CLUSTER ALGEBRAS I: FOUNDATIONS

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1 JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 15, Number, Pages S (01)00385-X Artcle electroncall publshed on December 8, 001 CLUSTER ALGEBRAS I: FOUNDATIONS SERGEY FOMIN AND ANDREI ZELEVINSKY To the memor of Serge Kerov 1. Introducton In ths paper, we ntate the stud of a new class of algebras, whch we call cluster algebras. Before gvng precse defntons, we present some of the man features of these algebras. For an postve nteger n, a cluster algebra A of rank n s a commutatve rng wth unt and no zero dvsors, equpped wth a dstngushed faml of generators called cluster varables. The set of cluster varables s the (non-dsont) unon of a dstngushed collecton of n-subsets called clusters. These clusters have the followng exchange propert: for an cluster x and an element x x, there s another cluster obtaned from x b replacng x wth an element x related to x b a bnomal exchange relaton (1.1) xx = M 1 + M, where M 1 and M are two monomals wthout common dvsors n the n 1varables x {x}. Furthermore, an two clusters can be obtaned from each other b a sequence of exchanges of ths knd. The prototpcal example of a cluster algebra of rank 1 s the coordnate rng A = C[SL ] of the[ group] SL, vewed n the followng wa. Wrtng a generc a b element of SL as, we consder the entres a and d as cluster varables, c d and the entres b and c as scalars. There are ust two clusters {a} and {d}, anda s the algebra over the polnomal rng C[b, c] generated b the cluster varables a and d subect to the bnomal exchange relaton ad =1+bc. Another mportant ncarnaton of a cluster algebra of rank 1 s the coordnate rng A = C[SL 3 /N ] of the base affne space of the specal lnear group SL 3 ;here N s the maxmal unpotent subgroup of SL 3 consstng of all unpotent upper trangular matrces. Usng the standard notaton (x 1,x,x 3,x 1,x 13,x 3 )forthe Plücker coordnates on SL 3 /N,wevewx and x 13 as cluster varables; then A s the algebra over the polnomal rng C[x 1,x 3,x 1,x 13 ] generated b the two cluster varables x and x 13 subect to the bnomal exchange relaton x x 13 = x 1 x 3 + x 3 x 1. Receved b the edtors Aprl 13, 001 and, n revsed form, October 6, Mathematcs Subect Classfcaton. Prmar 14M99; Secondar 17B99. Ke words and phrases. Cluster algebra, exchange pattern, Laurent phenomenon. The authors were supported n part b NSF grants #DMS , #DMS (S.F.), and #DMS (A.Z.). 497 c 001 b Serge Fomn and Andre Zelevnsk

2 498 SERGEY FOMIN AND ANDREI ZELEVINSKY Ths form of representng the algebra C[SL 3 /N ] s closel related to the choce of a lnear bass n t consstng of all monomals n the sx Plücker coordnates whch are not dvsble b x x 13. Ths bass was ntroduced and studed n [9] under the name canoncal bass. As a representaton of SL 3,thespaceC[SL 3 /N ]s the multplct-free drect sum of all rreducble fnte-dmensonal representatons, and each of the components s spanned b a part of the above bass. Thus, ths constructon provdes a canoncal bass n ever rreducble fnte-dmensonal representaton of SL 3. After Lusztg s work [13], ths bass had been recognzed as (the classcal lmt at q 1of)thedual canoncal bass,.e., the bass n the q-deformed algebra C q [SL 3 /N ] whch s dual to Lusztg s canoncal bass n the approprate q-deformed unversal envelopng algebra (a.k.a. quantum group). The dual canoncal bass n the space C[G/N] was later constructed explctl for a few other classcal groups G of small rank: for G = Sp 4 n [16] and for G = SL 4 n []. In both cases, C[G/N] can be seen to be a cluster algebra: there are 6 clusters of sze for G = Sp 4, and 14 clusters of sze 3 for G = SL 4. We conecture that the above examples can be extensvel generalzed: for an smpl-connected connected semsmple group G, the coordnate rngs C[G] and C[G/N], as well as coordnate rngs of man other nterestng varetes related to G, have a natural structure of a cluster algebra. Ths structure should serve as an algebrac framework for the stud of dual canoncal bases n these coordnate rngs and ther q-deformatons. In partcular, we conecture that all monomals n the varables of an gven cluster (the cluster monomals) belong to ths dual canoncal bass. A partcularl nce and well-understood example of a cluster algebra of an arbtrar rank n s the homogeneous coordnate rng C[Gr,n+3 ] of the Grassmannan of -dmensonal subspaces n C n+3. Ths rng s generated b the Plücker coordnates [], for 1 < n + 3, subect to the relatons [k][l]=[][kl]+[l][k], for all <<k<l. It s convenent to dentf the ndces 1,...,n+3 wth the vertces of a convex (n + 3)-gon, and the Plücker coordnates wth ts sdes and dagonals. We vew the sdes [1], [3],...,[n +,n+3], [1,n+3]asscalars, and the dagonals as cluster varables. The clusters are the maxmal famles of parwse noncrossng dagonals; thus, the are n a natural becton wth the trangulatons of ths polgon. It s known that the cluster monomals form a lnear bass n C[Gr,n+3 ]. To be more specfc, we note that ths rng s naturall dentfed wth the rng of polnomal SL -nvarants of an (n+3)-tuple of ponts n C. Under ths somorphsm, the bass of cluster monomals corresponds to the bass consdered n [11, 18]. (We are grateful to Bernd Sturmfels for brngng these references to our attenton.) An essental feature of the exchange relatons (1.1) s that the rght-hand sde does not nvolve subtracton. Recursvel applng these relatons, one can represent an cluster varable as a subtracton-free ratonal expresson n the varables of an gven cluster. Ths postvt propert s consstent wth a remarkable connecton between canoncal bases and the theor of total postvt, dscovered b G. Lusztg [14, 15]. Generalzng the classcal concept of totall postve matrces, he defned totall postve elements n an reductve group G, and proved that all elements of the dual canoncal bass n C[G] take postve values at them.

3 CLUSTER ALGEBRAS I 499 It was realzed n [15, 5] that the natural geometrc framework for total postvt s gven b double Bruhat cells, the ntersectons of cells of the Bruhat decompostons wth respect to two opposte Borel subgroups. Dfferent aspects of total postvt n double Bruhat cells were explored b the authors of the present paperandthercollaboratorsn[1,3,4,5,6,7,1,17,0]. Thebnomalexchange relatons of the form (1.1) plaed a crucal role n these studes. It was the desre to explan the ubqut of these relatons and to place them n a proper context that led us to the concept of cluster algebras. The crucal step n ths drecton was made n [0], where a faml of clusters and exchange relatons was explctl constructed n the coordnate rng of an arbtrar double Bruhat cell. However, ths faml was not complete: n general, some clusters were mssng, and not an member of a cluster could be exchanged from t. Thus, we started lookng for a natural wa to propagate exchange relatons from one cluster to another. The concept of cluster algebras s the result of ths nvestgaton. We conecture that the coordnate rng of an double Bruhat cell s a cluster algebra. Ths artcle, n whch we develop the foundatons of the theor, s conceved as the frst n a forthcomng seres. We attempt to make the exposton elementar and self-contaned; n partcular, no knowledge of semsmple groups, quantum groups or total postvt s assumed on the part of the reader. One of the man structural features of cluster algebras establshed n the present paper s the followng Laurent phenomenon: an cluster varable x vewed as a ratonal functon n the varables of an gven cluster s n fact a Laurent polnomal. Ths propert s qute surprsng: n most cases, the numerators of these Laurent polnomals contan a huge number of monomals, and the numerator for x moves nto the denomnator when we compute the cluster varable x obtaned from x b an exchange (1.1). The magc of the Laurent phenomenon s that, at ever stage of ths recursve process, a cancellaton wll nevtabl occur, leavng a sngle monomal n the denomnator. In vew of the postvt propert dscussed above, t s natural to expect that all Laurent polnomals for cluster varables wll have postve coeffcents. Ths seems to be a rather deep propert; our present methods do not provde a proof of t. On the brght sde, t s possble to establsh the Laurent phenomenon n man dfferent stuatons spreadng beond the cluster algebra framework. One such extenson s gven n Theorem 3.. B a modfcaton of the method developed here, a large number of addtonal nterestng nstances of the Laurent phenomenon are establshed n a separate paper [8]. The paper s organzed as follows. Secton contans an axomatc defnton, frst examples and the frst structural propertes of cluster algebras. One of the techncal dffcultes n settng up the foundatons nvolves the concept of an exchange graph whose vertces correspond to clusters, and the edges to exchanges among them. It s convenent to begn b takng the n-regular tree T n as our underlng graph. Ths tree can be vewed as a unversal cover for the actual exchange graph, whose appearance s postponed untl Secton 7. The Laurent phenomenon s establshed n Secton 3. In Sectons 4 and 5, we scrutnze the man defnton, obtan useful reformulatons, and ntroduce some mportant classes of cluster algebras. Secton 6 contans a detaled analss of cluster algebras of rank. Ths analss exhbts deep and somewhat msterous connectons between cluster algebras and Kac-Mood algebras. Ths s ust the tp of an ceberg: these connectons wll be

4 500 SERGEY FOMIN AND ANDREI ZELEVINSKY further explored (for cluster algebras of an arbtrar rank) n the sequel to ths paper. The man result of ths sequel s a complete classfcaton of cluster algebras of fnte tpe,.e., those wth fntel man dstnct clusters; cf. Example 7.6. Ths classfcaton turns out to be et another nstance of the famous Cartan-Kllng classfcaton.. Man defntons Let I be a fnte set of sze n; the standard choce wll be I =[n] ={1,,...,n}. Let T n denote the n-regular tree, whose edges are labeled b the elements of I, so that the n edges emanatng from each vertex receve dfferent labels. B a common abuse of notaton, we wll sometmes denote b T n the set of the tree s vertces. We wll wrte t t f vertces t, t T n are oned b an edge labeled b. To each vertex t T n, we wll assocate a cluster of n generators ( varables ) x(t) =(x (t)) I. All these varables wll commute wth each other and satsf the followng exchange relatons, for ever edge t t n T n : (.1) x (t) =x (t ) for an ; (.) x (t) x (t )=M (t)(x(t)) + M (t )(x(t )). Here M (t) andm (t ) are two monomals n the n varables x, I; we thnk of these monomals as beng assocated wth the two ends of the edge t t. To be more precse, let P be an abelan group wthout torson, wrtten multplcatvel. We call P the coeffcent group; a prototpcal example s a free abelan group of fnte rank. Ever monomal M (t) n (.) wll have the form (.3) M (t) =p (t) I x b, for some coeffcent p (t) P and some nonnegatve nteger exponents b. The monomals M (t) must satsf certan condtons (axoms). To state them, we wll need a lttle preparaton. Let us wrte P Q to denote that a polnomal P dvdes a polnomal Q. Accordngl, x M (t) means that the monomal M (t) contans the varable x. For a ratonal functon F = F (x,,...), the notaton F x g(x,,... ) wll denote the result of substtutng g(x,,...)forx nto F. To llustrate, f F (x, ) =x, thenf x = x x. Defnton.1. An exchange pattern on T n wth coeffcents n P s a faml of monomals M =(M (t)) t Tn, I of the form (.3) satsfng the followng four axoms: (.4) If t T n,thenx M (t). (.5) If t 1 t and x M (t 1 ), then x M (t ). (.6) (.7) If t 1 Let t 1 t t 3,then x M (t 1 ) f and onl f x M (t ). t t 3 t 4.Then M (t 3 ) where M 0 =(M (t )+M (t 3 )) x=0. M (t 4 ) = M (t ), M (t 1 ) x M 0/x

5 CLUSTER ALGEBRAS I 501 We note that n the last axom, the substtuton x M0 x s effectvel monomal, snce n the event that nether M (t )norm (t 3 )contanx, condton (.6) requres that both M (t )andm (t 1 ) do not depend on x, thus makng the whole substtuton rrelevant. One easl checks that axom (.7) s nvarant under the flp t 1 t 4, t t 3, so no restrctons are added f we appl t backwards. The axoms also mpl at once that settng (.8) M (t) =M (t ) for ever edge t t, we obtan another exchange pattern M ; ths gves a natural nvoluton M M on the set of all exchange patterns. Remark.. Informall speakng, axom (.7) descrbes the propagaton of an exchange pattern along the edges of T n. More precsel, let us fx the n exchange monomals for all edges emanatng from a gven vertex t. Ths choce unquel determnes the rato M (t )/M (t ) for an vertex t adacent to t and an edge t t (to see ths, take t = t and t 3 = t n (.7), and allow to var). In vew of (.5), ths rato n turn unquel determnes the exponents of all varables x k n both monomals M (t )andm (t ). There remans, however, one degree of freedom n determnng the coeffcents p (t )andp (t ) because onl ther rato s prescrbed b (.7). In Secton 5 we shall ntroduce an mportant class of normalzed exchange patterns for whch ths degree of freedom dsappears, and so the whole pattern s unquel determned b the n monomals assocated wth edges emanatng from a gven vertex. k Let ZP denote the group rng of P wth nteger coeffcents. For an edge t t, we refer to the bnomal P = M k (t) +M k (t ) ZP[x : I] astheexchange polnomal. We wll wrte t P t k or t P t to ndcate ths fact. Note that, n vew of the axom (.4), the rght-hand sde of the exchange relaton (.) can be wrtten as P (x(t)), whch s the same as P (x(t )). Let M be an exchange pattern on T n wth coeffcents n P. Note that snce P s torson-free, the rng ZP has no zero dvsors. For ever vertex t T n,let F(t) denote the feld of ratonal functons n the cluster varables x (t), I, wth coeffcents n ZP. For ever edge t P t, we defne a ZP-lnear feld somorphsm R tt : F(t ) F(t) b R tt (x (t )) = x (t) for k; (.9) R tt (x k (t )) = P (x(t)). x k (t) Note that propert (.4) ensures that R t t = R 1 tt.thetranston maps R tt enable us to dentf all the felds F(t) wth each other. We can then vew them as a sngle feld F that contans all the elements x (t), for all t T n and I. InsdeF, these elements satsf the exchange relatons (.1) (.). Defnton.3. Let A be a subrng wth unt n ZP contanng all coeffcents p (t) for I and t T n.thecluster algebra A = A A (M) ofrankn over A assocated

6 50 SERGEY FOMIN AND ANDREI ZELEVINSKY wth an exchange pattern M s the A-subalgebra wth unt n F generated b the unon of all clusters x(t), for t T n. The smallest possble ground rng A s the subrng of ZP generated b all the coeffcents p (t); the largest one s ZP tself. An ntermedate choce of A appears n Proposton.6 below. Snce A s a subrng of a feld F, t s a commutatve rng wth no zero dvsors. We also note that f M s obtaned from M b the nvoluton (.8), then the cluster algebra A A (M ) s naturall dentfed wth A A (M). 1 Example.4. Let n =1. ThetreeT 1 has onl one edge t t. The correspondng cluster algebra A has two generators x = x 1 (t) andx = x 1 (t ) satsfng the exchange relaton xx = p + p, where p and p are arbtrar elements of the coeffcent group P. In the unversal settng, we take P to be the free abelan group generated b p and p. Then the two natural choces for the ground rng A are the polnomal rng Z[p, p ], and the Laurent polnomal rng ZP = Z[p ±1,p ±1 ]. All other realzatons of A can be vewed as specalzatons of the unversal one. Despte the seemng trvalt of ths example, t covers several mportant algebras: the coordnate rng of each of the varetes SL, Gr,4 and SL 3 /B (cf. Secton 1) s a cluster algebra of rank 1, for an approprate choce of P, p, p and A. Example.5. Consder the case n =. ThetreeT s shown below: (.10) t 0 t 1 t t 3 t 4. Let us denote the cluster varables as follows: 1 = x 1 (t 0 )=x 1 (t 1 ), = x (t 1 )=x (t ), 3 = x 1 (t )=x 1 (t 3 ),... (the above equaltes among the cluster varables follow from (.1)). Then the clusters look lke 1, 0 1 t 0, 1 t 1 1 3, t 4, 3 t 3 1 5, 4 t 4. We clam that the exchange relatons (.) can be wrtten n the followng form: (.11) 0 = q 1 1 b + r 1, 1 3 = q c + r, 4 = q 3 3 b + r 3, 3 5 = q 4 4 c + r 4,..., where the ntegers b and c are ether both postve or both equal to 0, and the coeffcents q m and r m are elements of P satsfng the relatons (.1) q 0 q r1 c = r 0r, q 1 q 3 r b = r 1r 3, q q 4 r3 c = r r 4, q 3 q 5 r4 b = r 3 r 5,... Furthermore, an such choce of parameters b, c, (q m ), (r m ) results n a well-defned cluster algebra of rank. To prove ths, we notce that, n vew of (.4) (.5), both monomals M (t 0 )and M (t 1 ) do not contan the varable x, and at most one of them contans x 1.Ifx 1 enters nether M (t 0 )norm (t 1 ), then these two are smpl elements of P. But

7 CLUSTER ALGEBRAS I 503 then (.6) forces all monomals M (t m )tobeelementsofp, whle (.7) mples that t s possble to gve the names q m and r m to the two monomals correspondng to each edge t m t m+1 so that (.11) (.1) hold wth b = c =0. Next, consder the case when precsel one of the monomals M (t 0 )andm (t 1 ) contans x 1. Applng f necessar the nvoluton (.8) to our exchange pattern, we ma assume that M (t 0 )=q 1 x b 1 and M (t 1 )=r 1 for some postve nteger b and some q 1,r 1 P. Thus, the exchange relaton assocated to the edge t 0 t 1 takes the form 0 = q 1 1 b + r 1. B (.6), we have M 1 (t 1 )=q x c and M 1 (t )=r for some postve nteger c and some q,r P. Then the exchange relaton for the 1 edge t 1 t takes the form 1 3 = q c + r. At ths pont, we nvoke (.7): M (t ) M (t 3 ) = M (t 1 ) = r 1 M (t 0 ) x1 r /x 1 q 1 x b = r 1x b 1 1 x1 r /x 1 q 1 r b. B (.5), we have M (t )=q 3 x b 1 and M (t 3 )=r 3 for some q 3,r 3 P satsfng q 1 q 3 r b = r 1r 3. Contnung n the same wa, we obtan all relatons (.11) (.1). For fxed b and c, the unversal coeffcent group P s the multplcatve abelan group generated b the elements q m and r m for all m Z subect to the defnng relatons (.1). It s eas to see that ths s a free abelan group of nfnte rank. As a set of ts free generators, one can choose an subset of {q m,r m : m Z} that contans four generators q 0,r 0,q 1,r 1 and precsel one generator from each par {q m,r m } for m 0, 1. A nce specalzaton of ths setup s provded b the homogeneous coordnate rng of the Grassmannan Gr,5. Recall (cf. Secton 1) that ths rng s generated b the Plücker coordnates [k, l], where k and l are dstnct elements of the cclc group Z/5Z. We shall wrte m = m mod 5 Z/5Z for m Z, and adopt the conventon [k, l] =[l, k]; see Fgure 1. 3 q 5 q q 3 5 q 1 1 q 4 5 Fgure 1. The Grassmannan Gr,5 The deal of relatons among the Plücker coordnates s generated b the relatons [m, m +][m +1, m +3]=[m, m +1][m +, m +3]+[m, m +3][m +1, m +] for m Z. A drect check shows that these relatons are a specalzaton of the relatons (.11), f we set b = c =1, m =[m 1, m +1],q m =[m, m +],

8 504 SERGEY FOMIN AND ANDREI ZELEVINSKY and r m =[m, m 1][m +1, m +]=q m q m+ for all m Z. The coeffcent group P s the multplcatve free abelan group wth 5 generators q m. It s also mmedate that the elements q m and r m defned n ths wa satsf the relatons (.1). We conclude ths secton b ntroducng two mportant operatons on exchange patterns: restrcton and drect product. Let us start wth restrcton. Let M be an exchange pattern of rank n wth an ndex set I and coeffcent group P. Let J be a subset of sze m n I. Let us remove from T n all edges labeled b ndces n I J, and choose an connected component T of the resultng graph. Ths component s naturall dentfed wth T m. Let M denote the restrcton of M to T,.e., the collecton of monomals M (t) for all J and t T. ThenM s an exchange pattern on T whose coeffcent group P s the drect product of P wth the multplcatve free abelan group wth generators x, I J. We shall sa that M s obtaned from M b restrcton from I to J. NotethatM depends on the choce of a connected component T, so there can be several dfferent patterns obtaned from M b restrcton from I to J. (We thank the anonmous referee for pontng ths out.) Proposton.6. Let A = A A (M) be a cluster algebra of rank n assocated wth an exchange pattern M, andletm be obtaned from M b restrcton from I to J usng a connected component T. The A-subalgebra of A generated b t T x(t) s naturall dentfed wth the cluster algebra A A (M ),wherea s the polnomal rng A[x : I J]. Proof. If I J, then (.1) mples that x (t) stas constant as t vares over T. Therefore, we can dentf ths varable wth the correspondng generator x of the coeffcent group P, and the statement follows. Let us now consder two exchange patterns M 1 and M of ranks n 1 and n, respectvel, wth ndex sets I 1 and I, and coeffcent groups P 1 and P. We wll construct the exchange pattern M = M 1 M (the drect product of M 1 and M )ofrankn = n 1 + n, wth the ndex set I = I 1 I, and coeffcent group P = P 1 P. Consder the tree T n whose edges are colored b I, and, for ν {1, }, letπ ν : T n T nν be a map wth the followng propert: f t t n T n and I ν (resp., I I ν = I 3 ν ), then π ν (t) π ν (t )nt nν (resp., π ν (t) = π ν (t )). Clearl, such a map π ν exsts and s essentall unque: t s determned b specfng the mage of an vertex of T n. We now ntroduce the exchange pattern M on T n b settng, for ever t T n and I ν I, the monomal M (t) tobeequaltom (π ν (t)), the latter monomal comng from the exchange pattern M ν. The axoms (.4) (.7) for M are checked drectl. Proposton.7. Let A 1 = A A1 (M 1 ) and A = A A (M ) be cluster algebras. Let M = M 1 M and A = A 1 A. Then the cluster algebra A A (M) s canoncall somorphc to the tensor product of algebras A 1 A (all tensor products are taken over Z). Proof. Let us dentf each cluster varable x (t), for t T n and I 1 I (resp., I I), wth x (π 1 (t)) 1(resp.,1 x (π (t))). Under ths dentfcaton, the exchange relatons for the exchange pattern M become dentcal to the exchange relatons for M 1 and M.

9 CLUSTER ALGEBRAS I The Laurent phenomenon In ths secton we prove the followng mportant propert of cluster algebras. Theorem 3.1. In a cluster algebra, an cluster varable s expressed n terms of an gven cluster as a Laurent polnomal wth coeffcents n the group rng ZP. We conecture that each of the coeffcents n these Laurent polnomals s actuall a nonnegatve nteger lnear combnaton of elements n P. We wll obtan Theorem 3.1 as a corollar of a more general result, whch apples to more general underlng graphs and more general (not necessarl bnomal) exchange polnomals. Snce Theorem 3.1 s trval for n = 1, we shall assume that n. For ever m 1, let T n,m be a tree of the form shown n Fgure. The tree T n,m has m vertces of degree n n ts spne and m(n ) + vertces of degree 1. We label ever edge of the tree b an element of an n-element ndex set I, so that the n edges ncdent to each vertex on the spne receve dfferent labels. (The reader ma wsh to thnk of the tree T n,m as beng part of the n-regular tree T n of the cluster-algebra setup.) t tal t t head base Fgure. The caterpllar tree T n,m,forn =4,m =8 We fx two vertces t head and t tal of T n,m that do not belong to the spne and are connected to ts opposte ends. Ths gves rse to the orentaton on the spne: awa from t tal and towards t head (see Fgure ). As before, let P be an abelan group wthout torson, wrtten multplcatvel. Let Z 0 P denote the addtve semgroup generated b P n the nteger group rng ZP. Assume that a nonzero polnomal P n the varables x, I, wth coeffcents n Z 0 P, s assocated wth ever edge t t of T n,m. We call P an exchange polnomal, and wrte t P t to descrbe ths stuaton. Suppose that the exchange polnomals assocated wth the edges of T n,m satsf the followng condtons: (3.1) An exchange polnomal assocated wth an edge labeled b does not depend on x, and s not dvsble b an x, I. (3.) If t 0 P t 1 Q t R t 3,thenR = C (P x Q 0/x ), where Q 0 = Q x=0, and C s a Laurent polnomal wth coeffcents n Z 0 P. (Note the orentaton of the edge t 1 t n (3.).) For ever vertex t on the spne, let P(t) denote the faml of n exchange polnomals assocated wth the edges emanatng from t. Also, let C denote the collecton of all Laurent polnomals C that appear n condton (3.), for all possble choces

10 506 SERGEY FOMIN AND ANDREI ZELEVINSKY of t 0,t 1,t,t 3,andletA ZP denote the subrng wth unt generated b all coeffcents of the Laurent polnomals from P(t base ) C,wheret base s the vertex on the spne connected wth t tal. Asbefore,weassocateacluster x(t) ={x (t) : I} to each vertex t T n,m, and consder the feld F(t) of ratonal functons n these varables wth coeffcents n ZP. All these felds are dentfed wth each other b the transton somorphsms R tt : F(t ) F(t) defned as n (.9). We then vew the felds F(t) asasngle feld F that contans all the elements x (t), for t T n,m and I. These elements satsf the exchange relatons (.1) and the followng verson of (.): for an edge t P t n T n,m. x (t) x (t )=P (x(t)), Theorem 3.. If condtons (3.1) (3.) are satsfed, then each element of the cluster x(t head ) s a Laurent polnomal n the cluster x(t tal ), wth coeffcents n the rng A. We note that Theorem 3. s ndeed a generalzaton of Theorem 3.1, for the followng reasons: T n,m s naturall embedded nto T n ; condtons (3.1) (3.) are less restrctve than (.4) (.7); the clam beng made n Theorem 3. about coeffcents of the Laurent polnomals s stronger than that of Theorem 3.1, snce A ZP. Proof. We start wth some preparatons. We shall wrte an Laurent polnomal L n the varables x = {x : I} n the form L(x) = α S u α (L)x α, where all coeffcents u α (L) are nonzero, S s a fnte subset of the lattce Z I (.e., the lattce of rank n wth coordnates labeled b I), and x α s the usual shorthand for xα.thesetss called the support of L and denoted b S = S(L). Notce that once we fx the collecton C, condton (3.) can be used as a recursve rule for computng P(t )fromp(t), for an edge t t on the spne. It follows that the whole pattern of exchange polnomals s determned b the famles of polnomals P(t base )andc. Moreover, snce these polnomals have coeffcents n Z 0 P, and the expresson for R n (3.) does not nvolve subtracton, t follows that the support of an exchange polnomal s unquel determned b the supports of the polnomals from P(t base )andc. Note that condton (3.1) can be formulated as a set of restrctons on these supports. In partcular, t requres that n the stuaton of (3.), the Laurent polnomal C does not depend on x and s a polnomal n x ; n other words, ever α S(C) should have α =0andα 0. We now fx a faml of supports S(L), for all L P(t base ) C, and assume that ths faml comples wth (3.1). As s common n algebra, we shall vew the coeffcents u α (L), for all L P(t base ) C and α S(L), as ndetermnates. Then all the coeffcents n all exchange polnomals become canoncal (.e., ndependent of the choce of P) polnomals n these ndetermnates, wth postve nteger coeffcents.

11 CLUSTER ALGEBRAS I 507 The above dscusson shows that t suffces to prove our theorem n the followng unversal coeffcents setup: let P be the free abelan group (wrtten multplcatvel) wth generators u α (L), for all L P(t base ) C and α S(L). Under ths assumpton, A s smpl the nteger polnomal rng n the ndetermnates u α (L). Recall that we can vew all cluster varables x (t) aselementsofthefeldf(t tal ) of ratonal functons n the cluster x(t tal ) wth coeffcents n ZP. For t T n,m, let L(t) denote the rng of Laurent polnomals n the cluster x(t), wth coeffcents n A. WeveweachL(t) as a subrng of the ambent feld F(t tal ). In ths termnolog, our goal s to show that the cluster x(t head )scontaned n L(t tal ). We proceed b nducton on m, the sze of the spne. The clam s trval for m = 1, so let us assume that m, and furthermore assume that our statement s true for all caterpllars wth smaller spne. Let us abbrevate t 0 = t tal and t 1 = t base, and suppose that the path from t tal to t head starts wth the followng two edges: t 0 P t 1 Q t. Let t 3 T n,m be the vertex such that t R t 3. The followng lemma plas a crucal role n our proof. Lemma 3.3. The clusters x(t 1 ), x(t ),andx(t 3 ) are contaned n L(t 0 ).Furthermore, gcd(x (t 3 ),x (t 1 )) = gcd(x (t ),x (t 1 )) = 1 (as elements of L(t 0 )). Note that L 0 = L(t 0 ) s a unque factorzaton doman, so an two elements x, L 0 have a well-defned greatest common dvsor gcd(x, ),whchsanelement of L 0 defned up to a multple from the group L 0 of unts (that s, nvertble elements) of L 0. In our unversal stuaton, L 0 conssts of Laurent monomals n the cluster x(t 0 ) wth coeffcents ±1. Proof. The onl element from the clusters x(t 1 ), x(t ), and x(t 3 ) whose ncluson n L 0 s not mmedatel obvous s x (t 3 ). To smplf the notaton, let us denote x = x (t 0 ), = x (t 0 )=x (t 1 ), z = x (t 1 )=x (t ), u = x (t )=x (t 3 ), and v = x (t 3 ), so that these varables appear n the clusters at t 0,...,t 3,asshown below: x, t 0 P,z t 1 Q z,u t R u,v t 3. Note that the varables x k,fork / {, }, do not change as we move among the four clusters under consderaton. The lemma s then restated as sang that (3.3) (3.4) (3.5) v L 0 ; gcd(z,u) = 1 (as elements of L 0 ); gcd(z,v) = 1 (as elements of L 0 ). Another notatonal conventon wll be based on the fact that each of the polnomals P, Q, R has a dstngushed varable on whch t depends, namel x for P and R, and x for Q. (In vew of (3.1), P and R do not depend on x, whle Q does not depend on x.) Wth ths n mnd, we wll routnel wrte P, Q, andr as polnomals n one (dstngushed) varable. In the same sprt, the notaton Q, R, etc., wll refer to the partal dervatve wth respect to the dstngushed varable. We wll prove the statements (3.3), (3.4), and (3.5) one b one, n ths order.

12 508 SERGEY FOMIN AND ANDREI ZELEVINSKY B (3.), the polnomal R s gven b ( ) (3.6) R(u) =C(u)P Q(0) u, where C s an honest polnomal n u and a Laurent polnomal n the mute varables x k, k/ {, }. (Recall that C does not depend on x.) We then have: Snce and z = P () x ; u = Q(z) v = R(u) z R Q = R = ( ) P () x ( Q(z) z ( ) Q(0) C = z ; ) R = (3.3) follows. We next prove (3.4). We have u = Q(z) ( ) Q(z) R ( ) ( ) R Q(z) R Q(0) L 0 z z ( ) Q(0) P () = C z Q(0) ( ) Q(0) R + ( ) Q(0) x L 0, mod z. ( ) Q(0). z Snce x and arenvertblenl 0, we conclude that gcd(z,u) =gcd(p (),Q(0)). Now the trouble that we took n passng to unversal coeffcents fnall pas off: snce P () andq(0) are nonzero polnomals n the cluster x(t 0 ) whose coeffcents are dstnct generators of the polnomal rng A, t follows that gcd(p (),Q(0)) = 1, provng (3.4). It remans to prove (3.5). Let ( ) f(z) =R Q(z). Then f(z) f(0) ( ) v = + C Q(0) z x. Our goal s to show that gcd(z,v) = 1; to ths end, we are gong to compute v mod z as explctl as possble. We have, modz, f(z) f(0) ( ) f (0) = R Q(0) z Q (0). Hence ( ) v R Q(0) Q (0) + C ( ) Q(0) x mod z. Note that the rght-hand sde s a lnear polnomal n x, whose coeffcents are Laurent polnomals n the rest of the( varables ) of the) cluster x(t 0 ). Thus our clam wll follow f we show that gcd C,P() = 1. Ths, agan, s a ( Q(0)

13 CLUSTER ALGEBRAS I 509 consequence of our unversal coeffcents setup snce the coeffcents of C, P and Q are dstnct generators of the polnomal rng A. We can now complete the proof of Theorem 3.. We need to show that an varable x = x k (t head ) belongs to L(t 0 ). Snce both t 1 and t 3 are closer to t head than t 0, we can use the nductve assumpton to conclude that x belongs to both L(t 1 )andl(t 3 ). Snce x L(t 1 ), t follows from (.1) that x can be wrtten as x = f/x (t 1 ) a for some f L(t 0 )anda Z 0. On the other hand, snce x L(t 3 ), t follows from (.1) and from the ncluson x (t 3 ) L(t 0 ) guaranteed b Lemma 3.3 that x has the form x = g/x (t ) b x (t 3 ) c for some g L(t 0 )andsomeb, c Z 0. The ncluson x L(t 0 ) now follows from the fact that, b the last statement n Lemma 3.3, the denomnators n the two obtaned expressons for x are coprme n L(t 0 ). Several examples that can be vewed as applcatons of Theorem 3. are gven n [8]. 4. Exchange relatons: The exponents Let M =(M (t)) : t T n, I) be an exchange pattern (see Defnton.1). In ths secton we wll gnore the coeffcents n the monomals M (t) andtakea closer look at the dnamcs of ther exponents. (An alternatve pont of vew that the reader ma fnd helpful s to assume that all exchange patterns consdered n ths secton wll have all ther coeffcents p (t) equal to 1.) For ever edge t t n T n, let us wrte the rato M (t)/m (t ) of the correspondng monomals as (4.1) M (t) M (t ) = p (t) p (t ) b (t) x, where b (t) Z (cf. (.3)); we note that ratos of ths knd have alread appeared n (.7). Let us denote b B(t) =(b (t)) the n n nteger matrx whose entres are the exponents n (4.1). In vew of (.5), the exponents n M (t) andm (t )are recovered from B(t): (4.) M (t) = p (t) M (t ) = p (t ) : b (t)>0 : b (t)<0 b (t) x, b (t) x. Thus, the faml of matrces (B(t)) t Tn encodes all the exponents n all monomals of an exchange pattern. We shall descrbe the condtons on the faml of matrces (B(t)) mposed b the axoms of an exchange pattern. To do ths, we need some preparaton. Defnton 4.1. A square nteger matrx B =(b ) s called sgn-skew-smmetrc f, for an and, etherb = b =0,orelseb and b are of opposte sgn; n partcular, b =0forall. Defnton 4.. Let B =(b )andb =(b ) be square nteger matrces of the same sze. We sa that B s obtaned from B b the matrx mutaton n drecton k

14 510 SERGEY FOMIN AND ANDREI ZELEVINSKY and wrte B = µ k (B) f b f = k or = k; (4.3) b = b + b k b k + b k b k otherwse. An mmedate check shows that µ k s nvolutve,.e., ts square s the dentt transformaton. Proposton 4.3. Afamlofn n nteger matrces (B(t)) t Tn corresponds to an exchange pattern f and onl f the followng condtons hold: (1) B(t) s sgn-skew-smmetrc for an t T n. k () If t t,thenb(t )=µ k (B(t)). Proof. We start wth the onl f part,.e., we assume that the matrces B(t) are determned b an exchange pattern va (4.1) and check the condtons (1) (). The condton b (t) = 0 follows from (.4). The remanng part of (1) (dealng wth ), follows at once from (.6). Turnng to part (), the equalt b k = b k s mmedate from the defnton (4.1). Now suppose that k. In ths case, we k appl the axom (.7) to the edge t t taken together wth the two adacent edges emanatng from t and t and labeled b. Takng (4.) nto account, we obtan: (4.4) x b = x b, xk M/x k where M = :b k b k <0 x b k. Comparng the exponents of x k on both sdes of (4.4) elds b k = b k. Fnall, f k, then comparng the exponents of x on both sdes of (4.4) gves b f b k b k 0; b = b + b k b k otherwse. To complete the proof of (), t remans to notce that, n vew of the alread proven part (1), the condton b k b k 0sequvalenttob k b k 0, whch makes the last formula equvalent to (4.3). To prove the f part, t suffces to show that f the matrces B(t) satsf (1) (), then the monomals M (t) gven b the frst equalt n (4.) (wth p (t) =1) satsf the axoms of an exchange pattern. Ths s done b a drect check. Snce all matrx mutatons are nvolutve, an choce of an ntal vertex t 0 T n and an arbtrar n n nteger matrx B gves rse to a unque faml of nteger matrces B(t) satsfng condton () n Proposton 4.3 and such that B(t 0 )=B. Thus, the exponents n all monomals M (t) are unquel determned b a sngle matrx B = B(t 0 ). B Proposton 4.3, n order to determne an exchange pattern, B must be such that all matrces obtaned from t b a sequence of matrx mutatons are sgn-skew-smmetrc. Verfng that a gven matrx B has ths propert seems to be qute nontrval n general. Fortunatel, there s another restrcton on B that s much easer to check, whch mples the desred propert, and stll leaves us wth a large class of matrces suffcent for most applcatons.

15 CLUSTER ALGEBRAS I 511 Defnton 4.4. A square nteger matrx B =(b ) s called skew-smmetrzable f there exsts a dagonal skew-smmetrzng matrx D wth postve nteger dagonal entres d such that DB s skew-smmetrc,.e., d b = d b for all and. Proposton 4.5. For ever choce of a vertex t 0 T n and a skew-smmetrzable matrx B, there exsts a unque faml of matrces (B(t)) t Tn assocated wth an exchange pattern on T n and such that B(t 0 )=B. Furthermore, all the matrces B(t) are skew-smmetrzable, sharng the same skew-smmetrzng matrx. Proof. The proof follows at once from the followng two observatons: 1. Ever skew-smmetrzable matrx B s sgn-skew-smmetrc.. If B s skew-smmetrzable and B = µ k (B), then B s also skew-smmetrzable, wth the same skew-smmetrzng matrx. We call an exchange pattern and the correspondng cluster algebra skewsmmetrzable f all the matrces B(t) gven b (4.1) (equvalentl, one of them) are skew-smmetrzable. In partcular, all cluster algebras of rank n areskewsmmetrzable: for n =1wehaveB(t) (0), whle for n =, the calculatons n Example.5 show that one can take [ ] B(t m )=( 1) m 0 b (4.5) c 0 for all m Z, n the notaton of (.10) (.11). Remark 4.6. Skew-smmetrzable matrces are closel related to smmetrzable (generalzed) Cartan matrces appearng n the theor of Kac-Mood algebras. More generall, to ever sgn-skew-smmetrc matrx B =(b ) we can assocate a generalzed Cartan matrx A = A(B) =(a ) of the same sze b settng f = ; (4.6) a = b f. There seem to be deep connectons between the cluster algebra correspondng to B and the Kac-Mood algebra assocated wth A(B). We exhbt such a connecton for the rank case n Secton 6 below. Ths s however ust the tp of an ceberg: a much more detaled analss wll be presented n the sequel to ths paper. In order to show that non-skew-smmetrzable exchange patterns do exst, we conclude ths secton b exhbtng a 3-parameter faml of such patterns of rank 3. Proposton 4.7. Let α, β, andγ be three postve ntegers such that αβγ 3. There exsts a unque faml of matrces (B(t)) t T3 assocated wth a non-skewsmmetrzable exchange pattern on T 3 and such that the matrx B(t 0 ) at a gven vertex t 0 T 3 s equal to 0 α αβ (4.7) B(α, β, γ) = βγ 0 β. γ αγ 0 Proof. Frst of all, the matrx B(α, β, γ) s sgn-skew-smmetrc but not skewsmmetrzable. Indeed, an skew-smmetrzable matrx B = (b ) satsfes the equaton b 1 b 3 b 31 = b 1 b 3 b 13. However, ths equaton for B(α, β, γ) holds onl when αβγ s equal to 0 or.

16 51 SERGEY FOMIN AND ANDREI ZELEVINSKY For the purpose of ths proof onl, we refer to a 3 3matrxB as cclcal f ts entres follow one of the two sgn patterns , In partcular, B(α, β, γ) s cclcal; to prove the proposton, t suffces to show that an matrx obtaned from t b a sequence of matrx mutatons s also cclcal. The set of cclcal matrces s not stable under matrx mutatons. Let us defne some subsets of cclcal matrces that behave ncel wth respect to matrx mutatons. For a 3 3matrxB, wedenote c 1 = b 3 b 3,c = b 13 b 31, c 3 = b 1 b 1,r= b 1 b 3 b 31. For {1,, 3}, wesathatb s -based f we have r>c r/ c 6 for an {1,, 3}\{}. ForB(α, β, γ), we have r/ =c 1 = c = c 3 =αβγ 6, so t s -based for ever {1,, 3}. Our proposton becomes an mmedate consequence of the followng lemma. Lemma 4.8. Suppose B s cclcal and -based, and let. Then µ (B) s cclcal and -based. Proof. Wthout loss of generalt we can assume that =1and =. Denote B = µ (B), and let us wrte r = b 1 b 3 b 31, c 1 = b 3 b 3,etc. B(4.3),wehave b 1 = b 1,b 1 = b 1,b 3 = b 3,b 3 = b 3 ; therefore, c 1 = c 1 and c 3 = c 3.We also have b 13 = b 13 + b 1 b 3 + b 1 b 3, b 31 = b 31 + b 3 b 1 + b 3 b 1 (4.8). Snce B s cclcal, the two summands on the rght-hand sde of each of the equaltes n (4.8) have opposte sgns. Note that b 1 b 3 = r b 31 c b 31 = b 13 and b 3 b 1 = c 1 b 1 r b 1 b 3 b 3 = c 3 b 31 3 b 31. It follows that b 13 (resp., b 31 ) has the opposte sgn to b 13 (resp., b 31 ). Thus, B s cclcal, and t onl remans to show that B s -based. To ths effect, we note that b 31 = b 3b 1 b 31,andso r = b 1 b 3 b 31 = b 1 b 3 ( b 3 b 1 b 31 )=c 1 c 3 r (r/) 6 r =r ; therefore, both c 1 = c 1 and c 3 = c 3 do not exceed r /. As for c, we have b 13 = b 1 b 3 b 13,andso c = b 13b 31 =( b 1 b 3 b 13 )( b 3 b 1 b 31 )=r (1 c r ); snce c /r 1/, we conclude that r >r (1 c r )=c r /. Ths completes the proof that B s -based. Lemma 4.8 and Proposton 4.7 are proved.

17 CLUSTER ALGEBRAS I Exchange relatons: The coeffcents In ths secton we fx a faml of matrces B(t) satsfng the condtons n Proposton 4.3, and dscuss possble choces of coeffcents p (t) that can appear n the correspondng exchange pattern. We start wth the followng smple characterzaton. Proposton 5.1. Assume that matrces (B(t)) t Tn satsf condtons (1) () n Proposton 4.3. A faml of elements p (t) of a coeffcent group P gves rse, va (4.), to an exchange pattern f and onl f the satsf the followng relatons, whenever t 1 t t 3 t 4 : (5.1) p (t 1 )p (t 3 )p (t 3 ) max(b(t3),0) = p (t )p (t 4 )p (t ) max(b(t),0). Note that b (t )= b (t 3 ) b (4.3), so at most one of the p (t )andp (t 3 ) actuall enters (5.1). Proof. The onl axom of an exchange pattern that nvolves the coeffcents s (.7), and the relaton (5.1) s precsel what t prescrbes. Frst of all, let us menton the trval soluton of (5.1) when all the coeffcents p (t) areequalto1. Movng n the opposte drecton, we ntroduce the unversal coeffcent group (wth respect to a fxed faml (B(t))) as the abelan group P generated b the elements p (t), for all I and t T n, whch has (5.1) as the sstem of defnng relatons. The torson-freeness of ths group s guaranteed b the followng proposton whose straghtforward proof s omtted. Proposton 5.. The unversal coeffcent group P s a free abelan group. More specfcall, let t 0 T n,andlets be a collecton of pars (, t) that contans both (, t 0 ) and (, t) for an edge t 0 t, and precsel one of the pars (, t) and (, t ) for each edge t t wth t and t dfferent from t 0.Then{p (t) :(, t) S} s a set of free generators for P. We see that, n contrast to (4.3), relatons (5.1) leave nfntel man degrees of freedom n determnng the coeffcents p (t) (cf. Remark.). The rest of ths secton s devoted to mportant classes of exchange patterns wthn whch all the coeffcents are completel determned b specfng n of them correspondng to the edges emanatng from a gven vertex. Suppose that, n addton to the multplcatve group structure, the coeffcent group P s suppled wth a bnar operaton that we call auxlar addton. Furthermore, suppose that ths operaton s commutatve, assocatve, and dstrbutve wth respect to multplcaton; thus (P,, ) sasemfeld. (B the wa, under these assumptons P s automatcall torson-free as a multplcatve group: ndeed, f p m =1forsomep P and m, then p = pm p m 1 p p m 1 p m 1 = pm 1 p m 1 p m 1 p m 1 =1.) Defnton 5.3. An exchange pattern and the correspondng cluster algebra are called normalzed f P s a semfeld and p (t) p (t ) = 1 for an edge t t.

18 514 SERGEY FOMIN AND ANDREI ZELEVINSKY Proposton 5.4. Fx a vertex t 0 T n and n elements q and r ( I) of a semfeld P such that q r =1for all. Then ever faml of matrces B(t) satsfng the condtons n Proposton 4.3 gves rse to a unque normalzed exchange pattern such that, for ever edge t 0 t, we have p (t 0 )=q and p (t) =r. Thus a normalzed exchange pattern s completel determned b the n monomals M (t 0 ) and M (t), for all edges t 0 t. Proof. In a normalzed exchange pattern, the coeffcents p (t) andp (t ) correspondng to an edge t t are determned b ther rato (5.) va (5.3) Clearl, we have (5.4) p (t) = u (t) = p (t) p (t ) u (t) 1 u (t), p (t 1 )= 1 u (t). u (t)u (t )=1 for an edge t t. We can also rewrte the relaton (5.1) as follows: (5.5) u (t )=u (t)u (t) max(b(t),0) (1 u (t)) b(t) for an edge t t and an. Ths form of the relatons for the normalzed coeffcents makes our proposton obvous. Remark 5.5. It s natural to ask whether the normalzaton condton mposes addtonal multplcatve relatons among the coeffcents p (t)thatarenotconsequences of (5.1). In other words: can a normalzed sstem of coeffcents generate the unversal coeffcent group? In the next secton, we present a complete answer to ths queston n the rank case (see Remark 6.5 below). One example of a semfeld s the multplcatve group R >0 of postve real numbers, beng ordnar addton. However the followng example s more mportant for our purposes. Example 5.6. Let P be a free abelan group, wrtten multplcatvel, wth a fnte set of generators p ( I ), and wth auxlar addton defned b (5.6) p a p b = p mn(a,b). Then P s a semfeld; specfcall, t s a product of I copes of the tropcal semfeld (see, e.g., [1]). We denote ths semfeld b Trop(p : I ). Note that f all exponents a and b n (5.6) are nonnegatve, then the monomal on the rghthand sde s the gcd of the two monomals on the left. Defnton 5.7. We sa that an exchange pattern s of geometrc tpe f t s normalzed, has the coeffcent semfeld P =Trop(p : I ), and each coeffcent p (t) s a monomal n the generators p wth all exponents nonnegatve.

19 CLUSTER ALGEBRAS I 515 For an exchange pattern of geometrc tpe, the normalzaton condton smpl means that, for ever edge t t, the two monomals p (t) andp (t )nthe generators p are coprme,.e., have no varable n common. In all our examples of cluster algebras of geometrc orgn, ncludng those dscussed n the ntroducton, the exchange patterns turn out to be of geometrc tpe. These patterns have the followng useful equvalent descrpton. Proposton 5.8. Let P =Trop(p : I ). A faml of coeffcents p (t) P gves rse to an exchange pattern of geometrc tpe f and onl f the are gven b (5.7) p (t) = p max(c(t),0) I for some faml of ntegers (c (t) :t T n, I, I)) satsfng the followng k propert: for ever edge t t n T n,thematrcesc(t) =(c (t)) = (c ) and C(t )=(c (t )) = (c ) are related b c f = k; (5.8) c = c + c k b k (t)+c k b k (t) otherwse. Proof. As n the proof of Proposton 5.4, for ever edge t t we consder the rato u (t) =p (t)/p (t ). We ntroduce the matrces C(t) b settng u (t) = p c(t). The expresson (5.7) then becomes a specalzaton of the frst equalt n (5.3), wth auxlar addton gven b (5.6). To derve (5.8) from (5.5), frst replace b and b k, respectvel, then specalze, then pck up the exponent of p. Comparng (5.8) wth (4.3), we see that t s natural to combne a par of matrces (B(t),C(t)) nto one rectangular nteger matrx B(t) =(b ) I I, I b settng b = c for I and I. Then the matrces B(t) fort T n are related to each other b the same matrx mutaton rule (4.3), now appled to an I I and I. We refer to B(t) astheprncpal part of B(t). Combnng Propostons 5.8 and 4.5, we obtan the followng corollar. Corollar 5.9. Let B be an nteger matrx wth skew-smmetrzable prncpal part. There s a unque exchange pattern M = M( B) of geometrc tpe such that B(t 0 )= B at a gven vertex t 0 T n. In the geometrc tpe case, there s a dstngushed choce of ground rng for the correspondng cluster algebra: take A to be the polnomal rng Z[p : I ]. In the stuaton of Corollar 5.9, we wll denote the correspondng cluster algebra A A (M) smpl b A( B). In the notaton of Secton, A( B) s the subrng of the ambent feld F generated b cluster varables x (t) for all I and t T n together wth the generators p ( I )ofp. The set I s allowed to be empt: ths smpl means that all the coeffcents p (t) n the correspondng exchange pattern are equal to 1. In ths case, we have B(t) =B(t) for all t.

20 516 SERGEY FOMIN AND ANDREI ZELEVINSKY We note that the class of exchange patterns of geometrc tpe (and the correspondng cluster algebras A( B)) s stable under the operatons of restrcton and drect product ntroduced n Secton. The restrcton from I toasubsetj amounts to removng from B the columns labeled b I [ J; the drect ] product operaton replaces two matrces B 1 and B B1 0 b the matrx. 0 B 6. The rank case In ths secton, we llustrate the above results and constructons b treatng n detal the specal case n =. We label vertces and edges of the tree T as n (.10). For m Z, t wll be convenent to denote b m the element of {1, } m congruent to m modulo. Thus, T conssts of vertces t m and edges t m t m+1 for all m Z. We use the notaton of Example.5, so the clusters are of the form x(t m )={ m, m+1 }, and the exchange relatons are gven b (.11), wth coeffcents q m and r m satsfng (.1). More specfcall, we have (6.1) x m (t m )= m, x m+1 (t m )= m+1, p m (t m )=q m+1,p m+1 (t m )=r m for m Z. Choosex(t 1 )={ 1, } as the ntal cluster. Accordng to Theorem 3.1, each cluster varable m can be expressed as a Laurent polnomal n 1 and wth coeffcents n ZP. Let us wrte ths Laurent polnomal as m = N m( 1, ) (6.), d1(m) 1 d(m) where N m ( 1, ) ZP[ 1, ] s a polnomal not dvsble b 1 or. We wll nvestgate n detal the denomnators of these Laurent polnomals. Recall that for n = the matrces B(t) are gven b (4.5). Thus, all these matrces have the same assocated Cartan matrx [ ] b (6.3) A = A(B(t)) = c (see (4.6)). We wll show that the denomnators n (6.) have a nce nterpretaton n terms of the root sstem assocated to A. Let us recall some basc facts about ths root sstem (cf., e.g., [10]). Let Q = Z be a lattce of rank wth a fxed bass {α 1,α } of smple roots. The Wel group W = W (A) s a group of lnear transformatons of Q generated b two smple reflectons s 1 and s whose acton n the bass of smple roots s gven b [ ] [ ] 1 b 1 0 (6.4) s 1 =, s 0 1 =. c 1 Snce both s 1 and s are nvolutons, each element of W s one of the followng: w 1 (m) =s 1 s s 1 s m, w (m) =s s 1 s s m+1 ; here both products are of length m 0. It s well known that W s fnte f and onl f bc 3; we shall refer to ths as the fnte case. TheCoxeter number h of W s the order of s 1 s n W ; t s gven b Table 1. In the fnte case, W s the dhedral group of order h, and ts elements can be lsted as follows: w 1 (0) = w (0) = e

21 CLUSTER ALGEBRAS I 517 (the dentt element), w 1 (h) =w (h) =w 0 (the longest element), and h dstnct elements w 1 (m),w (m) for0<m<h. In the nfnte case, all elements w 1 (m) andw (m) form>0aredstnct. Table 1. The Coxeter number bc h A vector α Q s a real root for A f t s W -conugate to a smple root. Let Φ denote the set of real roots for A. ItsknownthatΦ=Φ + ( Φ + ), where Φ + = {α = d 1 α 1 + d α Φ: d 1,d 0} s the set of postve real roots. In the fnte case, Φ + has cardnalt h, andwe have Φ + = {w 1 (m)α m+1 :0 m<h}. In the nfnte case, we have Φ + = {w 1 (m)α m+1, w (m)α m+ : m 0}, wth all the roots w 1 (m)α m+1 and w (m)α m+ dstnct. We wll represent the denomnators n (6.) as vectors n the root lattce Q b settng δ(m) =d 1 (m)α 1 + d (m)α for all m Z. Inpartcular,wehaveδ(1) = α 1 and δ() = α. Theorem 6.1. In the rank case (fnte or nfnte alke), cluster varables are unquel up to a multple from P determned b ther denomnators n the Laurent expansons wth respect to a gven cluster. The set of these denomnators s naturall dentfed wth { α 1, α } Φ +.Moreprecsel: () In the nfnte case, we have (6.5) δ(m +3)=w 1 (m)α m+1, δ( m) =w (m)α m+ (m 0). In partcular, all m for m Z have dfferent denomnators d1(m) 1 d(m). () In the fnte case, we have (6.6) δ(m +3)=w 1 (m)α m+1 (h>m 0) and δ(m + h +) = δ(m) for all m Z, so the denomnators d1(m) 1 d(m) are perodc wth the perod h +. Moreover, m+h+ / m P for m Z. Proof. Before gvng a unfed proof of Theorem 6.1, we notce that part () can be proved b a drect calculaton. We wll express all coeffcents n terms of r 1, r,andq m for m Z (cf. Proposton 5.). Wth the help of Maple we fnd that n each fnte case, the elements m for 3 m h + 4 are the followng Laurent polnomals n 1 and. Tpe A 1 A 1 : b = c =0. 3 = q + r, 4 = q 3(q 1 + r 1 ), 5 = q 4 1, 6 = r 1q 5. 1 r 1 r q 1 q 3