Graph Theoretical Cluster Expansion Formulas

Transcription

1 Graph Theoretical Cluster Expansion Formulas Gregg Musiker MIT December 9, 008 http//math.mit.edu/ musiker/graphtalk.pdf Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

2 Outline. Introduction Snake Graphs for Surfaces without Punctures Graphs for the Classical Types (Bipartite Seeds) Other Examples of Graph Theoretic Interpretations Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

3 Cluster Expansions Definition (Sergey Fomin and Andrei Zelevinsky 00) A cluster algebra A is a certain subalgebra of k(x,...,x m ), the field of rational functions over {x,...,x m }. Generators constructed by a series of exchange relations, which in turn induce all relations satisfied by the generators. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

4 Cluster Expansions Definition (Sergey Fomin and Andrei Zelevinsky 00) A cluster algebra A is a certain subalgebra of k(x,...,x m ), the field of rational functions over {x,...,x m }. Generators constructed by a series of exchange relations, which in turn induce all relations satisfied by the generators. Theorem. (The Laurent Phenomenon FZ 00) For any cluster algebra defined by initial seed ({x,x,...,x m },B), all cluster variables of A(B) are Laurent polynomials in {x,x,...,x m } (with no coefficient x n+,...,x m in the denominator). Thus, any cluster variable x α = Pα(x,...,x m) x α xn αn where P α Z[x,...,x n ]. (We use the notation x α since we only consider cases in this talk where denominator defines cluster variable.) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

5 Cluster Expansions Definition (Sergey Fomin and Andrei Zelevinsky 00) A cluster algebra A is a certain subalgebra of k(x,...,x m ), the field of rational functions over {x,...,x m }. Generators constructed by a series of exchange relations, which in turn induce all relations satisfied by the generators. Theorem. (The Laurent Phenomenon FZ 00) For any cluster algebra defined by initial seed ({x,x,...,x m },B), all cluster variables of A(B) are Laurent polynomials in {x,x,...,x m } (with no coefficient x n+,...,x m in the denominator). Thus, any cluster variable x α = Pα(x,...,x m) x α xn αn where P α Z[x,...,x n ]. (We use the notation x α since we only consider cases in this talk where denominator defines cluster variable.) Conjecture. (Positivity Conjecture FZ 00) For any cluster variable x α the polynomial P α (x,...,x n ) has nonnegative integer coefficients. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

6 Some Prior Work on Positivity Conjecture Work of [Carroll-Price 00] gave expansion formulas for case of Ptolemy algeras, cluster algebras of type A n with boundary coefficients. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

7 Some Prior Work on Positivity Conjecture Work of [Carroll-Price 00] gave expansion formulas for case of Ptolemy algeras, cluster algebras of type A n with boundary coefficients. [FZ 00] proved positivity for finite type with bipartite seed. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

8 Some Prior Work on Positivity Conjecture Work of [Carroll-Price 00] gave expansion formulas for case of Ptolemy algeras, cluster algebras of type A n with boundary coefficients. [FZ 00] proved positivity for finite type with bipartite seed. [M-Propp 00, Sherman-Zelevinsky 00] proved positivity for rank two affine cluster algebras. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

9 Some Prior Work on Positivity Conjecture Work of [Carroll-Price 00] gave expansion formulas for case of Ptolemy algeras, cluster algebras of type A n with boundary coefficients. [FZ 00] proved positivity for finite type with bipartite seed. [M-Propp 00, Sherman-Zelevinsky 00] proved positivity for rank two affine cluster algebras. [Caldero-Zelevinsky 006] combinatorial formulas for Euler characteristic of Quiver Grassmannian of Kronecker Quiver. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

10 Some Prior Work on Positivity Conjecture Work of [Carroll-Price 00] gave expansion formulas for case of Ptolemy algeras, cluster algebras of type A n with boundary coefficients. [FZ 00] proved positivity for finite type with bipartite seed. [M-Propp 00, Sherman-Zelevinsky 00] proved positivity for rank two affine cluster algebras. [Caldero-Zelevinsky 006] combinatorial formulas for Euler characteristic of Quiver Grassmannian of Kronecker Quiver. Positivity also proven for those cluster variables for an acyclic seed [Caldero-Reineke 006], Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

11 Some Prior Work on Positivity Conjecture Work of [Carroll-Price 00] gave expansion formulas for case of Ptolemy algeras, cluster algebras of type A n with boundary coefficients. [FZ 00] proved positivity for finite type with bipartite seed. [M-Propp 00, Sherman-Zelevinsky 00] proved positivity for rank two affine cluster algebras. [Caldero-Zelevinsky 006] combinatorial formulas for Euler characteristic of Quiver Grassmannian of Kronecker Quiver. Positivity also proven for those cluster variables for an acyclic seed [Caldero-Reineke 006], as well as for Cluster algebras arising from unpunctured surfaces [Schiffler-Thomas 007, Schiffler 008], generalizing Trails model of Carroll-Price. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

12 Cluster Algebras of Triangulated Surfaces We follow (Fomin-Shapiro-Thurston) and have a surface (S,M). We assume marked points M S (no punctures). Recall an arc γ satisfies (we care about arcs up to isotopy) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

13 Cluster Algebras of Triangulated Surfaces We follow (Fomin-Shapiro-Thurston) and have a surface (S,M). We assume marked points M S (no punctures). Recall an arc γ satisfies (we care about arcs up to isotopy) The endpoints of γ are in M. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

14 Cluster Algebras of Triangulated Surfaces We follow (Fomin-Shapiro-Thurston) and have a surface (S,M). We assume marked points M S (no punctures). Recall an arc γ satisfies (we care about arcs up to isotopy) The endpoints of γ are in M. γ does not cross itself. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

15 Cluster Algebras of Triangulated Surfaces We follow (Fomin-Shapiro-Thurston) and have a surface (S,M). We assume marked points M S (no punctures). Recall an arc γ satisfies (we care about arcs up to isotopy) The endpoints of γ are in M. γ does not cross itself. relative interior of γ is disjoint from M and the boundary of S. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

16 Cluster Algebras of Triangulated Surfaces We follow (Fomin-Shapiro-Thurston) and have a surface (S,M). We assume marked points M S (no punctures). Recall an arc γ satisfies (we care about arcs up to isotopy) The endpoints of γ are in M. γ does not cross itself. relative interior of γ is disjoint from M and the boundary of S. γ does not cut out a monogon or digon. Seed Triangulation T = {,,..., n } Cluster Variable Arc γ x i i T. For γ T let e i (T : γ) be the minimal intersection number of i and γ. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

17 A Graph Theoretic Approach Recall from Ralf Schiffler s Talk: Theorem. (M-Schiffler 008) For every triangulation T (in a surface without punctures) and arc γ, we construct a snake graph G γ,t such that perfect matching M of G x γ = γ,t x(m)y(m) x e (T,γ) x e (T,γ) xn en(t,γ) where e i (T,γ) is the crossing number of i and γ, and x(m), y(m) are each monomials. (x γ is cluster variable with principal coefficients.) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

18 A Graph Theoretic Approach Recall from Ralf Schiffler s Talk: Theorem. (M-Schiffler 008) For every triangulation T (in a surface without punctures) and arc γ, we construct a snake graph G γ,t such that perfect matching M of G x γ = γ,t x(m)y(m) x e (T,γ) x e (T,γ) xn en(t,γ) where e i (T,γ) is the crossing number of i and γ, and x(m), y(m) are each monomials. (x γ is cluster variable with principal coefficients.) Definition. Given a simple undirected graph G = (V,E), a perfect matching M E is a set of distinguished edges so that every vertex of V is covered exactly once. (Each edge has weight x(e) where x(e) is allowed to be (unweighted) or some variable x i.) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

19 A Graph Theoretic Approach Recall from Ralf Schiffler s Talk: Theorem. (M-Schiffler 008) For every triangulation T (in a surface without punctures) and arc γ, we construct a snake graph G γ,t such that perfect matching M of G x γ = γ,t x(m)y(m) x e (T,γ) x e (T,γ) xn en(t,γ) where e i (T,γ) is the crossing number of i and γ, and x(m), y(m) are each monomials. (x γ is cluster variable with principal coefficients.) Definition. Given a simple undirected graph G = (V,E), a perfect matching M E is a set of distinguished edges so that every vertex of V is covered exactly once. (Each edge has weight x(e) where x(e) is allowed to be (unweighted) or some variable x i.) The weight of a matching M is the product of the weights of the constituent edges, i.e. x(m) = e M x(e). Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

20 Example of Octagon 8 9 γ Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

21 Example of Octagon 8 9 γ Recall that are 5 completed (T,γ)-paths of this octagon, with weights x + x x + x x + x x + x x 5 x x x 5. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

22 Example of Octagon (continued) 5 Consider the graph G TO,γ = G TO,γ has five perfect matchings (x 7,x 8,...,x = ): x (x 8 )x (x ), x (x 7 )x (x ), x (x 8 )x (x ), (x 7 )x x (x ), (x 7 )x x 5 (x ). Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

23 Example of Octagon (continued) 5 Consider the graph G TO,γ = G TO,γ has five perfect matchings (x 7,x 8,...,x = ): x (x 8 )x (x ), x (x 7 )x (x ), x (x 8 )x (x ), (x 7 )x x (x ), (x 7 )x x 5 (x ). Dividing each monomial by x x x 5, we obtain weights of (T,γ)-paths. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

24 How to construct G T,γ s (unpunctured surfaces) Definition. For i n (i.e. all i T), define Tile S i to be (weighted) triangulated quadrilateral homeomorphic to the quadrilateral bounding arc i in surface S. (Diagonal NW SE and opposite sides still opposite) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

25 How to construct G T,γ s (unpunctured surfaces) Definition. For i n (i.e. all i T), define Tile S i to be (weighted) triangulated quadrilateral homeomorphic to the quadrilateral bounding arc i in surface S. (Diagonal NW SE and opposite sides still opposite) Now given arc γ: Pick orientation of γ : s t. Label p 0 = s,p,...,p d,p d+ = t, the intersection points of γ with T (p j ij ). Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

26 How to construct G T,γ s (unpunctured surfaces) Definition. For i n (i.e. all i T), define Tile S i to be (weighted) triangulated quadrilateral homeomorphic to the quadrilateral bounding arc i in surface S. (Diagonal NW SE and opposite sides still opposite) Now given arc γ: Pick orientation of γ : s t. Label p 0 = s,p,...,p d,p d+ = t, the intersection points of γ with T (p j ij ). Let i (for j d ) denote the triangles bounded by arcs ij and ij+. ( 0 and d denote the first and last triangles that γ traverses.) Let [γ j ] denote the third side of j for j d. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

27 How to construct G T,γ s (unpunctured surfaces) Definition. For i n (i.e. all i T), define Tile S i to be (weighted) triangulated quadrilateral homeomorphic to the quadrilateral bounding arc i in surface S. (Diagonal NW SE and opposite sides still opposite) Now given arc γ: Pick orientation of γ : s t. Label p 0 = s,p,...,p d,p d+ = t, the intersection points of γ with T (p j ij ). Let i (for j d ) denote the triangles bounded by arcs ij and ij+. ( 0 and d denote the first and last triangles that γ traverses.) Let [γ j ] denote the third side of j for j d. 5 By convention Let G γ, := S i. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

28 How to construct G T,γ s (unpunctured surfaces) Definition. For i n (i.e. all i T), define Tile S i to be (weighted) triangulated quadrilateral homeomorphic to the quadrilateral bounding arc i in surface S. (Diagonal NW SE and opposite sides still opposite) Now given arc γ: Pick orientation of γ : s t. Label p 0 = s,p,...,p d,p d+ = t, the intersection points of γ with T (p j ij ). Let i (for j d ) denote the triangles bounded by arcs ij and ij+. ( 0 and d denote the first and last triangles that γ traverses.) Let [γ j ] denote the third side of j for j d. 5 By convention Let G γ, := S i. 6 Inductively attach tile S ij+ to graph G γ,j to obtain G γ,j+. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

29 How to construct G T,γ s (unpunctured surfaces) Definition. For i n (i.e. all i T), define Tile S i to be (weighted) triangulated quadrilateral homeomorphic to the quadrilateral bounding arc i in surface S. (Diagonal NW SE and opposite sides still opposite) Now given arc γ: Pick orientation of γ : s t. Label p 0 = s,p,...,p d,p d+ = t, the intersection points of γ with T (p j ij ). Let i (for j d ) denote the triangles bounded by arcs ij and ij+. ( 0 and d denote the first and last triangles that γ traverses.) Let [γ j ] denote the third side of j for j d. 5 By convention Let G γ, := S i. 6 Inductively attach tile S ij+ to graph G γ,j to obtain G γ,j+. (N or E edge of G γ,j agrees with tile S ij+ : N E and S W ) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

30 How to construct G T,γ s (unpunctured surfaces) Definition. For i n (i.e. all i T), define Tile S i to be (weighted) triangulated quadrilateral homeomorphic to the quadrilateral bounding arc i in surface S. (Diagonal NW SE and opposite sides still opposite) Now given arc γ: Pick orientation of γ : s t. Label p 0 = s,p,...,p d,p d+ = t, the intersection points of γ with T (p j ij ). Let i (for j d ) denote the triangles bounded by arcs ij and ij+. ( 0 and d denote the first and last triangles that γ traverses.) Let [γ j ] denote the third side of j for j d. 5 By convention Let G γ, := S i. 6 Inductively attach tile S ij+ to graph G γ,j to obtain G γ,j+. (N or E edge of G γ,j agrees with tile S ij+ : N E and S W ) 7 We define G T,γ to be G γ,d. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

31 How to construct G T,γ s (unpunctured surfaces) Definition. For i n (i.e. all i T), define Tile S i to be (weighted) triangulated quadrilateral homeomorphic to the quadrilateral bounding arc i in surface S. (Diagonal NW SE and opposite sides still opposite) Now given arc γ: Pick orientation of γ : s t. Label p 0 = s,p,...,p d,p d+ = t, the intersection points of γ with T (p j ij ). Let i (for j d ) denote the triangles bounded by arcs ij and ij+. ( 0 and d denote the first and last triangles that γ traverses.) Let [γ j ] denote the third side of j for j d. 5 By convention Let G γ, := S i. 6 Inductively attach tile S ij+ to graph G γ,j to obtain G γ,j+. (N or E edge of G γ,j agrees with tile S ij+ : N E and S W ) 7 We define G T,γ to be G γ,d. (Erase diagonals to obtain G T,γ.) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

32 Examples of G T,γ 8 9 γ 7 6 Example. Use above construction for 0 5 : Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

33 Examples of G T,γ 8 9 γ 7 6 Example. Use above construction for 0 5 : 7 8 Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

34 Examples of G T,γ 8 9 γ 7 6 Example. Use above construction for 0 5 : , 8 Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

35 Examples of G T,γ 8 9 γ 7 6 Example. Use above construction for 0 5 : , 8 7, Thus 5 G TO,γ = Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

36 Examples of G T,γ (continued) Example. We now construct graph G TA,γ. γ Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

37 Examples of G T,γ (continued) Example. We now construct graph G TA,γ. γ Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

38 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. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

39 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. Definiton. [W. Thurston-Conway] (Following description of [Elkies-Larsen-Kuperberg-Propp]) Given a snake graph G, up to orientation, there is a choice of minimal matching (M ) which consists of every-other edge on the boundary. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

40 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. Definiton. [W. Thurston-Conway] (Following description of [Elkies-Larsen-Kuperberg-Propp]) Given a snake graph G, up to orientation, there is a choice of minimal matching (M ) which consists of every-other edge on the boundary. Given any other matching M, let M M denote the symmetric difference. The height h M : Faces(G) Z 0 of matching M is a function recording which faces are enclosed by M M. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

41 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. Definiton. [W. Thurston-Conway] (Following description of [Elkies-Larsen-Kuperberg-Propp]) Given a snake graph G, up to orientation, there is a choice of minimal matching (M ) which consists of every-other edge on the boundary. Given any other matching M, let M M denote the symmetric difference. The height h M : Faces(G) Z 0 of matching M is a function recording which faces are enclosed by M M. For snake graphs, h M (F) {0,} and we obtain the formula y(m) := i y P Faces Labeled i h M(F) i. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

42 Height Function Examples Recall that G TO,γ has three faces, labeled, and 5. G TO,γ has five perfect matchings (x 7,x 8,...,x = ): x y y y 5, x x y y 5, x x y y, x x y, x x 5 (). ( This matching is M.) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

43 Height Function Examples Recall that G TO,γ has three faces, labeled, and 5. G TO,γ has five perfect matchings (x 7,x 8,...,x = ): x y y y 5, x x y y 5, x x y y, x x y, x x 5 (). ( This matching is M.) For example, we get heights y y, y, and y y 5 because of superpositions:,, and 5 Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

44 Height Function Examples (continued) For G TA,γ = 5, M is. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

45 Height Function Examples (continued) For G TA,γ = 5, M is. One of the 7 matchings, M, is, Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

46 Height Function Examples (continued) For G TA,γ = 5, M is. One of the 7 matchings, M, is, so M M =, Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

47 Height Function Examples (continued) For G TA,γ = 5, M is. One of the 7 matchings, M, is, so M M =, which has height y y. So one of the 7 terms in the cluster expansion of x x γ is x x x x (y y x ). Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

48 Summary Theorem. (M-Schiffler 008) For every triangulation T of unpunctured surface and arc γ, we construct a snake graph G γ,t such that perfect matching M of G x γ = γ,t x(m)y(m) x e (T,γ) x e (T,γ) xn en(t,γ) where e i (T,γ) is the crossing number of i and γ, x(m) is the edge-weight of perfect matching M, and y(m) is the height of perfect matching M. (x γ is cluster variable with principal coefficients.) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

49 Summary Theorem. (M-Schiffler 008) For every triangulation T of unpunctured surface and arc γ, we construct a snake graph G γ,t such that perfect matching M of G x γ = γ,t x(m)y(m) x e (T,γ) x e (T,γ) xn en(t,γ) where e i (T,γ) is the crossing number of i and γ, x(m) is the edge-weight of perfect matching M, and y(m) is the height of perfect matching M. (x γ is cluster variable with principal coefficients.) Corollary. The F-polynomial equals M y(m), is positive, and has constant term. The g-vector satisfies x g = x(m ). Corollary. The Laurent expansion of cluster variable x γ is positive for any cluster algebra (of geometric type) arising from a triangulated surface without punctures. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

50 Partially Generalizes Earlier Work on Classical Types Theorem. (M 007) For every classical root system, let B Φ denote the corresponding bipartite seed (without coefficients). Then there exists a family of graphs G Φ = {G α } α Φ+ such that x α, the cluster variable of A(B Φ ) corresponding to α Φ +, can be expressed as x α = P G α (x,...,x n ) x α. xαn n Further, we will construct the graphs in a very simple manner using the tiles T k. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

51 Tiles for the four classical types A x x x x x x x x C x x x x x x x x x Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

52 Graphs for A n and C n A Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

53 Graphs for A n and C n (cont.) C folds onto A 5 (Take right-half including middle) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

54 Tiles for the four classical types (cont.) x x x x x x 5 x 5 x B 5 x x x x x x x x D 5 x Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

55 The B n and D n cases B After mutating with respect to x and x (x and x ), we obtain Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

56 The B n and D n cases (cont.) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

57 The B n and D n cases (cont.) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

58 D 5 The B n and D n cases (cont.) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

59 The B n and D n cases (cont.) D 5 (cont.) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

60 The B n and D n cases (cont.) D 5 (cont.) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

61 The B n and D n cases (cont.) D 5 (cont.) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

62 G Seed matrix is B = [ ] 0 0 Hexagon has x on NW, NE, and S sides, Trapezoid has x on N side. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

63 Affine Rank Joint work with Jim Propp. [ ] [ ] 0 0 Let B = or. 0 0 Here we also exploit invariance of matrices B under mutation. So we are considering (b,c)-sequence { xn b + if n odd x n x n = + if n even for (b,c) = (,) or (,). x c n Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

64 Affine Rank (cont.) Since cluster algebra structure, (b,c) sequence consists of Laurent polynomials. Work of Sherman and Zelevinsky verifies positive coefficients for (,) and (,) using Newton polytope, and Caldero-Zelevinsky give another proof of positivity for (,) case via Quiver Grassmannians. This cluster algebra also comes from an annulus with one marked point on each boundary (no punctures). Equivalently, this is a cluster algebra of affine type Ã,. We give proof of positivity via graph theoretical interpretation similar to above. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

65 Affine Rank (cont.) (,): all cluster variables have denominators x dxd+ (resp. x d+ x d) We string together corresponding number of sqares x x x x in an intertwining fashion. Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

66 Affine Rank (cont.) (,): all cluster variables have denominators x dxd+ (resp. x d+ x d) We string together corresponding number of sqares x x x x in an intertwining fashion. Examples: x + x ++x x x x 6 + x + x + x x +x ++ x x x Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

67 Affine Rank (cont.) (,): Tiles are a square and an octagon: x 0 x Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

68 Sequnce Continues x 7 terms x 5 9 terms x 6 86 terms x 7 terms Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

69 Sequnce Continues (cont.) x terms x 9 06 terms x 0 0 terms x 987 terms Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9

70 Running the (, ) sequence backwards x terms x terms x terms x 97 terms Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

71 Running the (, ) sequence backwards (cont.) x 5 67 terms x terms x 7 terms x terms Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

72 Markoff polynomials Joint work by Carroll, Itsara, Le, M, Price, Thurston, and Viana under Propp in REACH program. 0 B = 0, Exchange graph is free ternary tree. 0 B invariant under mutation. All exchanges have form (x,y,z) (x,y,z) where xx = y + z. (Cluster algebra corresponds to once punctured torus.) Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

73 Markoff polynomials Joint work by Carroll, Itsara, Le, M, Price, Thurston, and Viana under Propp in REACH program. 0 B = 0, Exchange graph is free ternary tree. 0 B invariant under mutation. All exchanges have form (x,y,z) (x,y,z) where xx = y + z. (Cluster algebra corresponds to once punctured torus.) These also have graph theoretic interpretation: Snake Graphs,.e.g Polynomial(x,y,z) x y z x y x z x y x with tiles y x y z x z z y z x z y x y x Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

74 Work in Progress with Ralf Schiffler and Lauren Williams Theorem. Formulas for F-polynomials and g-vectors for types A, B, C, D with respect to any seed (not nec. acyclic). In Progress. Snake Graph Interpretations for Triangulated Surfaces (even in prescence of punctures). Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9

75 Thank You For Listening Cluster Expansion Formulas and Perfect Matchings (with Ralf Schiffler), arxiv:math.co/ A Graph Theoretic Expansion Formula for Cluster Algebras of Classical Type, musiker/finite.pdf, (To appear in the Annals of Combinatorics) Combinatorial Interpretations for Rank-Two Cluster Algebras of Affine Type (with Jim Propp), Electronic Journal of Combinatorics. Vol. (R5), 007. The Combinatorics of Frieze Patterns and Markoff Numbers (by Jim Propp), arxiv:math.co/056 Slides Available at http//math.mit.edu/ musiker/graphtalk.pdf Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, / 9