Positivity for Cluster Algebras from Surfaces

Positivity for cluster algebras from surfaces

Gregg Musiker (MIT)

(Joint work with Ralf Schiﬄer (University of Connecticut) and Lauren Williams (University of California, Berkeley))

Brown University Discrete Mathematics Seminar

February 9, 2010

http//math.mit.edu/∼musiker/ClusterSurface.pdf

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 1 / 29 Outline

1 Introduction: the Laurent phenomenon, and the positivity conjecture of Fomin-Zelevinsky.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 2 / 29 Outline

1 Introduction: the Laurent phenomenon, and the positivity conjecture of Fomin-Zelevinsky. 2 Cluster algebras arising from triangulated surfaces (following Fomin-Shapiro-Thurston’s terminology).

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 2 / 29 Outline

1 Introduction: the Laurent phenomenon, and the positivity conjecture of Fomin-Zelevinsky. 2 Cluster algebras arising from triangulated surfaces (following Fomin-Shapiro-Thurston’s terminology). 3 Graph theoretic construction for surfaces with or without punctures (joint work with Schiﬄer and Williams).

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 2 / 29 Outline

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 2 / 29 Introduction to Cluster Algebras

In the late 1990’s: Fomin and Zelevinsky were studying total positivity and canonical bases of algebraic groups. They noticed recurring combinatorial and algebraic structures.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 3 / 29 Introduction to Cluster Algebras

In the late 1990’s: Fomin and Zelevinsky were studying total positivity and canonical bases of algebraic groups. They noticed recurring combinatorial and algebraic structures.

Led them to deﬁne cluster algebras, which have now been linked to quiver representations, Poisson geometry, Teichm¨uller theory, tropical geometry, Lie groups, and other topics.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 3 / 29 Introduction to Cluster Algebras

In the late 1990’s: Fomin and Zelevinsky were studying total positivity and canonical bases of algebraic groups. They noticed recurring combinatorial and algebraic structures.

Led them to deﬁne cluster algebras, which have now been linked to quiver representations, Poisson geometry, Teichm¨uller theory, tropical geometry, Lie groups, and other topics.

Cluster algebras are a certain class of commutative rings which have a distinguished set of generators that are grouped into overlapping subsets, called clusters, each having the same cardinality.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 3 / 29 What is a Cluster Algebra?

Deﬁnition (Sergey Fomin and Andrei Zelevinsky 2001) A cluster algebra A (of geometric type) is a subalgebra of k(x1,..., xn, xn+1,..., xn+m) constructed cluster by cluster by certain exchange relations.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 4 / 29 What is a Cluster Algebra?

Specify an initial ﬁnite set of them, a Cluster, {x1, x2,..., xn+m}.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 4 / 29 What is a Cluster Algebra?

Specify an initial ﬁnite set of them, a Cluster, {x1, x2,..., xn+m}. Construct the rest via Binomial Exchange Relations:

+ − ′ di di xαxα = xγi + xγi . Y Y

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 4 / 29 What is a Cluster Algebra?

Specify an initial ﬁnite set of them, a Cluster, {x1, x2,..., xn+m}. Construct the rest via Binomial Exchange Relations:

+ − ′ di di xαxα = xγi + xγi . The set of all such generators areY known asYCluster Variables, and the initial pattern B of exchange relations determines the Seed.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 4 / 29 What is a Cluster Algebra?

Specify an initial ﬁnite set of them, a Cluster, {x1, x2,..., xn+m}. Construct the rest via Binomial Exchange Relations:

+ − ′ di di xαxα = xγi + xγi . The set of all such generators areY known asYCluster Variables, and the initial pattern B of exchange relations determines the Seed. Relations: Induced by the Binomial Exchange Relations.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 4 / 29 Example: Coordinate Ring of Grassmannian(2, n + 3)

n+3 Let Gr2,n+3 = {V |V ⊂ C , dim V = 2} planes in (n + 3)-space

Elements of Gr2,n+3 represented by 2-by-(n + 3) matrices of full rank.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 5 / 29 Example: Coordinate Ring of Grassmannian(2, n + 3)

n+3 Let Gr2,n+3 = {V |V ⊂ C , dim V = 2} planes in (n + 3)-space

Elements of Gr2,n+3 represented by 2-by-(n + 3) matrices of full rank.

Pl¨ucker coordinates pij (M) = det of 2-by-2 submatrices in columns i and j.

The coordinate ring C[Gr2,n+3] is generated by all the pij ’s for 1 ≤ i < j ≤ n +3 subject to the Pl¨ucker relations given by the 4-tuples

pik pjℓ = pij pkℓ + piℓpjk for i < j < k < ℓ.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 5 / 29 Example: Coordinate Ring of Grassmannian(2, n + 3)

n+3 Let Gr2,n+3 = {V |V ⊂ C , dim V = 2} planes in (n + 3)-space

Elements of Gr2,n+3 represented by 2-by-(n + 3) matrices of full rank.

Pl¨ucker coordinates pij (M) = det of 2-by-2 submatrices in columns i and j.

The coordinate ring C[Gr2,n+3] is generated by all the pij ’s for 1 ≤ i < j ≤ n +3 subject to the Pl¨ucker relations given by the 4-tuples

pik pjℓ = pij pkℓ + piℓpjk for i < j < k < ℓ.

Claim. C[Gr2,n+3] has the structure of a cluster algebra. Clusters are maximal algebraically independent sets of pij ’s.

Each have size (2n + 3) where (n + 3) of the variables are frozen and n of them are exchangeable.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 5 / 29 Example: Coordinate Ring of Grassmannian(2, n + 3)

Cluster algebra structure of Gr2,n+3 as a triangulated (n + 3)-gon.

Frozen Variables / Coeﬃcients ←→ sides of the (n + 3)-gon

Cluster Variables ←→ {pij : |i − j|= 6 1 mod (n + 3)} ←→ diagonals

Seeds ←→ triangulations of the (n + 3)-gon

Clusters ←→ Set of pij ’s corresponding to a triangulation

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 6 / 29 Example: Coordinate Ring of Grassmannian(2, n + 3)

Cluster algebra structure of Gr2,n+3 as a triangulated (n + 3)-gon.

Frozen Variables / Coeﬃcients ←→ sides of the (n + 3)-gon

Cluster Variables ←→ {pij : |i − j|= 6 1 mod (n + 3)} ←→ diagonals

Seeds ←→ triangulations of the (n + 3)-gon

Clusters ←→ Set of pij ’s corresponding to a triangulation

Can exchange between various clusters by ﬂipping between triangulations.

This is called mutation, and we will present a detailed example later.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 6 / 29 Another Example: Total Positivity

a b c Given a 3-by-3 matrix M = d e f ∈ SL , how do you check whether   3 g h i it is totally positive, meaning that all minors are positive?

(i.e. a > 0, b > 0, c > 0,..., ae − bd > 0,..., det M > 0.)

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 7 / 29 Another Example: Total Positivity

a b c Given a 3-by-3 matrix M = d e f ∈ SL , how do you check whether   3 g h i it is totally positive, meaning that all minors are positive?

(i.e. a > 0, b > 0, c > 0,..., ae − bd > 0,..., det M > 0.)

Answer: It is suﬃcient to check that c > 0, g > 0, bf − ce > 0, dh − eg > 0 and four other conditions

(for a total of 8 veriﬁcations rather than all 19 minors).

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 7 / 29 Another Example: Total Positivity

a b c Given a 3-by-3 matrix M = d e f ∈ SL , how do you check whether   3 g h i it is totally positive, meaning that all minors are positive?

(i.e. a > 0, b > 0, c > 0,..., ae − bd > 0,..., det M > 0.)

Answer: It is suﬃcient to check that c > 0, g > 0, bf − ce > 0, dh − eg > 0 and four other conditions

(for a total of 8 veriﬁcations rather than all 19 minors).

There are exactly 50 such overlapping sets of four conditions. These 50 algebraic elements generate a cluster algebra structure (i.e. binomial exchange relations among the elements).

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 7 / 29 Seeds and Mutation

Deﬁnition. A Seed is a pair (X, B), where X = {x1, x2,..., xn+m} is an initial Cluster, and B is an Exchange Matrix, i.e. a (n + m)-by-n skew-symmetrizable integral matrix. (di bij = −dj bji for di ∈ Z>0)

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 8 / 29 Seeds and Mutation

Columns of B encode the exchanges

′ |bik | |bik | xk xk = xi + xi bYik >0 bYik <0 for k ∈ {1, 2,... n}. Note: If only one sign occurs (e.g. bik > 0), we get a monomial of 1.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 8 / 29 Seeds and Mutation

Columns of B encode the exchanges

′ |bik | |bik | xk xk = xi + xi bYik >0 bYik <0 for k ∈ {1, 2,... n}. Note: If only one sign occurs (e.g. bik > 0), we get a monomial of 1. For all k ∈ {1, 2,..., n}, there exists another seed ′ consisting of cluster {x1,..., xk ,..., xn+m} ∪ {xk } and matrix µk (B).

−bbij if k = i or k = j bij if bik bkj ≤ 0 µk (B)ij =  bij + bik bkj if bik , bkj > 0  bij − bik bkj if bik , bkj < 0  Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 8 / 29  Finite Mutation Type and Finite Type

A priori, get a tree of exchanges.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 9 / 29 Finite Mutation Type and Finite Type

A priori, get a tree of exchanges.

In practice, often get identiﬁcations among seeds.

In extreme cases, get only a ﬁnite number of exchange patterns as tree closes up on itself. Such cluster algebras called ﬁnite mutation type.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 9 / 29 Finite Mutation Type and Finite Type

A priori, get a tree of exchanges.

In practice, often get identiﬁcations among seeds.

In extreme cases, get only a ﬁnite number of exchange patterns as tree closes up on itself. Such cluster algebras called ﬁnite mutation type.

Sometimes only a ﬁnite number of clusters. Called ﬁnite type.

Finite type =⇒ Finite mutation type.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 9 / 29 Finite Mutation Type and Finite Type

A priori, get a tree of exchanges.

In practice, often get identiﬁcations among seeds.

In extreme cases, get only a ﬁnite number of exchange patterns as tree closes up on itself. Such cluster algebras called ﬁnite mutation type.

Sometimes only a ﬁnite number of clusters. Called ﬁnite type.

Finite type =⇒ Finite mutation type.

Theorem. (FZ 2002) Finite type cluster algebras can be described via the Cartan-Killing classiﬁcation of Lie algebras.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 9 / 29 Example 2: Rank 2 Cluster Algebras 0 b Let B = , b, c ∈ Z . ({x , x }, B) is a seed for a cluster algebra −c 0 >0 1 2 of rank 2.

′ c ′ b µ1(B)= µ2(B)= −B and x1x1 = x2 + 1, x2x2 =1+ x1 . Thus the cluster variables in this case are xb + 1 if n is odd x n Z satisfying x x n−1 { n : ∈ } n n−2 = c . (xn−1 + 1 if n is even

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 10 / 29 Example 2: Rank 2 Cluster Algebras 0 b Let B = , b, c ∈ Z . ({x , x }, B) is a seed for a cluster algebra −c 0 >0 1 2 of rank 2.

′ c ′ b µ1(B)= µ2(B)= −B and x1x1 = x2 + 1, x2x2 =1+ x1 . Thus the cluster variables in this case are xb + 1 if n is odd x n Z satisfying x x n−1 { n : ∈ } n n−2 = c . (xn−1 + 1 if n is even Example (b = c = 1):

x2 + 1 x3 = . x1

x2 + 1 x3 + 1 x3 = . x4 = = x1 x2

x4 + 1 x5 = = x3 Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 10 / 29 Example 2: Rank 2 Cluster Algebras 0 b Let B = , b, c ∈ Z . ({x , x }, B) is a seed for a cluster algebra −c 0 >0 1 2 of rank 2.

x1+x2+1 + 1 x4 + 1 x1x2 x5 = = = x3 (x2 + 1)/x1 Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 10 / 29 Example 2: Rank 2 Cluster Algebras 0 b Let B = , b, c ∈ Z . ({x , x }, B) is a seed for a cluster algebra −c 0 >0 1 2 of rank 2.

x1+x2+1 + 1 x4 + 1 x1x2 x1(x1 + x2 +1+ x1x2) x5 = = = = x3 (x2 + 1)/x1 x1x2(x2 + 1) Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 10 / 29 Example 2: Rank 2 Cluster Algebras 0 b Let B = , b, c ∈ Z . ({x , x }, B) is a seed for a cluster algebra −c 0 >0 1 2 of rank 2.

Example (b = c = 1): (Finite Type, of Type A2) x2+1 + 1 x2 + 1 x3 + 1 x1 x1 + x2 + 1 x3 = . x4 = = = . x1 x2 x2 x1x2

x1+x2+1 + 1 x4 + 1 x1x2 x1(x1 + x2 +1+ x1x2) x1 + 1 x5 = = = = . x3 (x2 + 1)/x1 x1x2(x2 + 1) x2 Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 10 / 29 Example 2: Rank 2 Cluster Algebras 0 b Let B = , b, c ∈ Z . ({x , x }, B) is a seed for a cluster algebra −c 0 >0 1 2 of rank 2.

Example (b = c = 1): (Finite Type, of Type A2) x2+1 + 1 x2 + 1 x3 + 1 x1 x1 + x2 + 1 x3 = . x4 = = = . x1 x2 x2 x1x2

x1+x2+1 + 1 x4 + 1 x1x2 x1(x1 + x2 +1+ x1x2) x1 + 1 x5 = = = = . x6 = x1. x3 (x2 + 1)/x1 x1x2(x2 + 1) x2 Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 10 / 29 Cluster Expansions and the Laurent Phenomenon

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 11 / 29 Cluster Expansions and the Laurent Phenomenon

Example 3. Let A be the cluster algebra defined by the initial cluster {x1, x2, x3, y1, y2, y3} and the initial exchange pattern 0 1 0 −1 0 −1  0 1 0  x x′ = y + x , x x′ = x x y + 1, x x′ = y + x . 1 1 1 2 2 2 1 3 2 3 3 3 2  1 0 0     0 1 0     0 0 1      A is of finite type, type A3 and corresponds to a triangulated hexagon. y + x x x y + 1 y + x x x y y + y + x x , x , x , 1 2 , 1 3 2 , 3 2 , 1 3 1 2 1 2 , 1 2 3 x x x x x 1 2 3 1 2 x x y y + y + x x x y y y + y y + x y + x y + x2 1 3 2 3 3 2 , 1 3 1 2 3 1 3 2 3 2 1 2 . x x x x x 2 3 1 2 3 The yi ’s are known as principal coefficients. Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 11 / 29 The Positivity Conjecture of Fomin and Zelevinsky

Theorem. (The Laurent Phenomenon FZ 2001) For any cluster algebra deﬁned by initial seed ({x1, x2,..., xn+m}, B), all cluster variables of A(B) are Laurent polynomials in {x1, x2,..., xn+m} (with no coeﬃcient xn+1,..., xn+m in the denominator).

Because of the Laurent Phenomenon, any cluster variable xα can be Pα(x1,...,xn+m) Z Z expressed as α1 αn where Pα ∈ [x1,..., xn+m] and the αi ’s ∈ . x1 ···xn

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 12 / 29 The Positivity Conjecture of Fomin and Zelevinsky

Because of the Laurent Phenomenon, any cluster variable xα can be Pα(x1,...,xn+m) Z Z expressed as α1 αn where Pα ∈ [x1,..., xn+m] and the αi ’s ∈ . x1 ···xn

Conjecture. (FZ 2001) For any cluster variable xα and any initial seed (i.e. initial cluster {x1,..., xn+m} and initial exchange pattern B), the polynomial Pα(x1,..., xn) has nonnegative integer coeﬃcients.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 12 / 29 Some Prior Work on Positivity Conjecture

Work of [Carroll-Price 2002] gave expansion formulas for case of Ptolemy algebras, cluster algebras of type An with boundary coeﬃcients (Gr2,n+3).

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 13 / 29 Some Prior Work on Positivity Conjecture

Work of [Carroll-Price 2002] gave expansion formulas for case of Ptolemy algebras, cluster algebras of type An with boundary coefficients (Gr2,n+3). [FZ 2002] proved positivity for finite type with bipartite seed. [M-Propp 2003, Sherman-Zelevinsky 2003] proved positivity for rank two affine cluster algebras. Other rank two cases by [Dupont 2009]. Work towards positivity for acyclic seeds [Caldero-Reineke 2006]. Positivity for cluster algebras including a bipartite seed (which is necessarily acyclic) by [Nakajima 2009].

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 13 / 29 Some Prior Work on Positivity Conjecture

Positivity for arbitrary surfaces [M-Schiﬄer-Williams 2009].

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 13 / 29 Main Theorem

Theorem. (Positivity for cluster algebras from surfaces MSW 2009) Let A be any cluster algebra arising from a surface (with or without punctures), where the coeﬃcient system is of geometric type, and let Σ be any initial seed.

Then the Laurent expansion of every cluster variable with respect to the seed Σ has non-negative coeﬃcients.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 14 / 29 Main Theorem

Then the Laurent expansion of every cluster variable with respect to the seed Σ has non-negative coeﬃcients.

We prove this theorem by exhibiting a graph theoretic interpretation for the Laurent expansions corresponding to cluster variables.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 14 / 29 Main Theorem

Then the Laurent expansion of every cluster variable with respect to the seed Σ has non-negative coeﬃcients.

We prove this theorem by exhibiting a graph theoretic interpretation for the Laurent expansions corresponding to cluster variables.

Due to work of Felikson-Shapiro-Tumarkin, we get

Corollary. Positivity for any seed, for all but 11 skew-symmetric cluster algebras of ﬁnite mutation type. (Rank two skew-symmetric cases by Caldero-Reineke)

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 14 / 29 Cluster Algebras of Triangulated Surfaces

We follow (Fomin-Shapiro-Thurston), based on earlier work of Fock-Goncharov and Gekhtman-Shapiro-Vainshtein.

We have a surface S with a set of marked points M. (If P ∈ M is in the interior of S, i.e. S \ δS, then P is known an a puncture).

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 15 / 29 Cluster Algebras of Triangulated Surfaces

We follow (Fomin-Shapiro-Thurston), based on earlier work of Fock-Goncharov and Gekhtman-Shapiro-Vainshtein.

We have a surface S with a set of marked points M. (If P ∈ M is in the interior of S, i.e. S \ δS, then P is known an a puncture).

An arc γ satisﬁes (we care about arcs up to isotopy) 1 The endpoints of γ are in M. 2 γ does not cross itself. 3 except for the endpoints, γ is disjoint from M and the boundary of S. 4 γ does not cut out an unpunctured monogon or unpunctured bigon.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 15 / 29 Cluster Algebras of Triangulated Surfaces

We follow (Fomin-Shapiro-Thurston), based on earlier work of Fock-Goncharov and Gekhtman-Shapiro-Vainshtein.

We have a surface S with a set of marked points M. (If P ∈ M is in the interior of S, i.e. S \ δS, then P is known an a puncture).

Seed ↔ Triangulation T = {τ1,τ2,...,τn}

Cluster Variable ↔ Arc γ (xi ↔ τi ∈ T )

Cluster Mutation ↔ Ptolemy Exchanges (Flipping Diagonals).

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 15 / 29 Example of Hexagon

Consider the triangulated hexagon (S, M) with triangulation TH . τ τ 9 9 τ8 τ8

τ τ τ' 1 γ τ 1 1 τ 2 τ 2 τ 4 4 τ τ 7 7 τ τ3 3 τ τ 5 5 τ τ 6 6

′ x1x1 = y1(x7x9)+ x2(x8)

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 16 / 29 Example of Hexagon

Consider the triangulated hexagon (S, M) with triangulation TH . τ τ 9 9 τ 8 τ8

τ τ' τ' 1 1 τ 1 2 τ τ2'' τ 4 4 τ τ 7 7 τ τ τ 3 2 3 τ τ 5 5 τ τ 6 6

′ x1x1 = y1(x7x9)+ x2(x8) ′′ ′ x2x2 = y1y2x3(x9)+ x1(x4)

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 16 / 29 Example of Hexagon

Consider the triangulated hexagon (S, M) with triangulation TH . τ τ 9 9 τ8 τ8

τ' τ' 1 1 τ2'' τ2'' τ τ 4 4 γ = τ''' τ τ 3 7 7 τ τ τ 2 3 3 τ τ 5 5 τ τ 6 6

′ x1x1 = y1(x7x9)+ x2(x8) ′′ ′ x2x2 = y1y2x3(x9)+ x1(x4) ′′′ ′′ ′ x3x3 = y3x2 (x6)+ x1(x5)

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 16 / 29 Example of Hexagon (continued)

τ9 τ8 τ 1 γ τ 2 τ 4 τ 7 τ3 τ 5 τ 6

By using the Ptolemy relations on τ1, τ2, then τ3, we obtain 1 x′′′ = x = x2(x x )+ y x (x x x )+ y x (x x x ) 3 γ x x x 2 5 8 1 2 5 7 9 3 2 4 6 8 1 2 3

+ y1y3(x4x6x7x9)+ y1y2y3x1x3(x6x9) . Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 17 / 29 Example of Hexagon (continued) 2 3 5

9 1 7 24 3 6

Consider the graph GTH ,γ = 8 1 2

GTH ,γ has ﬁve perfect matchings (x4, x5,..., x9 = 1):

(x9)x1x3(x6), (x9x7x4x6),

x2(x8)(x4x6), (x9x7)x2(x5),

x2(x8)x2(x5). A perfect matching M ⊆ E is a set of distinguished edges so that every vertex of V is covered exactly once. The weight of a matching M is the

product of the weights of the constituent edges, i.e. x(M)= e∈M x(e). Musiker (MIT) Positivity for cluster algebras from surfaces FebruaryQ 9, 2010 18 / 29 Example of Hexagon (continued)

2 3 5

9 1 7 24 3 6

Consider the graph GTH ,γ = 8 1 2

GTH ,γ has ﬁve perfect matchings (x4, x5,..., x9 = 1):

(x9)x1x3(x6), (x9x7x4x6),

x2(x8)(x4x6), (x9x7)x2(x5),

2 x1x3y1y2y3+y1y3+x2y3+x2y1+x2 x2(x8)x2(x5). x1x2x3 These ﬁve monomials exactly match those appearing in the numerator of the expansion of xγ . The denominator of x1x2x3 corresponds to the labels of the three tiles. Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 18 / 29 A Graph Theoretic Approach

For every triangulation T (in a surface with or without punctures) and an ordinary arc γ through ordinary triangles, we construct a snake graph GT ,γ such that x(M)y(M) perfect matching M of GT ,γ xγ = . e1(T ,γ) e2(T ,γ) en(T ,γ) P x1 x2 · · · xn

xγ is cluster variable (corresp. to γ w.r.t. seed given by T ) with principal coeﬃcients.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 19 / 29 A Graph Theoretic Approach

xγ is cluster variable (corresp. to γ w.r.t. seed given by T ) with principal coeﬃcients.

ei (T , γ) is the crossing number of τi and γ (min. int. number), x(M) is the weight of M, y(M) is the height of M (to be deﬁned later), Similar formula will hold for non-ordinary arcs (or through self-folded triangles).

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 19 / 29 Examples of GT ,γ

Example 1. Using the above construction for τ9

τ8 τ 1 γ τ 2 τ 4 τ 7 τ 3 τ 5 τ 6 :

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 20 / 29 Examples of GT ,γ

Example 1. Using the above construction for τ9 τ9

τ8 τ8

τ τ τ' 1 γ τ 1 1 τ 2 τ 2 τ 4 4 τ τ 7 7 τ τ 3 3 τ τ 5 5 τ τ 6 : 6

9 1 7

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 20 / 29 Examples of GT ,γ

Example 1. Using the above construction for τ9 τ9 τ τ8 8 τ τ 1 γ τ 1 τ 2 τ' 2 τ 2 τ 4 4 τ τ 7 7 τ τ 3 3 τ τ 5 5 τ τ 6 : 6

2 2 3

9 1 7 9 1 7 2 4 8 , 8 1

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 20 / 29 Examples of GT ,γ

Example 1. Using the above construction for τ9 τ9

τ8 τ8 τ τ τ 2 1 γ τ 1 2 τ τ 4 τ' 4 τ τ 3 7 7 τ τ 3 3 τ τ 5 5 τ τ 6 : 6

2 2 3 2 3 5

9 1 7 9 1 7 2 4 9 1 7 24 3 6 8 , 8 1 , 8 1 2 . Thus 2 3 5

9 1 7 24 3 6 2 x1x3 +1+2x2 + x2 GTH ,γ = 8 1 2 . xγ|y1=y2=y3=1 = x1x2x3

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 20 / 29 Height Functions (of Perfect Matchings of Snake Graphs)

We now wish to give formula for y(M)’s, i.e. the terms in the F -polynomials.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 21 / 29 Height Functions (of Perfect Matchings of Snake Graphs)

We now wish to give formula for y(M)’s, i.e. the terms in the F -polynomials.

We use height functions which are due to William Thurston, and Conway-Lagarias.

Involves measuring contrast between a given perfect matching M and a ﬁxed minimal matching M−.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 21 / 29 Height Function Examples

Recall that GTH ,γ has three faces, labeled 1, 2 and 3. GTH ,γ has ﬁve perfect matchings (x4, x5,..., x9 = 1):

2 x1x3y1y2y3 + y1y3 + x2y3 + x2y1 + x2 xγ = x1x2x3

y1y2y3, y1y3,

y3, y1,

1 ←− This matching is M−.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 22 / 29 Height Function Examples

Recall that GTH ,γ has three faces, labeled 1, 2 and 3. GTH ,γ has ﬁve perfect matchings (x4, x5,..., x9 = 1):

2 x1x3y1y2y3 + y1y3 + x2y3 + x2y1 + x2 xγ = x1x2x3

y1y2y3, y1y3,

y3, y1,

1 ←− This matching is M−.

12 3 1 3 3 , , and

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 22 / 29 Height Function Examples (continued)

2 3

41 6 2 8 1 2 5 1 3 4 1 2 3 7 4

41 6 28 3 4 1 2 3 5 1 2 For GTA,γ = , M− is .

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 23 / 29 Height Function Examples (continued)

2 3

41 6 2 8 1 2 5 1 3 4 1 2 3 7 4

41 6 28 3 4 1 2 3 5 1 2 For GTA,γ = , M− is . One of the

1 2

1 2 3 17 matchings, M, is ,

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 23 / 29 Height Function Examples (continued)

2 3

41 6 2 8 1 2 5 1 3 4 1 2 3 7 4

41 6 28 3 4 1 2 3 5 1 2 For GTA,γ = , M− is . One of the

1 2 1 2

4 4

1 2 3 1 2 3 17 matchings, M, is , so M ⊖ M− = ,

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 23 / 29 Height Function Examples (continued)

2 3

41 6 2 8 1 2 5 1 3 4 1 2 3 7 4

41 6 28 3 4 1 2 3 5 1 2 For GTA,γ = , M− is . One of the

1 2 1 2

4 4

1 2 3 1 2 3 17 matchings, M, is , so M ⊖ M− = ,

2 which has height y1y2 . So one of the 17 terms in the cluster expansion of x4(x6x8)x4(x5)x2(x8) 2 xγ is (using FST convention) 2 2 (y1y2 ). x1 x2 x3x4 Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 23 / 29 Summary

Theorem. (M-Schiﬄer-Williams 2009) For every triangulation T of a surface (with or without punctures) and an ordinary arc γ, we construct a snake graph Gγ,T such that x(M)y(M) perfect matching M of Gγ,T xγ = . e1(T ,γ) e2(T ,γ) en(T ,γ) P x1 x2 · · · xn Here ei (T , γ) is the crossing number of τi and γ, x(M) is the edge-weight of perfect matching M, and y(M) is the height of perfect matching M. (xγ is cluster variable with principal coeﬃcients.)

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 24 / 29 Summary

Theorem. (M-Schiﬄer-Williams 2009) An analogous expansion formula holds for arcs with notches (only arise in the case of a punctured surface).

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 24 / 29 Summary

Theorem. (M-Schiﬄer-Williams 2009) An analogous expansion formula holds for arcs with notches (only arise in the case of a punctured surface).

Corollary. The F -polynomial equals M y(M), is positive, and has constant term 1. P g The g-vector satisﬁes x = x(M−).

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 24 / 29 Example 2 (Ordinary Arc through Self-folded Triangle)

(S, M)

11 14 7 γ1 6 9 4 5 ℓ 5 10 r 3 4 5 10 ℓ 9

8 ℓ 3 p r 11 3 12 4 14 5 6 r r 12 6 7 13 ℓ 3 4 ℓ r 11 r ℓ

3 ℓ

◦ ◦ Figure: Ideal Triangulation T of (S, M) and corresponding Snake Graph G T ,γ1 .

Note the three consecutive tiles of our snake graph with labels ℓ, r and ℓ, as γ1 traverses the loop ℓ twice and the enclosed radius r.

Musiker (MIT) Positivity for cluster algebras from surfaces February 9, 2010 25 / 29 Example 2 (Ordinary Arc through Self-folded Triangle)

(S, M)

11 14 7

γ1 6 9 5 ℓ 4 5 10 3 4 5 10 r ℓ 9 ℓ 3 8 r 3 4 5 p 11 12 14 6 r r 12 6 7 13 ℓ 3 4 ℓ r 11 r ℓ

3 ℓ