Soliton graphs and realizability: a gentle introductionjsidman/DMD Webpage/Karpman_slides.pdf ·...

Soliton graphs and realizability: a gentle introduction

Ray Karpman1 Yuji Kodama 2

1Otterbein University

2The Ohio State University

November 14, 2019

Karpman and Kodama (Otterbein, OSU) Soliton graphs November 14, 2019 1 / 33

Introduction

Outline

Setting the scene: pseudoline arrangements.

Soliton graphs.

The KP equation.

Contour plots.

Plabic graphs and realizability.

From graphs to triangulations.

Results.


Introduction

Outline


Soliton graphs.

The KP equation.

Contour plots.



Results.


Introduction

Outline


Soliton graphs.

The KP equation.

Contour plots.



Results.


Introduction

Outline


Soliton graphs.

The KP equation.

Contour plots.



Results.


Introduction

Outline


Soliton graphs.

The KP equation.

Contour plots.



Results.


Introduction

Outline


Soliton graphs.

The KP equation.

Contour plots.



Results.


Introduction

Outline


Soliton graphs.

The KP equation.

Contour plots.



Results.


Pseudolines and stretchability

Pseudoline arrangements

A pseudoline is a plane curve that does not intersect itself.

In a pseudoline arrangement, any two pseudolines cross exactly once.













Stretchability

Figure: The pseudoline arrangement at left is stretchable.



A non-stretchable arrangement

Figure: The smallest non-stretchable pseudoline arrangement [Ringel, 1956].


Soliton graphs

The KP equation

Non-linear wave equation

∂

∂x

(−4

∂u

∂t+ 6u

∂u

∂x+∂3u

∂x3

)+ 3

∂2u

∂y2= 0.

Line-soliton solutions of the KP equation model shallow-water waveswith peaks localized along straight lines.

Combinatorics of KP solitons studied in [Kodama and Williams, 2014].


Soliton graphs

The KP equation


∂

∂x

(−4

∂u

∂t+ 6u

∂u

∂x+∂3u

∂x3

)+ 3

∂2u

∂y2= 0.




Soliton graphs

The KP equation


∂

∂x

(−4

∂u

∂t+ 6u

∂u

∂x+∂3u

∂x3

)+ 3

∂2u

∂y2= 0.




Soliton graphs

Motivation: shallow-water waves

Figure: Photo by Michel Griffon - Own work, CC BY 3.0. View Original.


https://commons.wikimedia.org/w/index.php?curid=12598928

Soliton graphs

Constructing soliton solutions

Construct a soliton solution uA(x , y , t) of the KP equation from...

N ×M matrix A.

M linear forms:

θi = pix + qix + ωi (t), 1 ≤ i ≤ M.

The function uA(x , y , t) models the height of a wave at time t.

For fixed time t, wave peaks give a contour plot.


Soliton graphs



N ×M matrix A.

M linear forms:

θi = pix + qix + ωi (t), 1 ≤ i ≤ M.




Soliton graphs



N ×M matrix A.

M linear forms:

θi = pix + qix + ωi (t), 1 ≤ i ≤ M.




Soliton graphs



N ×M matrix A.

M linear forms:

θi = pix + qix + ωi (t), 1 ≤ i ≤ M.




Soliton graphs



N ×M matrix A.

M linear forms:

θi = pix + qix + ωi (t), 1 ≤ i ≤ M.




Soliton graphs

Contour plots

Goal: understand the combinatorics of contour plots.

100

200

-100

-200

0

100

200

-100

-200

0

-200 -100 100 2000 -200 -100 100 2000

t = 70t = 0

100

200

-100

-200

0

-200 -100 100 2000

t = -70

Figure: The contour plot evolves as t changes.


Soliton graphs

Simplifying assumptions

Assume all N × N minors of A are positive.

A represents point in the totally positive Grassmannian Gr>0(N,M).

Re-scale variables to get asymptotic contour plot.

Does not depend on choice of A, as long as all N × N minors of A arepositive.

Does depend on choice of forms θi (x , y , t), value of t.


Soliton graphs








Soliton graphs








Soliton graphs








Soliton graphs








Soliton graphs

Asymptotic contour plots: choosing parameters

Fix M “generic enough” points (pi , qi ) on the parabola q = p2, with

p1 < p2 < · · · < pM .

For 1 ≤ i ≤ M, fix linear form

θi = pix + qiy + ωi (t)

Here t is the multi-time parameter (t3, . . . , tM) and

ωi (t) =M−1∑k=3

pki tk .


Soliton graphs



p1 < p2 < · · · < pM .




ωi (t) =M−1∑k=3

pki tk .


Soliton graphs



p1 < p2 < · · · < pM .




ωi (t) =M−1∑k=3

pki tk .


Soliton graphs

Example

p1 = −2, p2 = 0, p3 = 1, p4 = 2, t = (1, 0)

Then we have

θ1(x , y , t) = −2x + 4y − 8

θ2(x , y , t) = 0

θ3(x , y , t) = x + y + 1

θ4(x , y , t) = 2x + 4y + 8


Soliton graphs

Example

p1 = −2, p2 = 0, p3 = 1, p4 = 2, t = (1, 0)

Then we have

θ1(x , y , t) = −2x + 4y − 8

θ2(x , y , t) = 0

θ3(x , y , t) = x + y + 1

θ4(x , y , t) = 2x + 4y + 8


Soliton graphs

Example Continued

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-4

-3

-2

-1

1

2

3

4

0

θ1 = −2x + 4y − 8

θ2 = 0

θ3 = x + y + 1

θ4 = 2x + 4y + 8

x

y

Figure: An asymptotic contour plot.


Soliton graphs

Asymptotic contour plots for Gr>0(1,M)

Divide the (x , y)-plane based into regions based on which of the θi isdominant over the others.

The asymptotic contour plot for fixed multi-time parameter t0 is thelocus where

f̂ = max{θi (x , y , t0) : 1 ≤ i ≤ M}

is non-linear.


Soliton graphs

Asymptotic contour plots for Gr>0(1,M)

Divide the (x , y)-plane based into regions based on which of the θi isdominant over the others.


f̂ = max{θi (x , y , t0) : 1 ≤ i ≤ M}

is non-linear.


Soliton graphs

Example: N = 2, M = 4

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-4

-3

-2

-1

1

2

3

4

Θ2,4Θ1,2

Θ2,3

Θ3,4

Θ1,4

x

y

Figure: An asymptotic contour plot.


Soliton graphs

Asymptotic contour plots for Gr>0(N ,M)

For each I -element subset of {1, 2, . . . ,M}, define

ΘI (x , y , t) =∑i∈I

θi (x , y , t).


f̂ = max{Θi (x , y , t0) : I ⊆ {1, 2, . . . ,M} and |I | = N}

is non-linear.


Soliton graphs

Asymptotic contour plots for Gr>0(N ,M)

For each I -element subset of {1, 2, . . . ,M}, define

ΘI (x , y , t) =∑i∈I

θi (x , y , t).


f̂ = max{Θi (x , y , t0) : I ⊆ {1, 2, . . . ,M} and |I | = N}

is non-linear.


Soliton graphs

Constructing soliton graphs

Embed contour plot in disk, view as purely combinatorial object.

Color an internal vertex white if the adjoining regions have N − 1indices in common, black otherwise.

Θ2,4Θ1,2

Θ2,3

Θ3,4

Θ1,4

2 4

1 4

1 2 3 4

2 3

Figure: From contour plot to soliton graph.

Goal: classify soliton graphs.


Soliton graphs




Θ2,4Θ1,2

Θ2,3

Θ3,4

Θ1,4

2 4

1 4

1 2 3 4

2 3




Soliton graphs




Θ2,4Θ1,2

Θ2,3

Θ3,4

Θ1,4

2 4

1 4

1 2 3 4

2 3




Soliton graphs




Θ2,4Θ1,2

Θ2,3

Θ3,4

Θ1,4

2 4

1 4

1 2 3 4

2 3




Plabic graphs and realizability

Plabic graphs

Every soliton graph is a plabic graph [Kodama and Williams, 2014].

Planar, bicolored graph embedded in a disk, satisfies some technicalconditions [Postnikov, 2006].

1

43

2

2 4

1 4

1 2 3 4

2 3



Plabic graphs



1

43

2

2 4

1 4

1 2 3 4

2 3



Plabic graphs



1

43

2

2 4

1 4

1 2 3 4

2 3



Face labels of plabic graphs

1

43

2

2 4

1 4

1 2 3 4

2 3

Label face F i if F is to the left of the zig-zag path Ti ending atboundary vertex i .

Face labels give clusters in the cluster algebra structure of Gr(N,M)[Scott, 2006, Oh et al., 2015].



Face labels of plabic graphs

1

43

2

2 4

1 4

1 2 3 4

2 3

Label face F i if F is to the left of the zig-zag path Ti ending atboundary vertex i .

Face labels give clusters in the cluster algebra structure of Gr(N,M)[Scott, 2006, Oh et al., 2015].



Realizability

Theorem (Kodama and Williams [2014])

Every soliton graph for Gr>0(N,M) is a plabic graph. Face labels of theplabic graph correspond to dominant planes of the soliton graph.

We say a plabic graph is realizable if it comes from a soliton graph.

Question: is every plabic graph realizable?



Realizability







Realizability







Summary of results

Question: does every plabic graph come from a soliton graph?

Answer is yes for...

Gr>0(2,M) [Kodama and Williams, 2014].

Gr>0(3, 6) [Huang, 2015].

Gr>0(3, 7),Gr>0(3, 8) [Kodama and K.]

In general, the answer is no [Kodama and K.]



Summary of results




Gr>0(3, 6) [Huang, 2015].

Gr>0(3, 7),Gr>0(3, 8) [Kodama and K.]




Summary of results




Gr>0(3, 6) [Huang, 2015].

Gr>0(3, 7),Gr>0(3, 8) [Kodama and K.]




Summary of results




Gr>0(3, 6) [Huang, 2015].

Gr>0(3, 7),Gr>0(3, 8) [Kodama and K.]




Summary of results




Gr>0(3, 6) [Huang, 2015].

Gr>0(3, 7),Gr>0(3, 8) [Kodama and K.]




Summary of results




Gr>0(3, 6) [Huang, 2015].

Gr>0(3, 7),Gr>0(3, 8) [Kodama and K.]



From soliton graphs to triangulations

The duality map

Map a plane to a point:

θi (x , y) = pix + qiy + ωi 7→ v̂i = (pi , qi , ωi )

Take convex hull of points, project from above to (p, q)-plane.

Get triangulation of the M-gon.



The duality map







The duality map







Example continued

v̂1 = (−2, 4,−8) v̂2 = (0, 0, 0) v̂3 = (1, 1, 1) v̂4 = (2, 4, 8).

-7 -6 -5 -4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

0

θ1

θ2

θ3

θ4

x

y

-2 -1 1 2

1

2

3

4v1

v2

v3

v4

p

q

0

Figure: A soliton tiling.



Example continued

v̂1 = (−2, 4,−8) v̂2 = (0, 0, 0) v̂3 = (1, 1, 1) v̂4 = (2, 4, 8).

-7 -6 -5 -4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

0

θ1

θ2

θ3

θ4

x

y

-2 -1 1 2

1

2

3

4v1

v2

v3

v4

p

q

0

Figure: A soliton tiling.



The N = 2 case

v̂12 = (−2, 4,−8) v̂2,3 = (1, 1, 1) v̂3,4 = (3, 5, 9) v̂1,4 = (0, 8, 0)

v̂2,4 = (2, 4, 8) v̂1,3 = (−1, 5,−7)

-2 -1 1 2

1

2

3

4v1

v2

v3

v4

p

q

0

-2 -1 1 2 3 4

1

2

3

4

5

6

7

8

v2 4v1 2

v2 3

v3 4

v1 4

p

q

0



The N = 2 case

v̂12 = (−2, 4,−8) v̂2,3 = (1, 1, 1) v̂3,4 = (3, 5, 9) v̂1,4 = (0, 8, 0)

v̂2,4 = (2, 4, 8) v̂1,3 = (−1, 5,−7)

-2 -1 1 2

1

2

3

4v1

v2

v3

v4

p

q

0

-2 -1 1 2 3 4

1

2

3

4

5

6

7

8

v2 4v1 2

v2 3

v3 4

v1 4

p

q

0



The N = 2 case

v̂12 = (−2, 4,−8) v̂2,3 = (1, 1, 1) v̂3,4 = (3, 5, 9) v̂1,4 = (0, 8, 0)

v̂2,4 = (2, 4, 8) v̂1,3 = (−1, 5,−7)

-2 -1 1 2

1

2

3

4v1

v2

v3

v4

p

q

0

-2 -1 1 2 3 4

1

2

3

4

5

6

7

8

v2 4v1 2

v2 3

v3 4

v1 4

p

q

0



Induction

Use induction algorithm to construct tiling for Gr>0(N + 1,M) fromtiling for Gr>0(N,M) [Huang, 2015].

1

2

3

4

5 6

7

8

9

10

11

2

1

5

2

9

11

8

6

4

1

2

3

4

5 6

7

8

9

10

11

Figure: Using the induction algorithm.



Induction

Use induction algorithm to construct tiling for Gr>0(N + 1,M) fromtiling for Gr>0(N,M) [Huang, 2015].

1

2

3

4

5 6

7

8

9

10

11

2

1

5

2

9

11

8

6

4

1

2

3

4

5 6

7

8

9

10

11

Figure: Using the induction algorithm.



Induction continued

Triangulation of the white polygons depends on the weights of ourpoints.

1 2

2 3

3 4

4 5

5 6

6 7

7 8

8 9

9 10

10 111 11

1 2 3

2 3 4

3 4 5

4 5 6 5 6 7

6 7 8

7 8 9

8 9 10

9 10 11

1 10 11

1 2 11

2 5 6

2 5 8

1 2 8

1 2 9

6 8 9

1 2 5

1 2 4

2 4 5

5 6 8

1 5 8

1 9 11

1 8 9

1 6 8

1 2 8

8 9 11

1 2 3

2 3 4

3 4 5

4 5 6 5 6 7

6 7 8

7 8 9

8 9 10

9 10 11

1 10 11

1 2 11

2 5 6

2 5 8

1 2 8

1 2 9

6 8 9

1 2 5

1 2 4

2 4 5

5 6 8

1 5 8

1 9 11

1 8 9

1 6 8

1 2 8

8 9 111 9 9 11

1 8

1 5

2 5

5 8

6 8

2 4

Figure: From N = 2 to N = 3.



Induction continued

Triangulation of the white polygons depends on the weights of ourpoints.

1 2

2 3

3 4

4 5

5 6

6 7

7 8

8 9

9 10

10 111 11

1 2 3

2 3 4

3 4 5

4 5 6 5 6 7

6 7 8

7 8 9

8 9 10

9 10 11

1 10 11

1 2 11

2 5 6

2 5 8

1 2 8

1 2 9

6 8 9

1 2 5

1 2 4

2 4 5

5 6 8

1 5 8

1 9 11

1 8 9

1 6 8

1 2 8

8 9 11

1 2 3

2 3 4

3 4 5

4 5 6 5 6 7

6 7 8

7 8 9

8 9 10

9 10 11

1 10 11

1 2 11

2 5 6

2 5 8

1 2 8

1 2 9

6 8 9

1 2 5

1 2 4

2 4 5

5 6 8

1 5 8

1 9 11

1 8 9

1 6 8

1 2 8

8 9 111 9 9 11

1 8

1 5

2 5

5 8

6 8

2 4

Figure: From N = 2 to N = 3.



Plabic graphs and tilings

Duality map gives correspondence between soliton graphs, tilings ofM-gon.

The planar dual of plabic graph is a plabic tiling [Oh et al., 2015].

To check if a plabic graph is realizable, check if the corresponding tilingis the dual of a soliton graph.














Results

Choices of parameters

For N = 3, M = 6, 7 or 8, every plabic graph is realizable for somechoice of parameters p1, p2, . . . , pM .

Which plabic graphs we can realize depends on our choice of pi .


Results

Choices of parameters

For N = 3, M = 6, 7 or 8, every plabic graph is realizable for somechoice of parameters p1, p2, . . . , pM .

Which plabic graphs we can realize depends on our choice of pi .


Results

Examples

1

2

34

5

61

2

3

4

5

6

136

235 145

135

123

234

345

456

156

126

134 356

125

135

123

234

345

456

156

126

Figure: Tilings which are only realizable for some choices of parameters.


Results

Classification for Gr(3, 6) and Gr(3, 7).

For Gr(3, 6), there are 34 possible graphs, each generic choice ofparameters lets us realize 32 of them [Huang, 2015].

For Gr(3, 7), there are 259 possible graphs, for each generic choice ofpi we can realize 231 of them [Kodama and K].

Main obstacle: same as in Gr(3, 6) case.

For Gr(3, 8), every graph is realizable for some choice of parameters.

Do not yet know which graphs realizable for each choice of parameters.


Results








Results








Results








Results

The general case

Not all plabic graphs are realizable.

Can build plabic graph from any simple, non-stretchable arrangementof pseudolines, which gives a counter-example [Kodama and K.]

Thanks to Hugh Thomas for suggesting this approach!

Smallest counter-example of this form is for Gr>0(9, 18)

Smaller counter-example exists for Gr>0(4, 8) [Galashin et al., 2019].


Results

The general case







Results

The general case







Results

The general case







Results

The general case







Results

References I

P. Galashin, A. Postnikov, and L. Williams. Higher secondary polytopesand regular plabic graphs. Preprint, 2019. arXiv:1909.05435 [math.CO].

J. Huang. Classification of Soliton Graphs on Totally PositiveGrassmannian. PhD thesis, The Ohio State University, 2015.

Y. Kodama and L. Williams. KP solitons and total positivity on theGrassmannian. Inventiones Mathematicae, 198, 2014. arXiv:1106.0023[math.CO].

S. Oh, D. Speyer, and A. Postnikov. Weak separation and plabic graphs.Proceedings of the London Mathematical Society, 110, 2015.arXiv:1109.4434 [math.CO].

A. Postnikov. Total positivity, Grassmannians and networks. Preprint,2006. arXiv:math/0609764 [math.CO].

G. Ringel. Teilungen der Ebene durch Geraden oder topologische Geraden.Math. Zeitschrift, 64:79–102, 1956.


Results

References II

J. S. Scott. Grassmannians and Cluster Algebras. Proceedings of theLondon Mathematical Society, 92(2):345–380, 03 2006. doi:10.1112/S0024611505015571. arXiv: arXiv:math/0311148 [math.CO].


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