Cluster algebras, snake graphs and continued fractions. Ralf Schiffler

Transcription

1 Cluster algebras, snake graphs and continued fractions Ralf Schiffler

2 Intro Cluster algebras Continued fractions Snake graphs

3 Intro Cluster algebras Continued fractions expansion formula via perfect matchings [Musiker-S-Williams ] Snake graphs

4 Intro Cluster algebras Continued fractions expansion formula via perfect matchings [Musiker-S-Williams ] bijection via perfect matchings [Çanakçı-S 6] Snake graphs

5 Intro expansion formula as continued fractions, asymptotic behavior [Çanakçı-S 6] Cluster algebras Continued fractions expansion formula via perfect matchings [Musiker-S-Williams ] bijection via perfect matchings [Çanakçı-S 6] Snake graphs

6 Intro expansion formula as continued fractions, asymptotic behavior [Çanakçı-S 6] Cluster algebras Continued fractions expansion formula via perfect matchings [Musiker-S-Williams ] bijection via perfect matchings [Çanakçı-S 6] Snake graphs

7 Intro expansion formula as continued fractions [Çanakçı-S 6] Cluster algebras Continued fractions expansion formula via perfect matchings [Musiker-S-Williams ] Snake graphs Combinatorial realization of continued fractions Applications in elementary Number Theory

8 Continued Fractions 8 = + 0

9 Continued Fractions 8 = + 0 = + 0

10 Continued Fractions 8 = + 0 = + 0 = + + 0

11 Continued Fractions 8 = + 0 = = + 0 = + + 0

12 Continued Fractions 8 = + 0 = = + 0 = = + + 0

13 Continued Fractions 8 = + 0 = + = = + 0 = = + + 0

14 Continued Fractions 8 = + 0 = + = = + 0 = = =: [,,,, ] +

15 Continued Fractions 8 = [,,,, ]

16 Continued Fractions - Euclidean algorithm 8 = + 0 = = + = + =

17 Continued Fractions - Euclidean algorithm 8 = + 0 = = + = + = 8 = [,,,, ]

18 Continued Fractions The division algorithm gives a bijection between Q and the set of continued fractions whose last coefficient is at least. Q {[a 0,..., a n ] a 0 Z, a,... a n, a n } note that [a 0,..., a n, ] = [a 0,..., a n + ], because a n + = an +.

19 Snake graphs A snake graph G is a connected planar graph consisting of a finite sequence of tiles G, G,..., G d such that G i and G i+ share exactly one edge e i and this edge is either the north edge of Gi and the south edge of G i+, or the east edge of Gi and the west edge of G i+.

20 Snake graphs

21 Snake graphs

22 Snake graphs

23 Snake graphs

24 Snake graphs

25 Snake graphs

26 Snake graphs

27 Snake graphs

28 Snake graphs

29 Snake graphs

30 Sign function

31 Sign function - constant on diagonals

32 Sign function - constant on diagonals

33 Sign function - constant on diagonals

34 Sign function - constant on diagonals

35 Sign function

36 Sign function sign sequence continued fraction,, +, +, +,, +, +,,, [,,,, ]

37 Theorem 8 perfect matchings [,,,, ] = 8 There is a bijection between snake graphs and continued fractions G[a, a,..., a n ] [a, a,..., a n ] Moreover, if m(g) denotes the number of perfect matchings of G then [a, a,..., a n ] = m(g[a, a,..., a n ]). m(g[a,..., a n ]) with a i, a n

38 Theorem perfect matchings [,,,, ] = 8 There is a bijection between snake graphs and continued fractions G[a, a,..., a n ] [a, a,..., a n ] Moreover, if m(g) denotes the number of perfect matchings of G then [a, a,..., a n ] = m(g[a, a,..., a n ]). m(g[a,..., a n ]) with a i, a n

39 Reflection flip [,,,, ] [,,,,, ]

40 Perfect matchings A perfect matching P of a graph G is a subset of the set of edges of G such that each vertex of G is incident to exactly one edge in P.

41 Perfect matchings A perfect matching P of a graph G is a subset of the set of edges of G such that each vertex of G is incident to exactly one edge in P.

42 Perfect matchings A perfect matching P of a graph G is a subset of the set of edges of G such that each vertex of G is incident to exactly one edge in P.

43 Perfect matchings A perfect matching P of a graph G is a subset of the set of edges of G such that each vertex of G is incident to exactly one edge in P.

44 Perfect matchings A perfect matching P of a graph G is a subset of the set of edges of G such that each vertex of G is incident to exactly one edge in P.

45 Tiles with sign changes

46 Tiles with sign changes removed

47 Tiles with sign changes removed

48 Tiles with sign changes removed

49 Tiles with sign changes removed

50 Subgraphs H i = G[ai ] H 5 H H H H

51 Division algorithm H 0 0 H 0 H H

52 Division algorithm Proof H 0 0 H 0 H H

53 Division algorithm Proof H 0 0 H 0 H H

54 Division algorithm Proof H 0 0 H 0 H H

55 Division algorithm Proof H 0 0 H 0 H H

56 Division algorithm Proof H 0 0 H 0 H H

57 Division algorithm Proof H 0 0 H 0 H H

58 Division algorithm Proof H 0 0 H 0 H H

59 Division algorithm Proof H 0 0 H 0 H H

60 Division algorithm Proof H 0 0 H 0 H H

61 Division algorithm Proof H 0 0 H 0 H H

62 Expansion formula for cluster variables Theorem (Musiker-S.-Williams) If γ is an arc in a triangulated surface (S, M), the cluster variable x γ is given by the formula x γ = cross(g γ ) P Match G γ x(p). b x a γ c d b c x a f d x e G γ x γ = x +x +x x x

63 Example a b a b a γ G γ b x x x x x γ = x ++x +x x x

64 Example Torus with puncture [,,,,,,, ] = The Laurent polynomial of this cluster variable has 985 terms.

65 Expansion formula as continued fraction Theorem (Çanakçı-S) The cluster variable x γ of the arc γ is the numerator of the continued fraction [L, L,..., L n ], where L i is a Laurent polynomial explicitly given by the subgraph H i. [Çanakçı-Felikson]

66 Example Torus with puncture [,,,,,,, ] = [a,..., a 8 ] = [,,,,,,, ], let x = x +x x, then L = x x, L = x x x, L = x x x x γ = numerator of [L, L,..., L n ],..., L 8 = x x x 8

67 Example, computation dispiace Salvatore, ho usato Mathematica + * / + * + * + * + * + * + * + *

68 What about the denominator? Torus with puncture [,,,,,,, ] = x γ = numerator of [L, L,..., L n ]

69 What about the denominator? Torus with puncture x γ = denominator of [L, L,..., L n ] = numerator of [L,..., L n ]

70 Asymptotic behavior of quotients x γ lim n x γ = x x + (x x ) + x x. where x = (x + x )/x x = (x + x )/x are obtained from the initial cluster by mutation in and then.

71 Applications to elementary Number Theory (skein) relations in the cluster algebras were expressed in terms of snake graphs [Çanakçı-S.] (snake graph calculus). This provides a long list of relations in terms of snake graphs. Translating into the language of continued fractions gives a long list of relations there.

72 Equations for continued fractions Theorem We have the following identities of numerators of continued fractions, where we set N [a,..., a 0] =, and N [a n+,..., a n] =. (a) For every i =,,..., n, N [a,..., a n] = N [a,..., a i ] N [a i+,..., a n] + N [a,..., a i ] N [a i+,..., a n]. (b) For every j 0 and i such that i + j n, N [a,..., a i+j ] N [a i,..., a n] = N [a,..., a n] N [a i,..., a i+j ] + ( ) j N [a,..., a i ] N [a i+j+,..., a n].

73 Equations for continued fractions (c) For continued fractions [a,..., a n] and [b,..., b m] such that [a i,..., a i+k ] = [b j,..., b j+k ] for certain i, j, k, we have N [a,..., a n] N [b,..., b m] = N [a,..., a i, b j,..., b m] N [b,..., b j, a i,..., a n] +( ) k N [a,..., a i,, b j a i, b j,..., b ] N where N = N [b m,..., b j+k+,, a i+k+ b j+k+, a i+k+,..., a n] if a i+k+ > b j+k+ ; N [b m,..., b j+k+, b j+k+ a i+k+,, a i+k+,,..., a n] if a i+k+ < b j+k+.

74 80 Rotation

75 80 Rotation

76 80 Rotation

77 80 Rotation

78 80 Rotation

79 80 Rotation

80 80 Rotation

81 80 Rotation rotation [,,,, ] reverse [,,,, ] Theorem The numerators of the continued fractions [a, a,..., a n ] and [a n,..., a, a ] are equal.

82 Palindromic snake graphs A continued fraction [a,..., a n ] is called even if n is even;

83 Palindromic snake graphs A continued fraction [a,..., a n ] is called even if n is even; palindromic if a i = a n i for all i.

84 Palindromic snake graphs A continued fraction [a,..., a n ] is called even if n is even; palindromic if a i = a n i for all i.. A snake graph is called palindromic if it is the snake graph of an even palindromic continued fraction;

85 Palindromic snake graphs A continued fraction [a,..., a n ] is called even if n is even; palindromic if a i = a n i for all i.. A snake graph is called palindromic if it is the snake graph of an even palindromic continued fraction; rotationally symmetric at a center tile if the rotation about 80 at the center of the central tile is an automorphism.

86 Palindromic snake graphs A continued fraction [a,..., a n ] is called even if n is even; palindromic if a i = a n i for all i.. A snake graph is called palindromic if it is the snake graph of an even palindromic continued fraction; rotationally symmetric at a center tile if the rotation about 80 at the center of the central tile is an automorphism. Theorem A snake graph is palindromic if and only if it has a rotational symmetry at its center tile.

87 Example [,,, ] = 9

88 Example N [,,, ] = 9

89 Example N [,,, ] = N [, ]N [, ] + N []N [] 9 = Theorem (Palindromification) Let [a, a,..., a n ] = p n q n. Then [a n,..., a, a, a, a,..., a n ] = p n + q n p n p n + q n q n.

90 Example N [,,, ] = N [, ]N [, ] + N []N [] 9 = Theorem (PalindromificationnoitacifimordnilaP) Let [a, a,..., a n ] = p n q n. Then [a n,..., a, a, a, a,..., a n ] = p n + q n p n p n + q n q n.

91 Sums of two squares An integer N is called a sum of two squares if there exist integers p > q with gcd(p, q) = such that N = p + q.

92 Sums of two squares An integer N is called a sum of two squares if there exist integers p > q with gcd(p, q) = such that N = p + q. Corollary Let N > 0. If N is a sum of two squares then there exists a palindromic snake graph G such that m(g) = N.

93 Sums of two squares An integer N is called a sum of two squares if there exist integers p > q with gcd(p, q) = such that N = p + q. Corollary Let N > 0. If N is a sum of two squares then there exists a palindromic snake graph G such that m(g) = N. The number of ways one can write N as a sum of two squares is equal to one half of the number of palindromic snake graphs with N perfect matchings. Example 5 can be written uniquely as sum of two squares as 5 = +. The even palindromic continued fractions with numerator 5 are [, ] and [,,, ].

94 Markov numbers A triple of positive integers (a, b, c) is called a Markov triple if a + b + c = abc. An integer is called a Markov number if it is a member of a Markov triple. (9,, 69) (9,, 5) (9,, 5) (,, ) (,, ) (,, 5) (,, 5) (9,, 5) Markov tree (,, ) Uniqueness Conjecture (Frobenius 9) The largest integer in a Markov triple determines the other two.

95 Markov numbers as numbers of perfect matchings of Markov snake graphs Markov triples are related to the clusters of the cluster algebra of the torus with one puncture [Beineke-Brüstle-Hille, Propp ]. A Markov snake graph is the snake graph of a cluster variable of the once punctured torus. The Markov numbers are precisely the number of perfect matchings of the Markov snake graphs. [Propp] Uniqueness Conjecture Any two Markov snake graphs have a different number of perfect matchings.

96 Markov tree snake graph version

97 Markov snake graphs (,) (,) The line with slope p/q = / with its lower Christoffel path in red, defining the Christoffel word x, x, x, y, x, x, y, x, x, y. The corresponding Markov snake graph is obtained by placing tiles of side length / on the Christoffel path leaving the first half step and the last half step empty.

98 Theorem Every Markov number is the numerator of an even palindromic continued fraction. Corollary Every Markov number, except, is a sum of two squares. In general, the decomposition of an integer as a sum of two squares is not unique.the smallest example among the Markov numbers is 60 = + 9 = +. / = [,,,,, ] and its palindromification [,,,,,,,,,,, ] is a Markov snake graph. /9 = [,,, ] and its palindromification [,,,,,,, ] is not Markov.

99 Chebyshev polynomials The (normalized) Chebyshev polynomials of the first kind T n are defined recursively by T 0 =, T = x, and T n = xt n T n. The first few polynomials are T = x T = x x T = x x + T 5 = x 5 x + x T 6 = x 6 5x + 6x

100 Chebyshev polynomials Let G n be the snake graph of [a, a,..., a n ] = [,,..., ]. Thus G n is a vertical straight snake graph with exactly n tiles. G 6 5

101 Chebyshev polynomials are specialized cluster variables i i i i i x x x x x x i i i i i Theorem Label all horizontal edges of G n by x. Label all vertical edges of G n by i =. Then P Match G n x(p) = T n

102 Corollaries lim n T [ n x = i T n i, x i,..., x ]. i T n = ( x + ) x T n. T n T n = T n.

103 Summary Cluster algebra side New expansion formula Study quotients of cluster algebra elements Study their asymptotic behavior Continued fraction side Combinatorial realization Intuition about continued fractions Additional structure on continued fractions (poset, x-grading, y-grading