Snake graphs from orbifolds Elizabeth Kelley (Joint work with - PowerPoint PPT Presentation

Snake graphs from orbifolds Elizabeth Kelley (Joint work with Esther Banaian) April 14, 2019 Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 1 / 25

Basic Definitions Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 2 / 25

Basic Definitions Fix a semifield ( P , ⊕ , · ). Let F be isomorphic to the field of rational functions in n independent variables with coefficients in QP . Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 2 / 25

Basic Definitions Fix a semifield ( P , ⊕ , · ). Let F be isomorphic to the field of rational functions in n independent variables with coefficients in QP . Definition: A labeled seed (of geometric type) in F is a triple Σ = ( x , y , B ) where: x = ( x 1 , . . . , x n ) is a free generating set for F ; y = ( y 1 , . . . , y n ) is an n -tuple with elements in P ; and B = ( b ij ) is a skew-symmetrizable n × n integer matrix. We call x the cluster , y the coefficient tuple , and B the exchange matrix of the seed ( x , y , B ). Cluster variables are related by standard binomial exchange relations. Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 2 / 25

Cluster algebras of surface type Some cluster algebras can be associated with triangulated surfaces, via the following dictionary: initial cluster seed Σ ↔ ideal triangulation T initial cluster variable x i ↔ initial arc τ i ∈ T other cluster variables ↔ other arcs in ( S , M ) mutation µ k ↔ “flipping” arc τ k Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 3 / 25

Cluster algebras of surface type Some cluster algebras can be associated with triangulated surfaces, via the following dictionary: initial cluster seed Σ ↔ ideal triangulation T initial cluster variable x i ↔ initial arc τ i ∈ T other cluster variables ↔ other arcs in ( S , M ) mutation µ k ↔ “flipping” arc τ k Σ 1 2 3 ↔ ↔ τ 1 τ 2 τ 3 Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 3 / 25

Example Some cluster algebras can be associated with triangulated surfaces, via the following dictionary: initial cluster seed Σ ↔ ideal triangulation T initial cluster variable x i ↔ initial arc τ i ∈ T other cluster variables ↔ other arcs in ( S , M ) mutation µ k ↔ “flipping” arc τ k τ 1 ′ Σ 1 2 3 ↔ ↔ τ 1 τ 2 τ 3 µ 1 Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 3 / 25

Example Some cluster algebras can be associated with triangulated surfaces, via the following dictionary: initial cluster seed Σ ↔ ideal triangulation T initial cluster variable x i ↔ initial arc τ i ∈ T other cluster variables ↔ other arcs in ( S , M ) mutation µ k ↔ “flipping” arc τ k τ 1 ′ Σ 1 2 3 ↔ ↔ τ 2 τ 3 µ 1 Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 4 / 25

Example Some cluster algebras can be associated with triangulated surfaces, via the following dictionary: initial cluster seed Σ ↔ ideal triangulation T initial cluster variable x i ↔ initial arc τ i ∈ T other cluster variables ↔ other arcs in ( S , M ) mutation µ k ↔ “flipping” arc τ k τ 1 ′ Σ 1 2 3 ↔ ↔ τ 2 τ 3 µ 1 τ 2 ′ Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 5 / 25

Example Some cluster algebras can be associated with triangulated surfaces, via the following dictionary: initial cluster seed Σ ↔ ideal triangulation T initial cluster variable x i ↔ initial arc τ i ∈ T other cluster variables ↔ other arcs in ( S , M ) mutation µ k ↔ “flipping” arc τ k τ 1 ′ Σ 1 2 3 ↔ ↔ τ 3 µ 1 τ 2 ′ Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 6 / 25

Cluster algebras of surface type For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs. Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs. c e b d 2 3 c f d τ 1 τ 2 τ 3 a 1 2 3 a e b 1 2 f Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs. c e b d 2 3 c f d τ 1 τ 2 τ 3 a 1 2 3 a e γ b 1 2 f Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs. c e b d 2 3 c f d τ 1 τ 2 τ 3 a 1 2 3 a e γ b 1 2 f Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs. c c e b d 3 τ 2 f d τ 1 τ 2 τ 3 b τ 1 2 3 a a e γ 1 2 + f Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs. c τ 2 e b d 3 c f d τ 1 τ 2 τ 3 a τ 1 2 3 a e γ b 1 2 − f Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs. c τ 2 τ 3 e b d c d τ 1 τ 2 τ 3 a c f τ 1 τ 2 3 τ 1 a e γ b 2 − + f Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs. c τ 2 τ 3 e b d c f d τ 1 τ 2 τ 3 a τ 1 τ 2 3 τ 1 a e γ b 2 − + f Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs. c τ 2 τ 3 d b d c f e τ 1 τ 2 τ 3 τ 2 a τ 1 τ 2 τ 3 τ 1 a e γ b f − + + f Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs. c τ 2 τ 3 e b d c f d τ 1 τ 2 τ 3 a f τ 1 τ 2 τ 3 τ 1 τ 2 a e γ b − − + f Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs. c τ 2 τ 3 e b d c f d τ 1 τ 2 τ 3 a τ 1 τ 2 τ 3 τ 1 τ 2 a e γ b − − + f Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Theorem: (Musiker-Schiffler-Williams, 2011) Let ( S , M ) be a bordered surface with triangulation T , A be the corresponding cluster algebra with principal coeficients, and γ be an ordinary arc on S . Then x γ can be written as a Laurent expansion in terms of the initial cluster variables as 1 � x γ = x ( P ) y ( P ) cross ( T , γ ) P where P is a perfect matching of G T ,γ . Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 8 / 25

Theorem: (Musiker-Schiffler-Williams, 2011) Let ( S , M ) be a bordered surface with triangulation T , A be the corresponding cluster algebra with principal coeficients, and γ be an ordinary arc on S . Then x γ can be written as a Laurent expansion in terms of the initial cluster variables as 1 � x γ = x ( P ) y ( P ) cross ( T , γ ) P where P is a perfect matching of G T ,γ . Note: Here, coefficients in the numerator are counting the number of perfect matchings with particular values of x ( P ) and y ( P ). Hence, the coefficients must be in Z ≥ 0 . Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 8 / 25

1 � x γ = x ( P ) y ( P ) cross ( T , γ ) P Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 9 / 25

1 � x γ = x ( P ) y ( P ) cross ( T , γ ) P cross ( T , γ ) = the crossing monomial of γ = x i 1 · · · x i d , for τ i 1 , . . . , τ i d crossed by γ Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 9 / 25

1 � x γ = x ( P ) y ( P ) cross ( T , γ ) P cross ( T , γ ) = the crossing monomial of γ = x i 1 · · · x i d , for τ i 1 , . . . , τ i d crossed by γ x ( P ) = (the weight of P ) = x i 1 · · · x i k , where τ i 1 , . . . , τ i k are the labels of edges of P . Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 9 / 25

1 � x γ = x ( P ) y ( P ) cross ( T , γ ) P cross ( T , γ ) = the crossing monomial of γ = x i 1 · · · x i d , for τ i 1 , . . . , τ i d crossed by γ x ( P ) = (the weight of P ) = x i 1 · · · x i k , where τ i 1 , . . . , τ i k are the labels of edges of P . y ( P ) = (the height of P ) = � n k =1 h m k τ k , where  y τ k if τ k is not an edge of a self-folded triangle    y r if τ k is the radius r to puncture p in a self-folded triangle h τ k = y r ( p )   if τ k is the loop in a self-folded triangle with radius r to p y r ( p )  Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 9 / 25