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Cluster Algebras Bernard Leclerca and Lauren K

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Cluster Algebras Bernard Leclerca and Lauren K SPECIAL FEATURE: INTRODUCTION Cluster algebras Bernard Leclerca and Lauren K. Williamsb,1 See Fig. 2 for a picture of the flip graph of aLaboratoire de Mathématiques Nicolas Oresme, Université de Caen, 14032 Caen, France; the hexagon. and bDepartment of Mathematics, University of California, Berkeley, CA 94720 Now we can associate a cluster variable (an element of F) to each of the diagonals of the n-gon by imposing a relation for every flip: What Is a Cluster Algebra? cluster x′ = (x − {x}) ∪ {x′}, with the new cluster variable x′ determined by an ex- Given a quadrilateral, with sides a, b, c,and Cluster algebras were conceived by Fomin d and diagonals e and f,westipulatethat and Zelevinsky (1) in the spring of 2000 as change relation of the form x x = x x + x x (Fig. 3). Using the fact a tool for studying dual canonical bases and e f a c b d + + − − that the flip graph is connected (i.e., we can total positivity in semisimple Lie groups. xx′ = y M + y M : get from the initial triangulation to any However, the theory of cluster algebras + − + other by a series of flips), it is clear that we has since taken on a life of its own, as Here y and y are coefficients, whereas M − can attach in this way an element x of F to connections and applications have been and M are monomials in the elements of i each diagonal i.Itisanexercisetoshowthat discovered in diverse areas of mathematics, x − {x}. this construction is well defined, that is, x including representation theory of quivers i Example: The Type A Cluster Algebra does not depend on the sequence of flips and finite dimensional algebras, cf., for used to pass from the initial triangulation example, refs. 2–15; Poisson geometry (16– Although we have not given the formal def- – inition of cluster algebra, we will nevertheless to any triangulation containing the diagonal i. 19); Teichmüller theory (20 24); string A – provide an example, which we hope will By definition, the cluster algebra n asso- theory (25 31); discrete dynamical systems F and integrability (6, 32–38); and combina- give the reader the flavor of the theory. The ciated with the n-gon is the subalgebra of – combinatorics of triangulations of an n-gon generated by all of the cluster variables as- torics (39 47). A Quite remarkably, cluster algebras pro- (a convex polygon with n vertices) will be sociated with its diagonals. A cluster of n is vide a unifying algebraic and combinatorial used to describe the example presented here. a subset of the set of cluster variables corre- framework for a wide variety of phenomena We will subsequently identify the resulting sponding to the diagonals of a triangulation A − in these and other settings. We refer the cluster algebra with the homogeneous co- of the n-gone. Thus, the rank of n is n 3. reader to the survey papers (36, 48–53) and ordinate ring of the Grassmannian Gr2,n of Note that our construction depends on a to the cluster algebras portal (www.math. 2-planes in an n-dimensional vector space. choice of initial triangulation. However, if lsa.umich.edu/~fomin/cluster.html) for var- Fig. 1 shows an example of a triangulation we choose two different triangulations, the = ious introductions to cluster algebras and T,withn 8. We have labeled the diagonals resulting cluster algebras will be isomorphic. ... their links with other subjects in mathe- of T by the numbers 1, 2, ,5andthesides Let Gr2,n be the Grassmann variety pa- ... matics (and physics). of the octagon by the numbers 6, 7, ,13. rametrizing 2-planes in an n-dimensional F = Q½ ; ...; ð ; ...; Þ In brief, a cluster algebra A of rank k is We now set x6 x13 x1 x5 . complex vector space. We now explain how F A a subring of an ambient field F of rational In other words, consists of rational func- the cluster algebra n is related to the ho- tions in the variables x , ..., x labeled by the C½ ; functions in k variables, say x1, ..., xk.Unlike 1 5 mogeneous coordinate ring Gr2 n of Gr2,n most commutative rings, a cluster algebra diagonals, with coefficients which are poly- in its Plücker embedding. ... C½ is not presented at the outset via a complete nomials in the variables x6, , x13 labeled by Recall that Gr2;n is generated by Plücker ... ≤ < ≤ set of generators and relations. Instead, from the sides. The variables x1, , x5 are the coordinates pij for 1 i j n. The ... the data of the initial seed—which includes initial cluster variables. The variables x6, , relations among the Plücker coordinates the k initial cluster variables x , ..., x ,plus x13 generate the ring of coefficients, and we are generated by three-term Plücker rela- 1 k F Q½ ; ...; an exchange matrix—one uses an iterative regard as an algebra over x6 x13 . tions: For any 1 ≤ i < j < k < ℓ ≤ n,onehas procedure called “mutation” to produce the We then use the combinatorics of trian- = + : rest of the cluster variables. In particular, each gulationsandflipsoftriangulationstodefine pikpjℓ pijpkℓ piℓpjk [1] new cluster variable is a rational expression the other cluster variables. Consider a tri- ... in x1, , xk. The cluster algebra is then de- angulation T containing a diagonal t.Within To make the connection with cluster fined to be the subring of F generated by all T,thediagonalt is the diagonal of some algebras, label the vertices of an n-gon from 1 cluster variables. quadrilateral. Then there is a new tri- to n in order around the boundary. Then The set of all cluster variables has a re- angulation T′ which is obtained by replac- each side and diagonal of the polygon is markable combinatorial structure: It is a ing t with the other diagonal of that quadri- uniquely identified by the labels of its end- union of overlapping algebraically inde- lateral. This local move is called a “flip.” points. This gives a bijection between the set pendent k subsets of F called “clusters,” Consider the graph whose vertex set is of Plücker coordinates and the set of sides which together have the structure of a the set of triangulations of an n-gon, with and diagonals of the n-gon (Fig. 1, Right). By simplicial complex called the “cluster com- an edge between two vertices whenever the plex.” The clusters are related to each other corresponding triangulations are related by Author contributions: B.L. and L.K.W. wrote the paper. by birational transformations of the fol- a flip. It is well-known that this flip graph is The authors declare no conflict of interest. lowing kind: For every cluster x and every connected, and moreover is the 1-skeleton of 1To whom correspondence should be addressed. Email: williams@ cluster variable x ∈ x, there is another a convex polytope called the “associahedron.” math.berkeley.edu. 9676–9679 | PNAS | July 8, 2014 | vol. 111 | no. 27 www.pnas.org/cgi/doi/10.1073/pnas.1410635111 Downloaded by guest on October 1, 2021 p Research Institute (MSRI) program in field of cluster algebras. We refer the reader 8 1212 INTRODUCTION p SPECIAL FEATURE: 7 9 18 p Berkeley on Cluster Algebras held during to the papers themselves for full details. 3 p 23 – 8 13 3 August 20 December 21, 2012 featured ex- 1 p Webs on Surfaces, Rings of Invariants, 6 2 p p tended stays of more than 50 mathema- 10 78 17 16 p p 4 36 34 ticians, ranging from graduate students and and Clusters 5 7 4 ≅ Ck p postdoctoral students to senior researchers. Let V be a vector space endowed with 13 11 p 35 p 67 45 Many additional mathematicians passed a volume form. The topic of the first paper 12 p 56 56 through the cluster algebras program for in our issue, by Fomin and Pylyavskyy (59), is rings of SL invariants of collec- Fig. 1. A triangulation T of an octagon (Left), and the shorter visits, for example, to participate in k labeling of T by Plücker coordinates (Right). one of the three focused workshops that tions of vectors, covectors, and matrices, in a took place as part of the program. k-dimensional vector space. More precisely, ð Þ = C½ð pÞa × b × ð ð ÞÞcSLðVÞ The program presented a broad panorama let Ra;b;c V V V SL V noting that the Plücker relations correspond of the current state of this rapidly expanding be the ring of SLðVÞ-invariant polynomials to exchange relations in A (Fig. 4), one p a c n subject, enabled many mathematicians to on ðV Þ × Vb × ðSLðVÞÞ . A generating set shows that there is a well-defined iso- broaden their understanding of the roles that for this ring has been known since 1976, but morphism from A ⊗ C to C½Gr ; map- n 2 n cluster algebras play in various active areas of the size of a minimal generating set, let alone ping the elements x associated with the i research, facilitated new interactions, and, an efficient description of the corresponding sides and diagonals to the corresponding more generally, provided the participants an ideal of relations, are, so far, unknown. Plücker coordinates. excellent opportunity to share and further In ref. 59, the authors study the case k = 3, One may generalize this example in several develop their ideas. This MSRI program was and give a series of remarkable results and ways. First, one may replace Gr2,n by an ar- conjectures about the structure of R ; ; ðVÞ.
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