure 11.Consequently, we proceed to prove the following three results for j = 3, then forj = 4, and so on by induction.

Lemma 8. x(2j+1)1 ←→

2

3

1

2j+1

for j ≥ 1,

x(2j)2 ←→ 1 1

33

2 2

4

2j+1

2j−1

for j ≥ 2,

x(i+2j)i ←→ 1 1

33

2 2

4

2k−1

2j+1

where k = i+j for i ≥ 2 and j ≥ 1.


Notice that in the case of B∞, as described by Lemmas 5-8, only graphs with towerswith odd heights appear. It is via excision, i.e. Proposition 2, that we can projectdown to the Bn-frieze pattern. Using Proposition 2, a tower of height (n+1), (resp.(n+2)) corresponds to a tower with height n (resp. (n−1)) if n is even (resp. odd),and larger towers of odd height correspond to even towers of smaller height. Thusby the definition of total order ω, Rule 2 accurately describes the possible graphsof type Bn. With the proof of Proposition 4 now broken down into manageablechunks, we proceed to prove the Lemmas.

Proof of Lemma 6. By Lemma 5, we have that the northeast portion of thefrieze pattern is filled in, and we use the entries on diagonal i− j = 2 and the base

cases of x(1)1 and x

(2)2 to extend to the rest of diagonal i − j = 0 by the diamond

condition. Assuming that cluster variables are

a ←→ 3 4 5 2i−1

b ←→

2i−1

1

3

21

c ←→ 3 4 5 2i+1

we wish to show that d ←→

2i+1

1

3

21

, given the diamond relationad = bc + 1. First of all, we see that the occurrences of tiles match up on eachside of the equal sign, which implies that the denominators agree appropriately.It suffices to show the weighted number of matchings also match up accordingly.

Any pair of matchings of the graphs 3 4 5 2i−1

and

2i+1

1

3

21 can be

decomposed into a pair of matchings on graphs 3 4 5 2i+1

and

2i−1

1

3

21

except for one matching. The logic is identical to that of Lemma 2. Here we swap

24 GREGG MUSIKER

the tops of the towers and note if the top horizontal edge of the hexagon is used(instead of the NW and NE diagonal edges), then completely swapping the twotowers is permissible.

This extraneous indecomposable pairing is the pair and it has ex-actly the correct weight (x2

1x2x23x

24 · · ·x

22i−1x2ix2i+1) and thus the Lemma is proved.

Proof of Lemma 7. The proof of this Lemma is analogous except that we shiftthe diamond pattern so that

a ←→

2i−1

1

3

21

, d ←→ 1 1

33

2 2

4

2i+1

b ←→ 1 1

33

2 2

4

2i−1

, c ←→

2i+1

1

3

21

.

(We use the base case x(4)2 for i = 2 and use the diamond condition for i ≥ 3.) The

top of the left tower in graphs associated to c and d have two tiles not appearing inthe graphs associated to a and b; and the right tower in graphs b and d correspondto two tiles which do not appear in graphs a and c. Consequently, we again have abijection by swapping the tops of the (left) towers, and there is exactly one extra-neous pair of matchings formed by taking an alternating pattern of distinguishededges:

. This pair has precisely the correct weight of x41x

32x

33x

24x

25 · · ·x

22i−1x2ix2i+1.

Proof of Lemma 8. To prove Lemma 8, we use the following observations. Let

graph G1 be 12

3

4

2i+1

12

3

2k+1

, G2 be 12

3

4

2i+1

12

3

2k+1

, J1 be2

3

1

4

2i+1

, J2 be2

3

1

2k+1

,


H1 be3

4

4 2i 2i+15

2k+1

, and let H2 be

3 4

4 2i 2i+15

2k+1

where i ≥ k,and the rightmost hexagons T2 in graphs G1 and G2 are rotated 120◦.

Any matching of the disjoint union J1 ⊔ J2 using the leftmost edge of T1 in thesecond tower or the bottom rightmost edge of T2 in the first tower corresponds toa matching of G2 by identifying the two mentioned edges together. Since both ofthese edges are unweighted, one copy of this edge can be removed, resulting in aperfect matching of G2. The rotation by 120◦ of the rightmost hexagon T2 of G2

does not affect the weight of such matchings since in the matching of a tower, thethree weighted edges of a hexagon either all appear or none of them appear.

Any matching of G1 which does not use the two mentioned edges must use theleftmost edge of T1 in the left tower, the three weighted edges of T2 in the lefttower, the two horizontal edges of T1 in the right tower, and the bottom rightmostedge of T2 in the right tower. We may then choose any matching of the left towerT4∪T5∪· · ·∪T2i+1 and any perfect matching of the right tower T3∪T4∪· · ·∪T2k+1.

On the other hand, a matching of G1 using the extraneous arc also must usethe leftmost edge of T1 on the left, the bottom edge of T2 on the left, the top edgeof T1 in the middle, and the top rightmost and bottom edge of T2 on the right.After including these forced edges, we may chose any perfect matching of the towerT3 ∪ T4 ∪ · · · ∪ T2i+1 and any perfect matching of the tower T4 ∪ T5 ∪ · · · ∪ T2k+1.To summarize,

P (G1) = P (G2) + x21x2x3P (H1)

P (J1)P (J2) = P (G2) + x21x2x3P (H2).

Putting these two equalities together we obtain

P (G1) = P (J1)P (J2) + x21x2x3

(

P (H2)− P (H1)

)

.(7)

We now use identity (7) to prove part 1 of Lemma 7. In this case, k = i− 1 and allbut one matching of H2 corresponds to a matching of H1 by the usual procedureof swapping the right-hand sides. The extraneous matching of H2 has the form

2i−1

2i−1

4 5

5 4 3

2i+1 2i

,

contributing a factor of x3x5x5x7 · · ·x2i−1x2i+1 ·x2x4x4x6 · · ·x2i−2x2i, and yieldingthe identity

P (G) = P (J1)P (J2) + x21x

22x

23 · · ·x

22i−1x2ix2i+1.

Since x(j−1)1 x

(j+1)1 = x

(j)2 +1 is satisfied by letting J1 ←→ x

(j−1)1 and J2 ←→ x

(j+1)1 ,

part one of Lemma 8 is proved.Part three is proved analogously to Lemma 7. In this case, we have a diamond

where all four entries are graphs consisting of two towers on the maximal base of twotrapezoids and two hexagons. We inductively know the validity of these graphs forLaurent polynomials a, b, and c, so it is sufficient to verify the diamond condition

26 GREGG MUSIKER

if d←→ 1 1

33

2 2

4

2k−1

2j+1

. For ease of notation, we temporarily let Ga, Gb, Gc,and Gd be the graphs corresponding to these particular Laurent polynomials. Thetwo towers of Ga have heights of (2k − 3) and (2j − 1) on the left- and right-handsides, respectively. Graph Gb has heights (2k − 3) and (2j + 1), Gc has heights(2k−1) and (2j−1), and as illustrated above, Gd has heights (2k−1) and (2j+1).

As before, we wish to present a bijection

φ : M(Ga)×M(Gd) \ {(m′a,m

′d)} →M(Gb)×M(Gc)

where (m′a,m

′d) is a specific pair of matchings. Map φ starts by swapping the two

left towers of Ga and Gd if able; this is analogous to the earlier cases. However, inthe case where these towers cannot be swapped (because the alternating patterncontinues down into the base) map φ then attempts to swap the right towers of Ga

and Gd. There is exactly one pair of matchings where both attempts at swappingfails. This is the pair of matchings where the alternating patterns appear on bothtowers down through the bases; by inspection, such a pair has precisely the weightof the extraneous monomial.

This leaves part two as the crux of the proof and the step which utilizes thediamond relation ad− b2c = 1 which makes B∞ different from the previous cases.We wish to show that the assignments

a ←→ 1 1

33

2 2

4

2i+1

2i−1

, d ←→ 1 1

33

2 2

4

2i+3

2i+1

b ←→2

3

1

2i+1

, c ←→ 1 1

33

2 2

4

2i+3

2i−1

satisfy ad − b2c = 1. We use identity (7) in the case k = i. Since the graphsH1 = H2 and J1 = J2 in this case, identity (7) yields a bijection between match-

ings of

2i+1

3 3

2i+1

2 21 1 and matchings of

3

2i+1

211

2i+1

3

2

where the darkened


edges denote the forced edges. Thus we obtain P (G1) = P (J1)2 for the case k = i.

Since the perfect matching enumerators P (G1) and P (J1)2 are equal, and the fre-

quencies of tiles are identical in both G1 and J1⊔J1, we have shown that if Laurent

polynomial b←→2

3

1

2i+1

then b2 ←→ 12

3

12

3

2i+1 2i+1

.With this substitution, a superposition argument analogous to that which just

proved part three demonstrates the validity of the standard diamond relation ad−(b′)c = 1 where we let b′ = b2 associate to the graph whose left and right towersboth have height 2i+ 1.

With the last step completed, these three Lemmas prove that the cluster variablesof the B∞-frieze pattern correspond exactly to the desired graphs. There is apattern inherent in the NW to SE and NE to SW diagonals once again. This time,this pattern manifests itself (in the region where i ≤ j) by dictating the choice ofright tower (NW to SE) and the choice of left tower (NE to SW). Recall that Lemma5 already described the pattern in the region where i > j using towers consistingof tiles Ta ∪ · · · ∪ Tb where a ≥ 3. We now turn to the problem of restricting tospecific Bn. We use the same methodology, i.e. excision via Proposition 2, as inthe An case.

Given that we did not include the right-hand boundary, graphs can contain tileTi for arbitrarily large i. We thus want to essentially let xn+1 = 1 to force tileTn to have the proper weights, as a tile in TBn

as opposed to TB∞. (By abuse of

notation, we will refer to the tiles of both B∞ and Bn as Ti and allow xi’s to be theweights of the edges in both cases.) However, unlike the An case, we cannot simplyapply this substitution and use horizontal periodicity to make sure the diamondcondition holds throughout. The problem is that the left-hand boundary satisfies adiamond condition of a different form. Nonetheless, the diagonals to the northeast,those where i− j ≥ 2, contain towers of the form Ta∪· · · ∪Tb where 3 ≤ a ≤ b, andany neighboring four entries satisfy the same diamond condition as the An case.Thus the logic of Proposition 2 carries over and we can excise connected subgraphsof type B∞ centered at tile Tn+1.

Using excision of type An, we are able to determine the northeast corner of thefinite Bn frieze pattern. Using these graphs during the inductive step of Lemma 6in lieu of the arbitrarily large towers of T3 ∪ · · · ∪ T2j−1 allows us to fill in the nextdiagonal of the Bn frieze pattern where the towers sitting on the base of T1∪T2∪T1

will consist exclusively of the tiles between T3 and Tn. Similarly, the recursive stepsof Lemmas 7 and 8 also follow with these truncated graphs, with tiles between T1

and Tn used instead.Applications of Lemma 7 and then successive applications of the third part of

Lemma 8 will allow this interpretation to extend down SW to all but a SE cornerof the frieze pattern. Note that the diagonals again determine the left-hand andright-hand towers of the Bn-frieze pattern since this property is inherited from theB∞-frieze pattern. We compute the SE corner by starting at the bottom withinitial row T2, T4, T6, . . . , and propagating upwards via the diamond condition.We get via Lemma 5, now with tile T2, instead of T3, as the smallest allowable

28 GREGG MUSIKER

tile, all but a single diagonal of the SE corner. The final diagonal has the form(Tower 1) ∪ T2 ∪ T1 ∪ T2 ∪ T1, proven by applying Lemma 7 upwards.

Notice that in the end, we obtain a frieze pattern where the NW to SE diagonalsdictate the right towers and NE to SW diagonals dictate the left towers. There isone caveat: the empty tower, Tow∅ is now allowed. Thus one has to determine fromcontext whether a graph consisting of a single tower is of the form Tow∅∪TowR orTowL∪Tow∅. Alternatively, we can picture the SE corner as sitting directly abovethe NE corner of the frieze pattern to form a half-diamond. Thus Proposition 4 isproven.

In Figure 12, we give the frieze pattern corresponding to B6. Notice that thereare six graphs in the northeast corner and four graphs in the southeast corner whichare also graphs corresponding to positive roots and cluster variables for the A6 case.In fact, if we decrease each label by 2 and horizontally reflect the southeast corner,we can fit these two pieces together to obtain the A4 frieze pattern of graphs exactly.

We were able to analyze Cn based on A2n−1 using a folding procedure. Analo-gously we can analyze Dn using Bn−1 and an unfolding procedure. We label theDynkin diagram for Dn starting with 1 and 1 on the left, and label the rest in aline from 2 to n− 1.

2 3 4 n−1

1

1Indexing the rows and columns in the order {1, 1, 2, 3, . . . , n−1}, the correspond-

ing exchange matrix is therefore

0 0 1 0 0 . . . 0 00 0 1 0 0 . . . 0 0−1 −1 0 −1 0 . . . 0 00 0 1 0 1 . . . 0 00 0 0 −1 0 . . . 0 0. . . . . . . . . . . . . . . . . . . . . . . .0 0 0 0 0 . . . (−1)n 0

.

We split the odd and even initial variables into the first two rows, in a zig-zaggingpattern, just as before. We then mutate in the order 1, 1, 3, 5, . . . , n (resp. n− 1)if n is odd (resp. even) to get the the third row, followed by mutation via 2 then4, 6, . . . n− 1 (resp. n) if n is odd (resp. even) to get the fourth row.

The advantage of such an ordering is that the mutated exchange matrix, whichwe use to encode the binomial exchanges, is always the same, up to sign. We notice

that the analogue of the diamond condition for this case is ad − bc = 1 if b = x(j)i

with i ≥ 2 and

x(j−1)2 x

(j+1)2 − x

(j)1 x

(j)

1x

(j)3 = 1(8)

x(j−1)1 x

(j+1)1 − x

(j)2 = 1(9)

x(j−1)

1x

(j+1)

1− x

(j)2 = 1(10)

on the western boundary.


1 3 5

12

3

1 3

4

5

5

6

2

3

1

5

12

1

3

4 6

3

4

5

1

3

1

3

2 2

4

5

6

12

1

3

4

5

4

3

21

3

4

5

1

3

1

3

2 2

4

5

6

4

12

1

3

1

3

1

3

2 2

4

5

6

4

5

1

3

1

3

2 2

4

21 1

21

3

4

5

6

1

3

21

3

2

4

5

4

1

3

12 2

1

3

1

3

2 2

4 4

5

6

1 12

3

2

4

5

2

3

21

3

4

1

3

12 2

4

5

6

5

2

3

4

1

3

12 2

4

6

2

3

4

5

5

4

21

2

3

4

4

5

6

24 6

Figure 12. The collection GB6 .

30 GREGG MUSIKER

x2

1

x2

11x

x234x 4 x32x3 x1

Figure 13. Tiles for D5.

We let TDnbe TBn−1 ∪ {T1} where T1 is the same tile as T1 except with a

different label. We also change tile T2 so that it is still a hexagon, but has weights1, x1, 1, x1, 1, and x3 going around clockwise from the top. Following the argumentsof Lemmas 5, 6, 7, and 8 result in the same graph theoretic interpretation and friezepattern structure. We use Rule 3 which is analogous to Rule 2 and describes thefamily of graphs GDn

.

Rule 3. Notice that when we apply Rule 1 to set of tiles TDn, we get a set of

graphs consisting of a base of T2 or T1 ∪ T2 adjoining a tower of Ta ∪ · · · ∪ Tb, asbefore. We enlarge the set of graphs by allowing a base of T1 ∪T2 (with or withoutan accompanying tower), and also allow both tile T1 and tile T1 to appear if andonly if it is of the following two forms:

12

1

3

m1

1

3

1

3

2 2

m

4

m

2

1

where 2 ≤ m1,m2 ≤ n − 1, ω(m2) > ω(m1), and in the second case, the leftmosttile T2 has the usual orientation and the rightmost tile T2 is rotated 120◦ clockwise,(so the weights are 1, x3, 1, x1, 1 and x1 starting from the top going clockwise).

Let TDnbe defined as above and GDn

be the set of graphs constructed accordingto Rules 1 and 3.

Proposition 5. The set GDnis in bijection with the set of non-initial cluster

variables for a coefficient-free cluster algebra of type Dn such that the statement ofTheorem 1 holds.

Remark 8. As indicated, the proof follows from the exact same logic as Lemmas 5through 8 used now to describe the D∞ rather than the B∞ case. The only caveat

is that as a consequence of the proof, x(j)1 will sometimes be a tower on base T1∪T2,


and sometimes contain base T1 ∪ T2. In particular, x(2j−1)1 contains T1 if and only

if j is odd, and so we get an alternating behavior.

In Figure 14, we give the frieze pattern for GD5 . We have the usual diamondcondition for four entries in three consecutive rows and three consecutive columns,

not including column one. We encode column one by placing x(j)1 on top of x

(j)

1,

and we have the exchange relations (8), (9), and (10).

5. G2

The case of G2 is the only cluster algebra of exceptional finite type for which wehave been able to extend our graph theoretic interpretation. We are able to do sosince the G2 root system is a folding of the D4 root system, just like the B3 rootsystem. We use collection TG2 = {T1, T2} with tile T1 as in the Bn case, and tile T2

is again a hexagon, but now has all three nontrivial weights being value x1. Whenwe deform T1 to be a square, the weight of x2 lies on the eastern side of T1. Thereare six possible graphs that correspond to the non-initial cluster variables.

G2 has Dynkin diagram

1 2and exchange matrix

[

0 1−3 0

]

.

6. Future Directions

Given the previous sections, coefficient-free cluster algebras of type An, Bn, Cn,Dn, or G2 have a combinatorial interpretation as a family of graphs such that thenumerators of the cluster variables enumerate the weighted number of matchingsand the denominators encode the occurrences of faces. Thus Theorem 1 is true inall of these cases. The next step would be to extend Theorem 1 to include clusteralgebras of type E6, E7, E8, and F4, and thus have the result for all cluster algebrasof finite type.

Remark 9. Even though the Dynkin diagrams for the En’s are simply laced, fittingcluster algebras of these three types into patterns analogous to those of the An’sand Dn’s has been notoriously hard. Such difficulties have arisen elsewhere such asin the original proof of positivity in [11], and also in recent models using T -pathson triangulated surfaces, for example in [13] or [14] among other work.

Additionally, in the work of Schiffler and Carroll-Price for An, the cluster algebraconsidered is specifically the Ptolemy algebra, a cluster algebra with coefficients.In the T -paths model, the boundary of the polygon gives rise to (n+ 3) additionalcoefficients which can be included in the exchange relations and cluster expansionformula. Since the graphs we obtain in the above combinatorial interpretations areweighted so sparsely, perhaps a certain number of coefficients can be handled bythe graph-model as well.

In [19], an analogous interpretation is given for rank 2 cluster algebras of affinetype and unpublished work [15, 18] done as a part of REACH, as described in [20],gives a graph theoretic interpretation for a totally cyclic rank 3 cluster algebra.This totally cyclic rank 3 cluster algebra corresponds to a triangulated surface ofgenus one with exactly one puncture (i.e. interior marked point). Such a clus-ter algebra has been studied geometrically including work of [3]. Perhaps thesegraph theoretical interpretations could be extended to other cluster algebras thus

32 GREGG MUSIKER

1

3

1

2

3

11

4

3

2

3

1

12

1

3

4

2

3

1

1

3

1

3

2 2

4

21 1

21

3

4

1

3

12 2

21

3

4

1

3

12 2

4

2

3

21

2

3

4

21

24

Figure 14. The collection GD5 .


12

1 1

211

2

1

21 11

2 2

1

Figure 15. Graphs for a cluster algebra of type G2.

providing proofs of Fomin and Zelevinsky’s positivity conjecture for even furthercases.

Lastly, we note that all the examples discussed above are families of planargraphs associated to generators of cluster algebras. When expanding our scopeto include more complicated cluster algebras, is the category of planar graphs toorestrictive? More specifically, why did we need the extra arcs in the Bn, Dn , G2,

and affine A(2)2 cases? Perhaps it is an artifice of taking a higher dimensional object

and projecting to two dimensions.

Postscript. Recent independent work of Karin Baur and Robert Marsh [2] hasgiven a combinatorial interpretation for the integers arising as the cluster variablesof a cluster algebra of type Dn with acyclic seed after specializing the initial clusterto be {1, 1, . . . , 1}. In particular, they consider ideal triangulations of the punc-tured n-gon, and provide a model involving (partial) matchings which yield a friezepattern of integers satisfying analogues of relations (8), (9), and (10). In contrast,our work in the present paper provides the cluster variables of a cluster algebrawith initial cluster {x1, x2, . . . , xn} and bipartite seed.

New work by Ralf Schiffler and Hugh Thomas has generalized the Carroll-Pricepaths model to give combinatorial expansion formulas using T -paths for clusteralgebras with coefficients arising from unpunctured surfaces [22].

Acknowledgments. The author would like to thank Andrei Zelevinsky for nu-merous helpful conversations including referring the author to [12] where Fibonaccipolynomials appear. Discussions with Sergey Fomin, Jim Propp, Ralf Schiffler,and Hugh Thomas have also been very useful including special thanks to HughThomas and Andrei Zelevinsky for their comments on an earlier draft of this pa-per. The author would also like to thank the anonymous referees whose commentshave helped improved this exposition. An abridged version of this work was pre-sented at FPSAC 2008. The author thanks the conference referees for their helpfulcomments.

34 GREGG MUSIKER

References

[1] I. Assem, D. Simson, and A. Skowronski, Elements of the representation theory of associated

algebras. Vol. 1. Techniques of representation theory. London Mathematical Society StudentTexts, 65. Cambridge University Press, Cambridge, 2006.

[2] K. Baur and R. Marsh, Ptolemy relations for punctured discs, preprint (2007)arXiv/math:CO/07111443

[3] A. Beineke, T. Brustle and L. Hille, Cluster-cyclic quivers with three vertices and the MarkovEquation, preprint (2006) arXiv/math:RA/0612213.

[4] P. Caldero and A. Zelevinsky, Laurent expansions in cluster algebras via quiver representa-tions, Moscow Math. J. 6 (2006), 411-429.

[5] P. Caldero and F. Chapoton, Cluster algebras as Hall algebras of quiver representations,Comment. Math. Helv. 81 (2006), 595-616.

[6] P. Caldero and M. Reineke, On the quiver Grassmannian in the acyclic case, preprint (2006)arXiv/math:RT/0611074.

[7] G. Carroll and G. Price, Two new combinatorial models for the Ptolemy recurrence, unpub-lished memo (2003).

[8] J. Conway and H.S.M. Coxeter, Triangulated polygons and frieze patterns, Math. Gaz. 57

(1973), 87-94.[9] S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002),

497-529.[10] S. Fomin and A. Zelevinsky, The Laurent phenomenon, Adv. in Applied Math. 28 (2002),

119-144.[11] S. Fomin and A. Zelevinsky, Cluster algebras II: Finite type classification, Invent. Math. 154

(2003), 63-121.[12] S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. 158

(2003), 977-1018.[13] S. Fomin and A. Zelevinsky, Cluster algebras IV: Coefficients, Compositio Mathematica 143

(2007), 112-164.[14] S. Fomin, M. Shapiro, and D. Thurston, Cluster algebras and triangulated surfaces. Part I:

Cluster complexes, preprint (2006), arXiv/math.RA/060836.[15] A. Itsara, G. Musiker, J. Propp, and R. Viana, Combinatorial interpretations for the Markoff

numbers, unpublished memo (2003).[16] V. Kac Infinite dimensional Lie algebras, 3rd edition, Cambridge University Press, 1990.[17] E. Kuo, Applications of graphical condensation for enumerating matchings and tilings, The-

oret. Comput. Sci. 319 (2004), 29-57.[18] G. Musiker, A conjectured combinatorial interpretation for the Markoff numbers, unpublished

memo (2002).[19] G. Musiker and J. Propp, Combinatorial interpretations for rank-two cluster algebras of affine

type, Electron. J. Combin. 14 no. 1, Research Paper 15 (2007), 1-23.[20] J. Propp, The combinatorics of frieze patterns and Markoff numbers, preprint (2005)

arXiv/math:CO/0511633.[21] R. Schiffler, A cluster expansion formula (An case), preprint (2006) arXiv/math:RT/0611956.[22] R. Schiffler and H. Thomas, On cluster algebras arising from unpunctured surfaces, (2007)

arXiv/math:0712.4131.[23] P. Sherman and A. Zelevinsky, Positivity and canonical bases in rank 2 cluster algebras of

finite and affine types, Moscow Math. J. 4 (2004), No. 4, 947-974.[24] H. Thomas, Personal communication.

[25] A. Zelevinsky, Semicanonical basis generators of the cluster algebra of type A(1)1 , Electron.

J. Combin. 14 no. 1, Note 5 (2007), 1-4.

Mathematics Department, Massachusetts Institute of Technology, Cambridge, MA

02139

E-mail address: [email protected]

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A GRAPH THEORETIC EXPANSION FORMULA FOR CLUSTER ALGEBRAS ...musiker/Finite.pdf · The ﬁnite type case is deﬁned as the case where the cluster variable generation procedure only