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Home , Pentagram map

The limit point of the

Max Glick Ohio State University

November 10, 2018 Three features of interest in a system

A

B = T (A) Geometric description

y1 1 −1 , , 3 2 , 4 (y1, y2, y3, y4) y2 y (1 + y ) y 1+y2   4 3 4 3 µ2

1 2 1 2

Integrability Cluster algebra structure The pentagram map

A

B = T (A)

T = the pentagram map Outline

1. Pentagram map background 2. Cluster (Y -Pattern) structure for the pentagram map. 3. The search for the limit point

b b b P b Prehistory [ ..., Clebsch 1800’s, ..., Schwartz 1992]

Theorem If A is a closed then there is a projective transformation of the taking A to T (A). Theorem If A is a closed then there is a projective transformation of the plane taking A to T 2(A). A A

2 T (A) T (A) Modern introduction [Schwartz 1992]

Theorem If A is a convex n-gon then the sequence of T k (A) converges exponentially to a point P =(X , Y ) ∈ R2 as k → ∞.

b b b P b Coordinate system [Schwartz 2008] ◮ Easier to work with twisted n-gons, e.g.

b b b

A =

b b b Coordinate system [Schwartz 2008] ◮ Easier to work with twisted n-gons, e.g.

b b b

A =

b b b

◮ Cross ratio (projectively natural) coordinates

A 7→ (x1,..., x2n). Coordinate system [Schwartz 2008] ◮ Easier to work with twisted n-gons, e.g.

b b b

A =

b b b

◮ Cross ratio (projectively natural) coordinates

A 7→ (x1,..., x2n). ◮ Evolution equation

1 − xj−3xj−2 xj−1 , j even ′ 1 − xj+1xj+2 xj =  1 − xj+3xj+2 xj+1 , j odd  1 − xj−1xj−2   Integrability of the pentagram map

The pentagram map is:

Twisted polygons Closed polygons Liouville integrable [OST 2010] [OST 2013] Algebro-geometric integrable [Soloviev 2013] [Soloviev 2013] “Cluster integrable” [GSTV 2012]

where ◮ OST = Ovsienko-Schwartz-Tabachnikov ◮ GSTV = Gekhtman-Shapiro-Tabachnikov-Vainshtein Other pentagram map work

1. Connection to cluster algebras (Part 2 of this talk) 2. Continuous limit n → ∞ (Boussinesq equation) ◮ Ovsienko-Schwartz-Tabachnikov 2010 3. Incidence theorems ◮ Schwartz 2001, 2008 ◮ G. 2015 ◮ Yao 2014 (preprint) 4. Generalizations ◮ Gekhtman-Shapiro-Tabachnikov-Vainshtein 2012 ◮ Khesin-Soloviev 2013, 2015, 2016 ◮ Mari Beffa 2013, 2015 ◮ Schwartz 2013 ◮ G.-Pylyavskyy 2016 The y-parameters

The y-parameters of a twisted A are the cross ratios

−1 y2k (A)= −(χ(Ak−1, B, C, Ak+1))

y2k+1(A)= −χ(D, Ak , Ak+1, E)

for B, C, D, E as below.

A +3 b k A +2 b k A +2 bb k bb b b Ak+1 E b C Ak+1 b B b b Ak Ak b bb

Ak−2 Ak−1 bb b b Ak−2 Ak−1 D Cluster structure for the pentagram map [G. 2011]

Theorem The y-parameters of a twisted n-gon A evolve under T as

(1 + yj−1)(1 + yj+1) yj − − , j even ′ (1 + y 1 )(1 + y 1 ) yj =  j−3 j+3 −1 odd yj , j

which agrees with the Y-pattern mutation dynamics of the quiver Qn.

To be defined: ◮ Y -pattern mutations ◮ the quiver Qn Y -patterns [Fomin, Zelevinsky 2007]

1 1 1 −1 , , 3 2 , 4 (y1, y2, y3, y4) (y y2 y (1 + y ) y ) 1+y2

4 3 4 3 µ2

1 2 1 2

Y -patterns are a family of discrete dynamical systems encoded by directed graphs (quivers) satisfying many nice properties: ◮ subtraction free rational expressions ◮ a natural Poisson structure [Gekhtman, Shapiro, Vainshtein] ◮ frequent cancellation of factors under iteration ◮ (sometimes) combinatorial formulas for iterates Y -seeds and mutations

A Y -seed is a pair (y, Q) where y =(y1,..., ym) is a collection of rational functions and Q is a quiver, i.e. a directed graph on set {1, 2,..., m} without oriented 2-cycles.

Given a Y -seed (y, Q) and some k ∈ {1,..., m}, the mutation ′ ′ µk (y, Q)=(y , Q ), where Y -seeds and mutations

A Y -seed is a pair (y, Q) where y =(y1,..., ym) is a collection of rational functions and Q is a quiver, i.e. a directed graph on vertex set {1, 2,..., m} without oriented 2-cycles.

Given a Y -seed (y, Q) and some k ∈ {1,..., m}, the mutation ′ ′ µk (y, Q)=(y , Q ), where ◮ The vector y′ is obtained from y via the following steps:

1. For each j → k in Q, multiply yj by 1+ yk . 1 2. For each k → j in Q, multiply yj by 1+ −1 . yk 3. Invert yk . Y -seeds and mutations

A Y -seed is a pair (y, Q) where y =(y1,..., ym) is a collection of rational functions and Q is a quiver, i.e. a directed graph on vertex set {1, 2,..., m} without oriented 2-cycles.

Given a Y -seed (y, Q) and some k ∈ {1,..., m}, the mutation ′ ′ µk (y, Q)=(y , Q ), where ◮ The vector y′ is obtained from y via the following steps:

1. For each j → k in Q, multiply yj by 1+ yk . 1 2. For each k → j in Q, multiply yj by 1+ −1 . yk 3. Invert yk . ◮ The quiver Q′ is obtained from Q via the following steps: 1. For every length 2 path i → k → j, add an arc from i to j. 2. Reverse the orientation of all arcs incident to k. 3. Remove all oriented 2-cycles. An example of a Y -seed mutation

1 1 1 −1 , , 3 2 , 4 (y1, y2, y3, y4) (y y2 y (1 + y ) y ) 1+y2

4 3 4 3 µ2

1 2 1 2 The pentagram quiver (n = 8)

4

6 2

5 3

7 1 8 16 9 15

11 13

10 14

12 Application: combinatorial formulas [G. 2011]

k k Fj−1,k Fj+1,k yj (T (A)) = yj+3i Fj−3,k Fj+3,k i=−k ! Y where

Fj,k = “Order ideal generating functions of Aztec diamond posets” Application: combinatorial formulas [G. 2011]

k k Fj−1,k Fj+1,k yj (T (A)) = yj+3i Fj−3,k Fj+3,k i=−k ! Y where

Fj,k = “Order ideal generating functions of Aztec diamond posets”

Example

y2 y4

y3

y0 y6

F3,2 =1+ y0 + y6 + y0y6 + y0y3y6(1 + y2 + y4 + y2y4) The limit point [Schwartz 1992]

Theorem If A is a convex n-gon then the sequence of polygons T k (A) converges exponentially to a point P =(X , Y ) ∈ R2 as k → ∞.

b b b P b

Question Are X and Y analytic functions of the coordinates x1, y1,..., xn, yn of the vertices of A? Example of limit point 3 A 2

1

1 2 3 Can estimate (X , Y ) ≈ (1.60947, 1.83760) Example of limit point 3 A 2

1

1 2 3 Can estimate (X , Y ) ≈ (1.60947, 1.83760) 482 − 49t 614 − 51t (X , Y )= , t2 − 27t + 178 t2 − 27t + 178   where t ≈−12.8777 Example of limit point 3 A 2

1

1 2 3 Can estimate (X , Y ) ≈ (1.60947, 1.83760) 482 − 49t 614 − 51t (X , Y )= , t2 − 27t + 178 t2 − 27t + 178   where t ≈−12.8777 t = largest magnitude root of λ3 − 7λ2 − 160λ + 1236 The main theorem

Question (Schwartz 1992)

Are X and Y analytic functions of the coordinates x1, y1,..., xn, yn of the vertices of A? Theorem (G. 2017) The answer to the question is “yes”, and even better, X and Y lie in a degree 3 extension of Q(x1, y1,..., xn, yn). Projective transformations

For φ˜ ∈ GL3(R), the linear map

φ˜ : R3 → R3

induces a projective transformation

φ : R2 → R2

φ11x + φ12y + φ13 φ21x + φ22y + φ23 (x, y) 7→ , φ31x + φ32y + φ33 φ31x + φ32y + φ33   of R2 ⊆ RP2.

◮ Write PGL3(R)= {projective transformations}. Schwartz’s question for and

◮ Let A be a convex pentagon. Then ∃φ ∈ PGL3(R) such that

T (A)= φ(A) Schwartz’s question for pentagons and hexagons

◮ Let A be a convex pentagon. Then ∃φ ∈ PGL3(R) such that

T (A)= φ(A) =⇒ T k (A)= φk (A) for all k Schwartz’s question for pentagons and hexagons

◮ Let A be a convex pentagon. Then ∃φ ∈ PGL3(R) such that

T (A)= φ(A) =⇒ T k (A)= φk (A) for all k =⇒ (X , Y ) a fixed point of φ Schwartz’s question for pentagons and hexagons

◮ Let A be a convex pentagon. Then ∃φ ∈ PGL3(R) such that

T (A)= φ(A) =⇒ T k (A)= φk (A) for all k =⇒ (X , Y ) a fixed point of φ X =⇒ Y an eigenvector of φ˜   1   Schwartz’s question for pentagons and hexagons

◮ Let A be a convex pentagon. Then ∃φ ∈ PGL3(R) such that

T (A)= φ(A) =⇒ T k (A)= φk (A) for all k =⇒ (X , Y ) a fixed point of φ X =⇒ Y an eigenvector of φ˜   1   ◮ It follows that X , Y are algebraic of degree 3 over Q(x1, y1,..., x5, y5). ◮ A similar argument works for hexagons. Recipe for (X , Y ) in general

T 3 Lift (xj , yj ) to vj =[xj yj 1] ∈ R . Define a linear map

L : R3 → R3 n |vj−1, u, vj+1| L(u)= − vj |vj−1, vj , vj+1| j=1 X where |·, ·, ·| denotes a 3-by-3 . Recipe for (X , Y ) in general

T 3 Lift (xj , yj ) to vj =[xj yj 1] ∈ R . Define a linear map

L : R3 → R3 n |vj−1, u, vj+1| L(u)= − vj |vj−1, vj , vj+1| j=1 X where |·, ·, ·| denotes a 3-by-3 determinant. Proposition For any convex n-gon, X Y   1 is an eigenvector of L.   Properties of LA

3 3 LA : R → R n |vj−1, u, vj+1| LA(u)= − vj |vj−1, vj , vj+1| j=1 X Proposition Let A be a convex n-gon, B = T (A).

1. LA is independent of the choice of lifts v1,..., vn

2. Q ∈ conv(B)=⇒ LA(Q) ∈ conv(A)

3. LA = LB Picture of (projectivization of) LA

Aj Aj−1 Aj+1

b Q

n 1 (△QAj−1Aj+1) LA(Q)= Aj Λ Area(△Aj−1Aj Aj+1) j=1 X (Λ = normalization constant) Underlying

There are identifications

{polygons} {edge weightings of Γ} {vertex weightings of Q}

for which

T Postnikov transformations Y -pattern mutations

where

Γ=

Q = dual quiver Underlying combinatorics (continued)

Method 1 (G. 2011, Gekhtman et al. 2012, ...)

◮ edge weights = A1A2A3, A1A2A4, A1A3A4, A2A3A4, ...

◮ y-variables = cross ratios [A1, B1, B2, A3],....

A2 A1 A3 b b b b B1 B2 B3 A4 Underlying combinatorics (continued)

Method 2 (based on Kenyon)

◮ (complex) edge weights = displacements A2 − B0, A2 − B1, A2 − B2, A2 − B3, A3 − B1, ... ◮ y-variables = double ratios (not projectively natural)

A2 b A1 A3 b b B1 B2 A2 − B0 A2 − B3 b y = b A2 − B1 A2 − B2 B0 B3     A duality on polygons

A B

◮ sides of A parallel to sides of B A duality on polygons

A B

◮ sides of A parallel to sides of B ◮ sides of T (A) parallel to sides of T −1(B) A duality on polygons

A B

◮ sides of A parallel to sides of B ◮ sides of T (A) parallel to sides of T −1(B) ◮ sides of T 2(A) parallel to sides of T −2(B) A duality on polygons

A B

◮ sides of A parallel to sides of B ◮ sides of T (A) parallel to sides of T −1(B) ◮ sides of T 2(A) parallel to sides of T −2(B) ◮ ... Say A and B are dual about the at infinity. A duality on polygons (continued)

Proposition A closed has a (possibly twisted) dual about a given line or through a given point.

A B A duality on polygons (continued)

Proposition A closed convex polygon has a (possibly twisted) dual about a given line or through a given point.

A B

Theorem A closed convex polygon has a closed dual through its limit point.