Positivity for cluster algebras from surfaces

Transcription

1 Positivity fo custe agebas fom sufaces Gegg Musike (MIT) (Joint wok with Raf Schiffe (Univesity of Connecticut) and Lauen Wiiams (Univesity of Caifonia, Bekeey)) Bown Univesity Discete Mathematics Semina Febuay 9, 2010 http//math.mit.edu/ musike/custesuface.pdf Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

2 Outine 1 Intoduction: the Lauent phenomenon, and the positivity conjectue of Fomin-Zeevinsky. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

3 Outine 1 Intoduction: the Lauent phenomenon, and the positivity conjectue of Fomin-Zeevinsky. 2 Custe agebas aising fom tianguated sufaces (foowing Fomin-Shapio-Thuston s teminoogy). Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

4 Outine 1 Intoduction: the Lauent phenomenon, and the positivity conjectue of Fomin-Zeevinsky. 2 Custe agebas aising fom tianguated sufaces (foowing Fomin-Shapio-Thuston s teminoogy). 3 Gaph theoetic constuction fo sufaces with o without punctues (joint wok with Schiffe and Wiiams). Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

5 Outine 1 Intoduction: the Lauent phenomenon, and the positivity conjectue of Fomin-Zeevinsky. 2 Custe agebas aising fom tianguated sufaces (foowing Fomin-Shapio-Thuston s teminoogy). 3 Gaph theoetic constuction fo sufaces with o without punctues (joint wok with Schiffe and Wiiams). Exampes of this constuction. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

6 Intoduction to Custe Agebas In the ate 1990 s: Fomin and Zeevinsky wee studying tota positivity and canonica bases of agebaic goups. They noticed ecuing combinatoia and agebaic stuctues. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

7 Intoduction to Custe Agebas In the ate 1990 s: Fomin and Zeevinsky wee studying tota positivity and canonica bases of agebaic goups. They noticed ecuing combinatoia and agebaic stuctues. Led them to define custe agebas, which have now been inked to quive epesentations, Poisson geomety, Teichmüe theoy, topica geomety, Lie goups, and othe topics. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

8 Intoduction to Custe Agebas In the ate 1990 s: Fomin and Zeevinsky wee studying tota positivity and canonica bases of agebaic goups. They noticed ecuing combinatoia and agebaic stuctues. Led them to define custe agebas, which have now been inked to quive epesentations, Poisson geomety, Teichmüe theoy, topica geomety, Lie goups, and othe topics. Custe agebas ae a cetain cass of commutative ings which have a distinguished set of geneatos that ae gouped into oveapping subsets, caed custes, each having the same cadinaity. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

9 What is a Custe Ageba? Definition (Segey Fomin and Andei Zeevinsky 2001) A custe ageba A (of geometic type) is a subageba of k(x 1,...,x n,x n+1,...,x n+m ) constucted custe by custe by cetain exchange eations. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, 2010 / 29

10 What is a Custe Ageba? Definition (Segey Fomin and Andei Zeevinsky 2001) A custe ageba A (of geometic type) is a subageba of k(x 1,...,x n,x n+1,...,x n+m ) constucted custe by custe by cetain exchange eations. Geneatos: Specify an initia finite set of them, a Custe, {x 1,x 2,...,x n+m }. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, 2010 / 29

11 What is a Custe Ageba? Definition (Segey Fomin and Andei Zeevinsky 2001) A custe ageba A (of geometic type) is a subageba of k(x 1,...,x n,x n+1,...,x n+m ) constucted custe by custe by cetain exchange eations. Geneatos: Specify an initia finite set of them, a Custe, {x 1,x 2,...,x n+m }. Constuct the est via Binomia Exchange Reations: x α x α = x d+ i γ i + x d i γ i. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, 2010 / 29

12 What is a Custe Ageba? Definition (Segey Fomin and Andei Zeevinsky 2001) A custe ageba A (of geometic type) is a subageba of k(x 1,...,x n,x n+1,...,x n+m ) constucted custe by custe by cetain exchange eations. Geneatos: Specify an initia finite set of them, a Custe, {x 1,x 2,...,x n+m }. Constuct the est via Binomia Exchange Reations: x α x α = x d+ i γ i + x d i γ i. The set of a such geneatos ae known as Custe Vaiabes, and the initia patten B of exchange eations detemines the Seed. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, 2010 / 29

13 What is a Custe Ageba? Definition (Segey Fomin and Andei Zeevinsky 2001) A custe ageba A (of geometic type) is a subageba of k(x 1,...,x n,x n+1,...,x n+m ) constucted custe by custe by cetain exchange eations. Geneatos: Specify an initia finite set of them, a Custe, {x 1,x 2,...,x n+m }. Constuct the est via Binomia Exchange Reations: x α x α = x d+ i γ i + x d i γ i. The set of a such geneatos ae known as Custe Vaiabes, and the initia patten B of exchange eations detemines the Seed. Reations: Induced by the Binomia Exchange Reations. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, 2010 / 29

14 Exampe: Coodinate Ring of Gassmannian(2, n + 3) Let G 2,n+3 = {V V C n+3, dimv = 2} panes in (n + 3)-space Eements of G 2,n+3 epesented by 2-by-(n + 3) matices of fu ank. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

15 Exampe: Coodinate Ring of Gassmannian(2, n + 3) Let G 2,n+3 = {V V C n+3, dimv = 2} panes in (n + 3)-space Eements of G 2,n+3 epesented by 2-by-(n + 3) matices of fu ank. Pücke coodinates p ij (M) = det of 2-by-2 submatices in coumns i and j. The coodinate ing C[G 2,n+3 ] is geneated by a the p ij s fo 1 i < j n + 3 subject to the Pücke eations given by the -tupes p ik p j = p ij p k + p i p jk fo i < j < k <. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

16 Exampe: Coodinate Ring of Gassmannian(2, n + 3) Let G 2,n+3 = {V V C n+3, dimv = 2} panes in (n + 3)-space Eements of G 2,n+3 epesented by 2-by-(n + 3) matices of fu ank. Pücke coodinates p ij (M) = det of 2-by-2 submatices in coumns i and j. The coodinate ing C[G 2,n+3 ] is geneated by a the p ij s fo 1 i < j n + 3 subject to the Pücke eations given by the -tupes p ik p j = p ij p k + p i p jk fo i < j < k <. Caim. C[G 2,n+3 ] has the stuctue of a custe ageba. Custes ae maxima agebaicay independent sets of p ij s. Each have size (2n + 3) whee (n + 3) of the vaiabes ae fozen and n of them ae exchangeabe. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

17 Exampe: Coodinate Ring of Gassmannian(2, n + 3) Custe ageba stuctue of G 2,n+3 as a tianguated (n + 3)-gon. Fozen Vaiabes / Coefficients sides of the (n + 3)-gon Custe Vaiabes {p ij : i j 1 mod (n + 3)} diagonas Seeds tianguations of the (n + 3)-gon Custes Set of p ij s coesponding to a tianguation Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

18 Exampe: Coodinate Ring of Gassmannian(2, n + 3) Custe ageba stuctue of G 2,n+3 as a tianguated (n + 3)-gon. Fozen Vaiabes / Coefficients sides of the (n + 3)-gon Custe Vaiabes {p ij : i j 1 mod (n + 3)} diagonas Seeds tianguations of the (n + 3)-gon Custes Set of p ij s coesponding to a tianguation Can exchange between vaious custes by fipping between tianguations. This is caed mutation, and we wi pesent a detaied exampe ate. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

19 Anothe Exampe: Tota Positivity a b c Given a 3-by-3 matix M = d e f SL 3, how do you check whethe g h i it is totay positive, meaning that a minos ae positive? (i.e. a > 0, b > 0, c > 0,...,ae bd > 0,...,det M > 0.) Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

20 Anothe Exampe: Tota Positivity a b c Given a 3-by-3 matix M = d e f SL 3, how do you check whethe g h i it is totay positive, meaning that a minos ae positive? (i.e. a > 0, b > 0, c > 0,...,ae bd > 0,...,det M > 0.) Answe: It is sufficient to check that c > 0, g > 0, bf ce > 0, dh eg > 0 and fou othe conditions (fo a tota of 8 veifications athe than a 19 minos). Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

21 Anothe Exampe: Tota Positivity a b c Given a 3-by-3 matix M = d e f SL 3, how do you check whethe g h i it is totay positive, meaning that a minos ae positive? (i.e. a > 0, b > 0, c > 0,...,ae bd > 0,...,det M > 0.) Answe: It is sufficient to check that c > 0, g > 0, bf ce > 0, dh eg > 0 and fou othe conditions (fo a tota of 8 veifications athe than a 19 minos). Thee ae exacty 50 such oveapping sets of fou conditions. These 50 agebaic eements geneate a custe ageba stuctue (i.e. binomia exchange eations among the eements). Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

22 Seeds and Mutation Definition. A Seed is a pai (X,B), whee X = {x 1,x 2,...,x n+m } is an initia Custe, and B is an Exchange Matix, i.e. a (n + m)-by-n skew-symmetizabe intega matix. (d i b ij = d j b ji fo d i Z >0 ) Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

23 Seeds and Mutation Definition. A Seed is a pai (X,B), whee X = {x 1,x 2,...,x n+m } is an initia Custe, and B is an Exchange Matix, i.e. a (n + m)-by-n skew-symmetizabe intega matix. (d i b ij = d j b ji fo d i Z >0 ) Coumns of B encode the exchanges x k x k = b ik >0 x b ik i + b ik <0 x b ik i fo k {1,2,... n}. Note: If ony one sign occus (e.g. b ik > 0), we get a monomia of 1. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

24 Seeds and Mutation Definition. A Seed is a pai (X,B), whee X = {x 1,x 2,...,x n+m } is an initia Custe, and B is an Exchange Matix, i.e. a (n + m)-by-n skew-symmetizabe intega matix. (d i b ij = d j b ji fo d i Z >0 ) Coumns of B encode the exchanges x k x k = b ik >0 x b ik i + b ik <0 x b ik i fo k {1,2,... n}. Note: If ony one sign occus (e.g. b ik > 0), we get a monomia of 1. Fo a k {1, 2,..., n}, thee exists anothe seed consisting of custe {x 1,..., x k,...,x n+m } {x k } and matix µ k(b). b ij if k = i o k = j b ij if b ik b kj 0 µ k (B) ij = b ij + b ik b kj if b ik,b kj > 0 b ij b ik b kj if b ik,b kj < 0 Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

25 Finite Mutation Type and Finite Type A pioi, get a tee of exchanges. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

26 Finite Mutation Type and Finite Type A pioi, get a tee of exchanges. In pactice, often get identifications among seeds. In exteme cases, get ony a finite numbe of exchange pattens as tee coses up on itsef. Such custe agebas caed finite mutation type. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

27 Finite Mutation Type and Finite Type A pioi, get a tee of exchanges. In pactice, often get identifications among seeds. In exteme cases, get ony a finite numbe of exchange pattens as tee coses up on itsef. Such custe agebas caed finite mutation type. Sometimes ony a finite numbe of custes. Caed finite type. Finite type = Finite mutation type. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

28 Finite Mutation Type and Finite Type A pioi, get a tee of exchanges. In pactice, often get identifications among seeds. In exteme cases, get ony a finite numbe of exchange pattens as tee coses up on itsef. Such custe agebas caed finite mutation type. Sometimes ony a finite numbe of custes. Caed finite type. Finite type = Finite mutation type. Theoem. (FZ 2002) Finite type custe agebas can be descibed via the Catan-Kiing cassification of Lie agebas. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

29 Exampe 2: Rank 2 Custe Agebas [ ] 0 b Let B =, b,c Z c 0 >0. ({x 1,x 2 },B) is a seed fo a custe ageba of ank 2. µ 1 (B) = µ 2 (B) = B and x 1 x 1 = x c 2 + 1, x 2 x 2 = 1 + x b 1. Thus the custe vaiabes in this case ae { {x n : n Z} satisfying x n x n 2 = xn 1 b + 1 if n is odd xn 1 c + 1 if n is even. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

30 Exampe 2: Rank 2 Custe Agebas [ ] 0 b Let B =, b,c Z c 0 >0. ({x 1,x 2 },B) is a seed fo a custe ageba of ank 2. µ 1 (B) = µ 2 (B) = B and x 1 x 1 = x c 2 + 1, x 2 x 2 = 1 + x b 1. Thus the custe vaiabes in this case ae { {x n : n Z} satisfying x n x n 2 = xn 1 b + 1 if n is odd xn 1 c + 1 if n is even. Exampe (b = c = 1): x 3 = x x 1. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

31 Exampe 2: Rank 2 Custe Agebas [ ] 0 b Let B =, b,c Z c 0 >0. ({x 1,x 2 },B) is a seed fo a custe ageba of ank 2. µ 1 (B) = µ 2 (B) = B and x 1 x 1 = x c 2 + 1, x 2 x 2 = 1 + x b 1. Thus the custe vaiabes in this case ae { {x n : n Z} satisfying x n x n 2 = xn 1 b + 1 if n is odd xn 1 c + 1 if n is even. Exampe (b = c = 1): x 3 = x x 1. x = x x 2 = Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

32 Exampe 2: Rank 2 Custe Agebas [ ] 0 b Let B =, b,c Z c 0 >0. ({x 1,x 2 },B) is a seed fo a custe ageba of ank 2. µ 1 (B) = µ 2 (B) = B and x 1 x 1 = x c 2 + 1, x 2 x 2 = 1 + x b 1. Thus the custe vaiabes in this case ae { {x n : n Z} satisfying x n x n 2 = xn 1 b + 1 if n is odd xn 1 c + 1 if n is even. Exampe (b = c = 1): x 3 = x x 1. x = x x 2 = x 2 +1 x x 2 = Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

33 Exampe 2: Rank 2 Custe Agebas [ ] 0 b Let B =, b,c Z c 0 >0. ({x 1,x 2 },B) is a seed fo a custe ageba of ank 2. µ 1 (B) = µ 2 (B) = B and x 1 x 1 = x c 2 + 1, x 2 x 2 = 1 + x b 1. Thus the custe vaiabes in this case ae { {x n : n Z} satisfying x n x n 2 = xn 1 b + 1 if n is odd xn 1 c + 1 if n is even. Exampe (b = c = 1): x 3 = x x 1. x = x x 2 = x 2 +1 x x 2 = x 1 + x x 1 x 2. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

34 Exampe 2: Rank 2 Custe Agebas [ ] 0 b Let B =, b,c Z c 0 >0. ({x 1,x 2 },B) is a seed fo a custe ageba of ank 2. µ 1 (B) = µ 2 (B) = B and x 1 x 1 = x c 2 + 1, x 2 x 2 = 1 + x b 1. Thus the custe vaiabes in this case ae { {x n : n Z} satisfying x n x n 2 = xn 1 b + 1 if n is odd xn 1 c + 1 if n is even. Exampe (b = c = 1): x 3 = x x 1. x = x x 2 = x 2 +1 x x 2 = x 1 + x x 1 x 2. x 5 = x + 1 x 3 = Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

35 Exampe 2: Rank 2 Custe Agebas [ ] 0 b Let B =, b,c Z c 0 >0. ({x 1,x 2 },B) is a seed fo a custe ageba of ank 2. µ 1 (B) = µ 2 (B) = B and x 1 x 1 = x c 2 + 1, x 2 x 2 = 1 + x b 1. Thus the custe vaiabes in this case ae { {x n : n Z} satisfying x n x n 2 = xn 1 b + 1 if n is odd xn 1 c + 1 if n is even. Exampe (b = c = 1): x 3 = x x 1. x = x x 2 = x 2 +1 x x 2 = x 1 + x x 1 x 2. x 5 = x + 1 x 3 = x 1 +x 2 +1 x 1 x (x 2 + 1)/x 1 = Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

36 Exampe 2: Rank 2 Custe Agebas [ ] 0 b Let B =, b,c Z c 0 >0. ({x 1,x 2 },B) is a seed fo a custe ageba of ank 2. µ 1 (B) = µ 2 (B) = B and x 1 x 1 = x c 2 + 1, x 2 x 2 = 1 + x b 1. Thus the custe vaiabes in this case ae { {x n : n Z} satisfying x n x n 2 = xn 1 b + 1 if n is odd xn 1 c + 1 if n is even. Exampe (b = c = 1): x 3 = x x 1. x = x x 2 = x 2 +1 x x 2 = x 1 + x x 1 x 2. x 5 = x + 1 x 3 = x 1 +x 2 +1 x 1 x = x 1(x 1 + x x 1 x 2 ) = (x 2 + 1)/x 1 x 1 x 2 (x 2 + 1) Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

37 Exampe 2: Rank 2 Custe Agebas [ ] 0 b Let B =, b,c Z c 0 >0. ({x 1,x 2 },B) is a seed fo a custe ageba of ank 2. µ 1 (B) = µ 2 (B) = B and x 1 x 1 = x c 2 + 1, x 2 x 2 = 1 + x b 1. Thus the custe vaiabes in this case ae { {x n : n Z} satisfying x n x n 2 = xn 1 b + 1 if n is odd xn 1 c + 1 if n is even. Exampe (b = c = 1): (Finite Type, of Type A 2 ) x 3 = x x 1. x = x x 2 = x 2 +1 x x 2 = x 1 + x x 1 x 2. x 5 = x + 1 x 3 = x 1 +x 2 +1 x 1 x = x 1(x 1 + x x 1 x 2 ) = x (x 2 + 1)/x 1 x 1 x 2 (x 2 + 1) x 2 Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

38 Exampe 2: Rank 2 Custe Agebas [ ] 0 b Let B =, b,c Z c 0 >0. ({x 1,x 2 },B) is a seed fo a custe ageba of ank 2. µ 1 (B) = µ 2 (B) = B and x 1 x 1 = x c 2 + 1, x 2 x 2 = 1 + x b 1. Thus the custe vaiabes in this case ae { {x n : n Z} satisfying x n x n 2 = xn 1 b + 1 if n is odd xn 1 c + 1 if n is even. Exampe (b = c = 1): (Finite Type, of Type A 2 ) x 3 = x x 1. x = x x 2 = x 2 +1 x x 2 = x 1 + x x 1 x 2. x 5 = x + 1 x 3 = x 1 +x 2 +1 x 1 x = x 1(x 1 + x x 1 x 2 ) = x x 6 = x 1. (x 2 + 1)/x 1 x 1 x 2 (x 2 + 1) x 2 Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

39 Custe Expansions and the Lauent Phenomenon Exampe 3. Let A be the custe ageba defined by the initia custe {x 1,x 2,x 3,y 1,y 2,y 3 } and the initia exchange patten x 1 x 1 = y 1 + x 2, x 2 x 2 = x 1x 3 y 2 + 1, x 3 x 3 = y 3 + x Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

40 Custe Expansions and the Lauent Phenomenon Exampe 3. Let A be the custe ageba defined by the initia custe {x 1,x 2,x 3,y 1,y 2,y 3 } and the initia exchange patten x 1 x 1 = y 1 + x 2, x 2 x 2 = x 1x 3 y 2 + 1, x 3 x 3 = y 3 + x A is of finite type, type A 3 and coesponds to a tianguated hexagon. { x 1,x 2,x 3, y 1 + x 2, x 1x 3 y 2 + 1, y 3 + x 2, x 1x 3 y 1 y 2 + y 1 + x 2, x 1 x 2 x 3 x 1 x 2 x 1 x 3 y 2 y 3 + y 3 + x 2 x 2 x 3, x 1x 3 y 1 y 2 y 3 + y 1 y 3 + x 2 y 3 + x 2 y 1 + x 2 2 x 1 x 2 x 3 The y i s ae known as pincipa coefficients. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29 }.

41 The Positivity Conjectue of Fomin and Zeevinsky Theoem. (The Lauent Phenomenon FZ 2001) Fo any custe ageba defined by initia seed ({x 1,x 2,...,x n+m },B), a custe vaiabes of A(B) ae Lauent poynomias in {x 1,x 2,...,x n+m } (with no coefficient x n+1,...,x n+m in the denominato). Because of the Lauent Phenomenon, any custe vaiabe x α can be expessed as Pα(x 1,...,x n+m) whee P α Z[x 1,...,x n+m ] and the α i s Z. x α 1 1 xn αn Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

42 The Positivity Conjectue of Fomin and Zeevinsky Theoem. (The Lauent Phenomenon FZ 2001) Fo any custe ageba defined by initia seed ({x 1,x 2,...,x n+m },B), a custe vaiabes of A(B) ae Lauent poynomias in {x 1,x 2,...,x n+m } (with no coefficient x n+1,...,x n+m in the denominato). Because of the Lauent Phenomenon, any custe vaiabe x α can be expessed as Pα(x 1,...,x n+m) whee P α Z[x 1,...,x n+m ] and the α i s Z. x α 1 1 xn αn Conjectue. (FZ 2001) Fo any custe vaiabe x α and any initia seed (i.e. initia custe {x 1,...,x n+m } and initia exchange patten B), the poynomia P α (x 1,...,x n ) has nonnegative intege coefficients. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

43 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 2002] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G 2,n+3 ). Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

44 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 2002] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G 2,n+3 ). [FZ 2002] poved positivity fo finite type with bipatite seed. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

45 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 2002] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G 2,n+3 ). [FZ 2002] poved positivity fo finite type with bipatite seed. [M-Popp 2003, Sheman-Zeevinsky 2003] poved positivity fo ank two affine custe agebas. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

46 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 2002] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G 2,n+3 ). [FZ 2002] poved positivity fo finite type with bipatite seed. [M-Popp 2003, Sheman-Zeevinsky 2003] poved positivity fo ank two affine custe agebas. Othe ank two cases by [Dupont 2009]. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

47 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 2002] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G 2,n+3 ). [FZ 2002] poved positivity fo finite type with bipatite seed. [M-Popp 2003, Sheman-Zeevinsky 2003] poved positivity fo ank two affine custe agebas. Othe ank two cases by [Dupont 2009]. Wok towads positivity fo acycic seeds [Cadeo-Reineke 2006]. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

48 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 2002] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G 2,n+3 ). [FZ 2002] poved positivity fo finite type with bipatite seed. [M-Popp 2003, Sheman-Zeevinsky 2003] poved positivity fo ank two affine custe agebas. Othe ank two cases by [Dupont 2009]. Wok towads positivity fo acycic seeds [Cadeo-Reineke 2006]. Positivity fo custe agebas incuding a bipatite seed (which is necessaiy acycic) by [Nakajima 2009]. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

49 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 2002] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G 2,n+3 ). [FZ 2002] poved positivity fo finite type with bipatite seed. [M-Popp 2003, Sheman-Zeevinsky 2003] poved positivity fo ank two affine custe agebas. Othe ank two cases by [Dupont 2009]. Wok towads positivity fo acycic seeds [Cadeo-Reineke 2006]. Positivity fo custe agebas incuding a bipatite seed (which is necessaiy acycic) by [Nakajima 2009]. Custe agebas aising fom unpunctued sufaces [Schiffe-Thomas 2007, Schiffe 2008], geneaizing Tais mode of Cao-Pice. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

50 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 2002] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G 2,n+3 ). [FZ 2002] poved positivity fo finite type with bipatite seed. [M-Popp 2003, Sheman-Zeevinsky 2003] poved positivity fo ank two affine custe agebas. Othe ank two cases by [Dupont 2009]. Wok towads positivity fo acycic seeds [Cadeo-Reineke 2006]. Positivity fo custe agebas incuding a bipatite seed (which is necessaiy acycic) by [Nakajima 2009]. Custe agebas aising fom unpunctued sufaces [Schiffe-Thomas 2007, Schiffe 2008], geneaizing Tais mode of Cao-Pice. Gaph theoetic intepetation fo unpunctued sufaces [M-Schiffe 2008]. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

51 Some Pio Wok on Positivity Conjectue Wok of [Cao-Pice 2002] gave expansion fomuas fo case of Ptoemy agebas, custe agebas of type A n with bounday coefficients (G 2,n+3 ). [FZ 2002] poved positivity fo finite type with bipatite seed. [M-Popp 2003, Sheman-Zeevinsky 2003] poved positivity fo ank two affine custe agebas. Othe ank two cases by [Dupont 2009]. Wok towads positivity fo acycic seeds [Cadeo-Reineke 2006]. Positivity fo custe agebas incuding a bipatite seed (which is necessaiy acycic) by [Nakajima 2009]. Custe agebas aising fom unpunctued sufaces [Schiffe-Thomas 2007, Schiffe 2008], geneaizing Tais mode of Cao-Pice. Gaph theoetic intepetation fo unpunctued sufaces [M-Schiffe 2008]. Positivity fo abitay sufaces [M-Schiffe-Wiiams 2009]. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

52 Main Theoem Theoem. (Positivity fo custe agebas fom sufaces MSW 2009) Let A be any custe ageba aising fom a suface (with o without punctues), whee the coefficient system is of geometic type, and et Σ be any initia seed. Then the Lauent expansion of evey custe vaiabe with espect to the seed Σ has non-negative coefficients. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

53 Main Theoem Theoem. (Positivity fo custe agebas fom sufaces MSW 2009) Let A be any custe ageba aising fom a suface (with o without punctues), whee the coefficient system is of geometic type, and et Σ be any initia seed. Then the Lauent expansion of evey custe vaiabe with espect to the seed Σ has non-negative coefficients. We pove this theoem by exhibiting a gaph theoetic intepetation fo the Lauent expansions coesponding to custe vaiabes. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

54 Main Theoem Theoem. (Positivity fo custe agebas fom sufaces MSW 2009) Let A be any custe ageba aising fom a suface (with o without punctues), whee the coefficient system is of geometic type, and et Σ be any initia seed. Then the Lauent expansion of evey custe vaiabe with espect to the seed Σ has non-negative coefficients. We pove this theoem by exhibiting a gaph theoetic intepetation fo the Lauent expansions coesponding to custe vaiabes. Due to wok of Feikson-Shapio-Tumakin, we get Cooay. Positivity fo any seed, fo a but 11 skew-symmetic custe agebas of finite mutation type. (Rank two skew-symmetic cases by Cadeo-Reineke) Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

55 Custe Agebas of Tianguated Sufaces We foow (Fomin-Shapio-Thuston), based on eaie wok of Fock-Gonchaov and Gekhtman-Shapio-Vainshtein. We have a suface S with a set of maked points M. (If P M is in the inteio of S, i.e. S \ δs, then P is known an a punctue). Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

56 Custe Agebas of Tianguated Sufaces We foow (Fomin-Shapio-Thuston), based on eaie wok of Fock-Gonchaov and Gekhtman-Shapio-Vainshtein. We have a suface S with a set of maked points M. (If P M is in the inteio of S, i.e. S \ δs, then P is known an a punctue). An ac γ satisfies (we cae about acs up to isotopy) 1 The endpoints of γ ae in M. 2 γ does not coss itsef. 3 except fo the endpoints, γ is disjoint fom M and the bounday of S. γ does not cut out an unpunctued monogon o unpunctued bigon. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

57 Custe Agebas of Tianguated Sufaces We foow (Fomin-Shapio-Thuston), based on eaie wok of Fock-Gonchaov and Gekhtman-Shapio-Vainshtein. We have a suface S with a set of maked points M. (If P M is in the inteio of S, i.e. S \ δs, then P is known an a punctue). An ac γ satisfies (we cae about acs up to isotopy) 1 The endpoints of γ ae in M. 2 γ does not coss itsef. 3 except fo the endpoints, γ is disjoint fom M and the bounday of S. γ does not cut out an unpunctued monogon o unpunctued bigon. Seed Tianguation T = { 1, 2,..., n } Custe Vaiabe Ac γ (x i i T) Custe Mutation Ptoemy Exchanges (Fipping Diagonas). Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

58 Exampe of Hexagon Conside the tianguated hexagon (S,M) with tianguation T H γ x 1 x 1 = y 1 (x 7 x 9 ) + x 2 (x 8 ) Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

59 Exampe of Hexagon Conside the tianguated hexagon (S,M) with tianguation T H x 1 x 1 = y 1 (x 7 x 9 ) + x 2 (x 8 ) x 2 x 2 = y 1 y 2 x 3 (x 9 ) + x 1 (x ) Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

60 Exampe of Hexagon Conside the tianguated hexagon (S,M) with tianguation T H γ = x 1 x 1 = y 1 (x 7 x 9 ) + x 2 (x 8 ) x 2 x 2 = y 1 y 2 x 3 (x 9 ) + x 1 (x ) x 3 x 3 = y 3 x 2(x 6 ) + x 1(x 5 ) Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

61 Exampe of Hexagon (continued) γ 2 By using the Ptoemy eations on 1, 2, then 3, we obtain x 3 = x γ = 1 ( x x 1 x 2 x 2(x 2 5 x 8 ) + y 1 x 2 (x 5 x 7 x 9 ) + y 3 x 2 (x x 6 x 8 ) 3 ) + y 1 y 3 (x x 6 x 7 x 9 ) + y 1 y 2 y 3 x 1 x 3 (x 6 x 9 ). 6 Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, /

62 Exampe of Hexagon (continued) Conside the gaph G TH,γ = G TH,γ has five pefect matchings (x,x 5,...,x 9 = 1): (x 9 )x 1 x 3 (x 6 ), (x 9 x 7 x x 6 ), x 2 (x 8 )(x x 6 ), (x 9 x 7 )x 2 (x 5 ), x 2 (x 8 )x 2 (x 5 ). A pefect matching M E is a set of distinguished edges so that evey vetex of V is coveed exacty once. The weight of a matching M is the poduct of the weights of the constituent edges, i.e. x(m) = e M x(e). Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

63 Exampe of Hexagon (continued) Conside the gaph G TH,γ = G TH,γ has five pefect matchings (x,x 5,...,x 9 = 1): (x 9 )x 1 x 3 (x 6 ), (x 9 x 7 x x 6 ), x 2 (x 8 )(x x 6 ), (x 9 x 7 )x 2 (x 5 ), x 1 x 3 y 1 y 2 y 3 +y 1 y 3 +x 2 y 3 +x 2 y 1 +x2 x 2 (x 8 )x 2 (x 5 ). 2 x 1 x 2 x 3 These five monomias exacty match those appeaing in the numeato of the expansion of x γ. The denominato of x 1 x 2 x 3 coesponds to the abes of the thee ties. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

64 A Gaph Theoetic Appoach Fo evey tianguation T (in a suface with o without punctues) and an odinay ac γ though odinay tianges, we constuct a snake gaph G T,γ such that pefect matching M of G x γ = T,γ x(m)y(m) x e 1(T,γ) 1 x e 2(T,γ) 2 xn en(t,γ). x γ is custe vaiabe (coesp. to γ w..t. seed given by T) with pincipa coefficients. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

65 A Gaph Theoetic Appoach Fo evey tianguation T (in a suface with o without punctues) and an odinay ac γ though odinay tianges, we constuct a snake gaph G T,γ such that pefect matching M of G x γ = T,γ x(m)y(m) x e 1(T,γ) 1 x e 2(T,γ) 2 xn en(t,γ). x γ is custe vaiabe (coesp. to γ w..t. seed given by T) with pincipa coefficients. e i (T,γ) is the cossing numbe of i and γ (min. int. numbe), x(m) is the weight of M, y(m) is the height of M (to be defined ate), Simia fomua wi hod fo non-odinay acs (o though sef-foded tianges). Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

66 Exampes of G T,γ Exampe 1. Using the above constuction fo γ : Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

67 Exampes of G T,γ Exampe 1. Using the above constuction fo γ : Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

68 Exampes of G T,γ Exampe 1. Using the above constuction fo 8 γ : , Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

69 Exampes of G T,γ Exampe 1. Using the above constuction fo γ : , , Thus G TH,γ = x γ y1 =y 2 =y 3 =1 = x 1x x 2 + x 2 2 x 1 x 2 x 3 Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

70 Height Functions (of Pefect Matchings of Snake Gaphs) We now wish to give fomua fo y(m) s, i.e. the tems in the F-poynomias. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

71 Height Functions (of Pefect Matchings of Snake Gaphs) We now wish to give fomua fo y(m) s, i.e. the tems in the F-poynomias. We use height functions which ae due to Wiiam Thuston, and Conway-Lagaias. Invoves measuing contast between a given pefect matching M and a fixed minima matching M. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

72 Height Function Exampes Reca that G TH,γ has thee faces, abeed 1, 2 and 3. G TH,γ has five pefect matchings (x,x 5,...,x 9 = 1): x γ = x 1x 3 y 1 y 2 y 3 + y 1 y 3 + x 2 y 3 + x 2 y 1 + x 2 2 x 1 x 2 x 3 y 1 y 2 y 3, y 1 y 3, y 3, y 1, 1 This matching is M. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

73 Height Function Exampes Reca that G TH,γ has thee faces, abeed 1, 2 and 3. G TH,γ has five pefect matchings (x,x 5,...,x 9 = 1): x γ = x 1x 3 y 1 y 2 y 3 + y 1 y 3 + x 2 y 3 + x 2 y 1 + x 2 2 x 1 x 2 x 3 y 1 y 2 y 3, y 1 y 3, y 3, y 1, 1 This matching is M ,, and 3 Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

74 Height Function Exampes (continued) Fo G TA,γ = 5 1 2, M is Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

75 Height Function Exampes (continued) Fo G TA,γ = 5 1 2, M is One of the matchings, M, is 1 2 3, Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

76 Height Function Exampes (continued) Fo G TA,γ = 5 1 2, M is One of the matchings, M, is 1 2 3, so M M = 1 2 3, Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

77 Height Function Exampes (continued) Fo G TA,γ = 5 1 2, M is One of the matchings, M, is 1 2 3, so M M = 1 2 3, which has height y 1 y2 2. So one of the 17 tems in the custe expansion of x x γ is (using FST convention) (x 6 x 8 )x (x 5 )x 2 (x 8 ) x1 2x2 2 x (y 1 y 2 3x 2 ). Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

78 Summay Theoem. (M-Schiffe-Wiiams 2009) Fo evey tianguation T of a suface (with o without punctues) and an odinay ac γ, we constuct a snake gaph G γ,t such that pefect matching M of G γ,t x(m)y(m) x γ = x e 1(T,γ) 1 x e 2(T,γ) 2 xn en(t,γ) Hee e i (T,γ) is the cossing numbe of i and γ, x(m) is the edge-weight of pefect matching M, and y(m) is the height of pefect matching M. (x γ is custe vaiabe with pincipa coefficients.). Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

79 Summay Theoem. (M-Schiffe-Wiiams 2009) Fo evey tianguation T of a suface (with o without punctues) and an odinay ac γ, we constuct a snake gaph G γ,t such that pefect matching M of G γ,t x(m)y(m) x γ = x e 1(T,γ) 1 x e 2(T,γ) 2 xn en(t,γ) Hee e i (T,γ) is the cossing numbe of i and γ, x(m) is the edge-weight of pefect matching M, and y(m) is the height of pefect matching M. (x γ is custe vaiabe with pincipa coefficients.) Theoem. (M-Schiffe-Wiiams 2009) An anaogous expansion fomua hods fo acs with notches (ony aise in the case of a punctued suface).. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

80 Summay Theoem. (M-Schiffe-Wiiams 2009) Fo evey tianguation T of a suface (with o without punctues) and an odinay ac γ, we constuct a snake gaph G γ,t such that pefect matching M of G γ,t x(m)y(m) x γ = x e 1(T,γ) 1 x e 2(T,γ) 2 xn en(t,γ) Hee e i (T,γ) is the cossing numbe of i and γ, x(m) is the edge-weight of pefect matching M, and y(m) is the height of pefect matching M. (x γ is custe vaiabe with pincipa coefficients.) Theoem. (M-Schiffe-Wiiams 2009) An anaogous expansion fomua hods fo acs with notches (ony aise in the case of a punctued suface). Cooay. The F-poynomia equas M y(m), is positive, and has constant tem 1. The g-vecto satisfies x g = x(m ). Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29.

81 Exampe 2 (Odinay Ac though Sef-foded Tiange) (S, M) 11 1 γ p Figue: Idea Tianguation T of (S, M) and coesponding Snake Gaph G T,γ 1. Note the thee consecutive ties of ou snake gaph with abes, and, as γ 1 taveses the oop twice and the encosed adius. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

82 Exampe 2 (Odinay Ac though Sef-foded Tiange) (S, M) γ p Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

83 Exampe 2 (Odinay Ac though Sef-foded Tiange) (S, M) γ p Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

84 Exampe 2 (Odinay Ac though Sef-foded Tiange) (S, M) γ p Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

85 Exampe 2 (Odinay Ac though Sef-foded Tiange) (S, M) γ p Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

86 Exampe 2 (Odinay Ac though Sef-foded Tiange) (S, M) γ p Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

87 Exampe 2 (Odinay Ac though Sef-foded Tiange) (S, M) γ p Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

88 Exampe 2 (Odinay Ac though Sef-foded Tiange) (S, M) γ p Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

89 Exampe 3 (Notched Ac in Punctued Suface) (S, M) γ p Figue: Idea Tianguation T of (S, M) and coesponding Snake Gaph G T,γ 2. We obtain the Lauent expansion fo x γ2 by summing ove so caed γ-symmetic matchings of G T,γ 2, those that agee on the two bod ends. Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

90 Thank You Fo Listening Positivity fo Custe Agebas fom Sufaces (with Raf Schiffe and Lauen Wiiams), axiv:math.co/ Custe Expansion Fomuas and Pefect Matchings (with Raf Schiffe), musike/pm.pdf (To appea in the Jouna of Agebaic Combinatoics) A Gaph Theoetic Expansion Fomua fo Custe Agebas of Cassica Type, musike/finite.pdf (To appea in the Annas of Combinatoics) Fomin, Shapio, and Thuston. Custe Agebas and Tianguated Sufaces I: Custe Compexes, Acta Math. 201 (2008), no. 1, Fomin and Zeevinsky. Custe Agebas IV: Coefficients, Compos. Math. 13 (2007), no. 1, Sides Avaiabe at http//math.mit.edu/ musike/custesuface.pdf Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

91 Encoe Exampe of G T,γ : Annuus Exampe. We now constuct gaph G TA,γ. 1 γ Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29

92 Encoe Exampe of G T,γ : Annuus Exampe. We now constuct gaph G TA,γ. γ Musike (MIT) Positivity fo custe agebas fom sufaces Febuay 9, / 29