Cluster algebras, snake graphs and continued fractions Ralf - - PowerPoint PPT Presentation

Cluster algebras, snake graphs and continued fractions

Ralf Schiffler

Intro

Cluster algebras Continued fractions Snake graphs

Intro

Cluster algebras

• expansion formula via

perfect matchings [Musiker-S-Williams 11]

• ■

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

Continued fractions Snake graphs

Intro

Cluster algebras

• expansion formula via

perfect matchings [Musiker-S-Williams 11]

• ■

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

Continued fractions Snake graphs

• bijection via

perfect matchings [C ¸anak¸ cı-S 16]

• s

s s s s s s s s s s s s s s s s s s s

Intro

Cluster algebras

expansion formula as continued fractions, asymptotic behavior [C ¸anak¸ cı-S 16]

• expansion formula via

perfect matchings [Musiker-S-Williams 11]

• ■

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

Continued fractions Snake graphs

• bijection via

perfect matchings [C ¸anak¸ cı-S 16]

• s

s s s s s s s s s s s s s s s s s s s

Intro

Cluster algebras

expansion formula as continued fractions, asymptotic behavior [C ¸anak¸ cı-S 16]

• expansion formula via

perfect matchings [Musiker-S-Williams 11]

• ■

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

Continued fractions Snake graphs

• bijection via

perfect matchings [C ¸anak¸ cı-S 16]

• s

s s s s s s s s s s s s s s s s s s s

Intro

Cluster algebras

expansion formula as continued fractions [C ¸anak¸ cı-S 16]

• expansion formula via

perfect matchings [Musiker-S-Williams 11]

• ■

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

Continued fractions Snake graphs

• Combinatorial realization
• f continued fractions

Applications in elementary Number Theory

• s

s s s s s s s s s s s s s s s s s s s

Continued Fractions

84 37 = 2 + 10 37

Continued Fractions

84 37 = 2 + 10 37 = 2 + 1

37 10

Continued Fractions

84 37 = 2 + 10 37 = 2 + 1

37 10

= 2 + 1 3 + 7 10

Continued Fractions

84 37 = 2 + 10 37 = 2 + 1

37 10

= 2 + 1 3 + 7 10 = 2 + 1 3 + 1

10 7

Continued Fractions

84 37 = 2 + 10 37 = 2 + 1

37 10

= 2 + 1 3 + 7 10 = 2 + 1 3 + 1

10 7

= 2 + 1 3 + 1 1 + 3 7

Continued Fractions

84 37 = 2 + 10 37 = 2 + 1

37 10

= 2 + 1 3 + 7 10 = 2 + 1 3 + 1

10 7

= 2 + 1 3 + 1 1 + 3 7 = 2 + 1 3 + 1 1 + 1 2 + 1 3

Continued Fractions

84 37 = 2 + 10 37 = 2 + 1

37 10

= 2 + 1 3 + 7 10 = 2 + 1 3 + 1

10 7

= 2 + 1 3 + 1 1 + 3 7 = 2 + 1 3 + 1 1 + 1 2 + 1 3 =: [2, 3, 1, 2, 3]

Continued Fractions

84 37 = [2, 3, 1, 2, 3]

Continued Fractions - Euclidean algorithm

84 = 2 · 37 + 10 37 = 3 · 10 + 7 10 = 1 · 7 + 3 7 = 2 · 3 + 1 3 = 3 · 1

Continued Fractions - Euclidean algorithm

84 = 2 · 37 + 10 37 = 3 · 10 + 7 10 = 1 · 7 + 3 7 = 2 · 3 + 1 3 = 3 · 1 84 37 = [2, 3, 1, 2, 3]

Continued Fractions

The division algorithm gives a bijection between Q and the set of continued fractions whose last coefficient is at least 2. Q → {[a0, . . . , an] | a0 ∈ Z, a1, . . . an−1 ≥ 1, an ≥ 2}

note that [a0, . . . , an−1, 1] = [a0, . . . , an−1 + 1], because an−1 + 1 1 = an−1 + 1.

Snake graphs

A snake graph G is a connected planar graph consisting of a finite sequence of tiles G1, G2, . . . , Gd such that

◮ Gi and Gi+1 share exactly one edge ei and this edge is

◮ either the north edge of Gi and the south edge of Gi+1, ◮ or the east edge of Gi and the west edge of Gi+1.

Snake graphs

Snake graphs

Snake graphs

Snake graphs

Snake graphs

Snake graphs

Snake graphs

Snake graphs

Snake graphs

Snake graphs

Sign function

Sign function - constant on diagonals

Sign function - constant on diagonals

Sign function - constant on diagonals

Sign function - constant on diagonals

Sign function

Sign function – sign sequence – continued fraction

−, −, +, +, +, −, +, +, −, −, − [2, 3, 1, 2, 3]

Theorem

← → [2, 3, 1, 2, 3] = 84

37

84 perfect matchings

There is a bijection between snake graphs and continued fractions 1 G[a1, a2, . . . , an] ← → [a1, a2, . . . , an] Moreover, if m(G) denotes the number of perfect matchings of G then [a1, a2, . . . , an] = m(G[a1, a2, . . . , an]) m(G[a2, . . . , an]) .

1with ai ≥ 1, an ≥ 2

Theorem

← → [2, 3, 1, 2, 3] = 84

37

37 perfect matchings

There is a bijection between snake graphs and continued fractions 2 G[a1, a2, . . . , an] ← → [a1, a2, . . . , an] Moreover, if m(G) denotes the number of perfect matchings of G then [a1, a2, . . . , an] = m(G[a1, a2, . . . , an]) m(G[a2, . . . , an]) .

2with ai ≥ 1, an ≥ 2

Reflection

flip

[2, 3, 1, 2, 3]

[1, 1, 3, 1, 2, 3]

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.

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.

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.

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.

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.

Tiles with sign changes

Tiles with sign changes removed

Tiles with sign changes removed

Tiles with sign changes removed

Tiles with sign changes removed

Subgraphs Hi ∼ = G[ai]

H1 H2 H3 H4 H5

Division algorithm

2 3 4 7 10 17 27 37 47 84 2 2 3 4 7 10 17 27 37 2 3 4 7 10 2 3 4 7 10 17 27 37 2 3 2 3 4 7 10 2 3 4 7 2 3 4 7 10 2 3 4 7 1 2 3 2 3 4 7 2 3 2 1 H1 H2 H3 H4

Division algorithm – Proof

2 3 4 7 10 17 27 37 47 84 2 2 3 4 7 10 17 27 37 2 3 4 7 10 2 3 4 7 10 17 27 37 2 3 2 3 4 7 10 2 3 4 7 2 3 4 7 10 2 3 4 7 1 2 3 2 3 4 7 2 3 2 1 H1 H2 H3 H4

Division algorithm – Proof

2 3 4 7 10 17 27 37 47 84 2 2 3 4 7 10 17 27 37 2 3 4 7 10 2 3 4 7 10 17 27 37 2 3 2 3 4 7 10 2 3 4 7 2 3 4 7 10 2 3 4 7 1 2 3 2 3 4 7 2 3 2 1 H1 H2 H3 H4

Division algorithm – Proof

2 3 4 7 10 17 27 37 47 84 2 2 3 4 7 10 17 27 37 2 3 4 7 10 2 3 4 7 10 17 27 37 2 3 2 3 4 7 10 2 3 4 7 2 3 4 7 10 2 3 4 7 1 2 3 2 3 4 7 2 3 2 1 H1 H2 H3 H4

Division algorithm – Proof

2 3 4 7 10 17 27 37 47 84 2 2 3 4 7 10 17 27 37 2 3 4 7 10 2 3 4 7 10 17 27 37 2 3 2 3 4 7 10 2 3 4 7 2 3 4 7 10 2 3 4 7 1 2 3 2 3 4 7 2 3 2 1 H1 H2 H3 H4

Division algorithm – Proof

2 3 4 7 10 17 27 37 47 84 2 2 3 4 7 10 17 27 37 2 3 4 7 10 2 3 4 7 10 17 27 37 2 3 2 3 4 7 10 2 3 4 7 2 3 4 7 10 2 3 4 7 1 2 3 2 3 4 7 2 3 2 1 H1 H2 H3 H4

Division algorithm – Proof

2 3 4 7 10 17 27 37 47 84 2 2 3 4 7 10 17 27 37 2 3 4 7 10 2 3 4 7 10 17 27 37 2 3 2 3 4 7 10 2 3 4 7 2 3 4 7 10 2 3 4 7 1 2 3 2 3 4 7 2 3 2 1 H1 H2 H3 H4

Division algorithm – Proof

2 3 4 7 10 17 27 37 47 84 2 2 3 4 7 10 17 27 37 2 3 4 7 10 2 3 4 7 10 17 27 37 2 3 2 3 4 7 10 2 3 4 7 2 3 4 7 10 2 3 4 7 1 2 3 2 3 4 7 2 3 2 1 H1 H2 H3 H4

Division algorithm – Proof

2 3 4 7 10 17 27 37 47 84 2 2 3 4 7 10 17 27 37 2 3 4 7 10 2 3 4 7 10 17 27 37 2 3 2 3 4 7 10 2 3 4 7 2 3 4 7 10 2 3 4 7 1 2 3 2 3 4 7 2 3 2 1 H1 H2 H3 H4

Division algorithm – Proof

2 3 4 7 10 17 27 37 47 84 2 2 3 4 7 10 17 27 37 2 3 4 7 10 2 3 4 7 10 17 27 37 2 3 2 3 4 7 10 2 3 4 7 2 3 4 7 10 2 3 4 7 1 2 3 2 3 4 7 2 3 2 1 H1 H2 H3 H4

Division algorithm – Proof

2 3 4 7 10 17 27 37 47 84 2 2 3 4 7 10 17 27 37 2 3 4 7 10 2 3 4 7 10 17 27 37 2 3 2 3 4 7 10 2 3 4 7 2 3 4 7 10 2 3 4 7 1 2 3 2 3 4 7 2 3 2 1 H1 H2 H3 H4

Expansion formula for cluster variables

Theorem (Musiker-S.-Williams)

If γ is an arc in a triangulated surface (S, M), the cluster variable xγ is given by the formula xγ = 1 cross(Gγ)

• P∈Match Gγ

x(P).

1 3 2 γ a b c d e f x2 x3 x1 1 2 1 2 1 2

1 2

1 1 2 2 3 a b d c Gγ

xγ = x1+x2+x3

x1x2

Example 2

2 1 b a γ 2 b 1 2 a 1 2 1 2 2 1 b a 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2

1 1 1 1 2

x4

2

x2

2

x2

2

1 x2

1

Gγ

xγ = x2

1 +1+2x2 2 +x4 2

x2

1 x2

Example 3

Torus with 1 puncture [2, 2, 2, 2, 2, 2, 2, 2] = 985 408

2 1 1 2 3

3 3 3 3 3 3 3 3 2 2 2 2 1 1 1

The Laurent polynomial of this cluster variable has 985 terms.

Expansion formula as continued fraction

Theorem (C ¸anak¸ cı-S)

The cluster variable xγ of the arc γ is the numerator of the continued fraction [L1, L2, . . . , Ln], where Li is a Laurent polynomial explicitly given3 by the subgraph Hi.

3[C

¸anak¸ cı-Felikson]

Example 3

Torus with 1 puncture [2, 2, 2, 2, 2, 2, 2, 2] = 985 408

2 1 1 2 3

3 3 3 3 3 3 3 3 2 2 2 2 1 1 1

[a1, . . . , a8] = [2, 2, 2, 2, 2, 2, 2, 2], let x′

3 = x2

1 +x2 2

x3

, then L1 = x′

3

1 x2 , L2 = x′

3

x2 x2

1

, L3 = x′

3

x2

1

x3

2

, . . . , L8 = x′

3

x7

2

x8

1

xγ = numerator of [L1, L2, . . . , Ln]

Example 3, computation dispiace Salvatore, ho usato Mathematica



 +   * /  +   *   +   *   +   *   +   *   +   *   +   *   +   *  

 +  +  +  +  +

 + +  +  + + +  +  + + +  +  + + +  +  + + +  +  + + + +     +   +  +  +  +  +  + +  +  + +  +  + + + 

What about the denominator?

Torus with 1 puncture [2, 2, 2, 2, 2, 2, 2, 2] = 985 408

2 1 1 2 3

3 3 3 3 3 3 3 3 2 2 2 2 1 1 1

xγ = numerator of [L1, L2, . . . , Ln]

What about the denominator?

Torus with 1 puncture

2 1 1 2 3

3 3 3 3 3 3 3 2 2 2 1 1 1

xγ′ = denominator of [L1, L2, . . . , Ln] = numerator of [L2, . . . , Ln]

Asymptotic behavior of quotients

lim

n→∞

xγ xγ′ = x′

1 − x1 +

• (x′

1 − x1)2 + 4x′2 3

2x′

3

.

where x′

3 = (x2 1 + x2 2)/x3

x′

1 = (x2 2 + x′2 3 )/x1

are obtained from the initial cluster by mutation in 3 and then 1.

Applications to elementary Number Theory

◮ (skein) relations in the cluster algebras were expressed in

terms of snake graphs [C ¸anak¸ cı-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.

Equations for continued fractions

Theorem

We have the following identities of numerators of continued fractions, where we set N[a1, . . . , a0] = 1, and N[an+1, . . . , an] = 1. (a) For every i = 1, 2, . . . , n, N[a1, . . . , an] = N[a1, . . . , ai] N[ai+1, . . . , an] + N[a1, . . . , ai−1] N[ai+2, . . . , an]. (b) For every j ≥ 0 and i such that 1 ≤ i + j ≤ n − 1, N[a1, . . . , ai+j] N[ai, . . . , an] = N[a1, . . . , an] N[ai, . . . , ai+j] + (−1)j N[a1, . . . , ai−2] N[ai+j+2, . . . , an].

Equations for continued fractions

(c) For continued fractions [a1, . . . , an] and [b1, . . . , bm] such that

[ai, . . . , ai+k] = [bj, . . . , bj+k] for certain i, j, k, we have N[a1, . . . , an] N[b1, . . . , bm] = N[a1, . . . , ai−

1, bj, . . . , bm] N[b1, . . . , bj− 1, ai, . . . , an]

+(−1)k N[a1, . . . , ai−2−1, 1, bj−

1 − ai− 1−1, bj− 2, . . . , b1] N ′

where N ′ =            N[bm, . . . , bj+

k+ 2−1, 1, ai+ k+ 1 − bj+ k+ 1−1, ai+ k+ 2, . . . , an]

if ai+

k+1 > bj+ k+1;

N[bm, . . . , bj+

k+ 2, bj+ k+ 1− ai+ k+ 1−1, 1, ai+ k+ 2−1, , . . . , an]

if ai+

k+1 < bj+ k+1.

180◦ Rotation

180◦ Rotation

180◦ Rotation

180◦ Rotation

180◦ Rotation

180◦ Rotation

180◦ Rotation

180◦ Rotation

rotation

• [2, 3, 1, 2, 3]

reverse

[3, 2, 1, 3, 2]

Theorem

The numerators of the continued fractions [a1, a2, . . . , an] and [an, . . . , a2, a1] are equal.

Palindromic snake graphs

A continued fraction [a1, . . . , an] is called

◮ even if n is even;

Palindromic snake graphs

A continued fraction [a1, . . . , an] is called

◮ even if n is even; ◮ palindromic if ai = an−i for all i.

Palindromic snake graphs

A continued fraction [a1, . . . , an] is called

◮ even if n is even; ◮ palindromic if ai = an−i for all i.

. A snake graph is called

◮ palindromic if it is the snake graph of an even palindromic

continued fraction;

Palindromic snake graphs

A continued fraction [a1, . . . , an] is called

◮ even if n is even; ◮ palindromic if ai = an−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

180◦ at the center of the central tile is an automorphism.

Palindromic snake graphs

A continued fraction [a1, . . . , an] is called

◮ even if n is even; ◮ palindromic if ai = an−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

180◦ 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.

Example

[2, 2, 2, 2] = 29

12

Example

N[2, 2, 2, 2] = 29

Example

N[2, 2, 2, 2] = N[2, 2]N[2, 2] + N[2]N[2] 29 = 5 ∗ 5 + 2 ∗ 2

Theorem (Palindromification)

Let [a1, a2, . . . , an] = pn qn . Then [an, . . . , a2, a1, a1, a2, . . . , an] = p2

n + q2 n

pn−1pn + qn−1qn .

Example

N[2, 2, 2, 2] = N[2, 2]N[2, 2] + N[2]N[2] 29 = 5 ∗ 5 + 2 ∗ 2

Theorem (PalindromificationnoitacifimordnilaP)

Let [a1, a2, . . . , an] = pn qn . Then [an, . . . , a2, a1, a1, a2, . . . , an] = p2

n + q2 n

pn−1pn + qn−1qn .

Sums of two squares

An integer N is called a sum of two squares if there exist integers p > q ≥ 1 with gcd(p, q) = 1 such that N = p2 + q2.

Sums of two squares

An integer N is called a sum of two squares if there exist integers p > q ≥ 1 with gcd(p, q) = 1 such that N = p2 + q2.

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.

Sums of two squares

An integer N is called a sum of two squares if there exist integers p > q ≥ 1 with gcd(p, q) = 1 such that N = p2 + q2.

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 = 22 + 12. The even palindromic continued fractions with numerator 5 are [2, 2] and [1, 1, 1, 1].

Markov numbers

A triple of positive integers (a, b, c) is called a Markov triple if a2 + b2 + c2 = 3abc. An integer is called a Markov number if it is a member of a Markov triple.

(29, 2, 169) (29, 2, 5) ❦ ❦ ❦ ❦ (29, 433, 5) (1, 1, 1) (1, 2, 1) (1, 2, 5) ❧ ❧ ❧ ❘ ❘ ❘ (1, 13, 5) ❙ ❙ ❙ ❙ (194, 13, 5) Markov tree (1, 13, 34)

Uniqueness Conjecture (Frobenius 1913)

The largest integer in a Markov triple determines the other two.

Markov numbers as numbers of perfect matchings of Markov snake graphs

◮ Markov triples are related to the clusters of the cluster algebra

• f the torus with one puncture [Beineke-Br¨

ustle-Hille 11, Propp –].

◮ A Markov snake graph is the snake graph of a cluster variable

• f 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.

Markov tree — snake graph version

Markov snake graphs

(7,3) 1 2 3 4 5 6 7 1 2 3 (7,3) 1 2 3 4 5 6 7 1 2 3

The line with slope p/q = 3/7 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 1/2 on the Christoffel path leaving the first half step and the last half step empty.

Theorem

Every Markov number is the numerator of an even palindromic continued fraction.

Corollary

Every Markov number, except 1, 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 610 = 232 + 92 = 212 + 132. 21/13 = [1, 1, 1, 1, 1, 2] and its palindromification [2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2] is a Markov snake graph. 23/9 = [2, 1, 1, 4] and its palindromification [4, 1, 1, 2, 2, 1, 1, 4] is not Markov.

Chebyshev polynomials

The (normalized) Chebyshev polynomials of the first kind Tn are defined recursively by T0 = 1, T1 = x, and Tn = xTn−1 − Tn−2. The first few polynomials are T2 = x2 − 1 T3 = x3 − 2x T4 = x4 − 3x2 + 1 T5 = x5 − 4x3 + 3x T6 = x6 − 5x4 + 6x2 − 1

Chebyshev polynomials

Let Gn be the snake graph of [a1, a2, . . . , an] = [1, 1, . . . , 1]. Thus Gn is a vertical straight snake graph with exactly n − 1 tiles. 2 3 4 5 1 G6

Chebyshev polynomials are specialized cluster variables

x i i x x x x x i i i i i i i i

Theorem

Label all horizontal edges of Gn by x. Label all vertical edges of Gn by i = √−1. Then

• P∈Match Gn

x(P) = Tn

Corollaries

Tn Tn−1 = i x i , x i , . . . , x i

• .

lim

n→∞

Tn Tn−1 = 1 2

• x +
• x2 − 4
• .

T 2

n − T 2 n−1 = T2n.

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