R -systems Pavel Galashin MIT galashin@mit.edu AMS Joint Meeting, - - PowerPoint PPT Presentation

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R -systems Pavel Galashin MIT galashin@mit.edu AMS Joint Meeting, - - PowerPoint PPT Presentation

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R -systems Pavel Galashin MIT galashin@mit.edu AMS Joint Meeting, San Diego, CA, January 13, 2018 Joint work with Pavlo Pylyavskyy Pavel Galashin (MIT) R -systems San Diego, CA, 01/13/2018 1 / 23 Pavel Galashin (MIT) R -systems San Diego,

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R-systems

Pavel Galashin

MIT galashin@mit.edu

AMS Joint Meeting, San Diego, CA, January 13, 2018 Joint work with Pavlo Pylyavskyy

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 1 / 23

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Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 2 / 23

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Part 1: Definition

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A system of equations

Let G = (V , E) be a strongly connected digraph.

a b c d bc c(c + d) d(c + d) ac X X ′

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 4 / 23

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A system of equations

Let G = (V , E) be a strongly connected digraph.

a b c d bc c(c + d) d(c + d) ac X X ′

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 4 / 23

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A system of equations

Let G = (V , E) be a strongly connected digraph.

a b c d bc c(c + d) d(c + d) ac a′ b′ c′ d′ X X ′

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 4 / 23

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A system of equations

Let G = (V , E) be a strongly connected digraph.

a b c d bc c(c + d) d(c + d) ac a′ b′ c′ d′ X X ′

∀ v ∈ V , XvX ′

v = v→w

Xw

u→v

1 X ′

u

−1

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 4 / 23

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SLIDE 8

A system of equations

Let G = (V , E) be a strongly connected digraph.

a b c d bc c(c + d) d(c + d) ac X X ′

∀ v ∈ V , XvX ′

v = v→w

Xw

u→v

1 X ′

u

−1

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 4 / 23

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A system of equations

Let G = (V , E) be a strongly connected digraph.

a b c d bc c(c + d) d(c + d) ac X X ′

∀ v ∈ V , XvX ′

v = v→w

Xw

u→v

1 X ′

u

−1 d · ac = (a)

  • 1

c(c + d) + 1 d(c + d) −1

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 4 / 23

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Solution

Theorem (G.-Pylyavskyy, 2017)

Let G = (V , E) be a strongly connected digraph. Then there exists a birational map φ : PV PV such that X, X ′ ∈ PV give a solution ⇐ ⇒ X ′ = φ(X).

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 5 / 23

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Periodic examples (exercise)

a b c d e f

a b c d e f g

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 6 / 23

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Arborescence formula

a b c d a b c d a b c d wt = acd wt = ad2 wt = abd a b c d a b c d a b c d wt = abc wt = bd2 wt = bcd

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 7 / 23

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Map

R-systems Superpotential & Mirror symmetry Birational toggling Geometric RSK Birational rowmotion Zamolodchikov periodicity Cluster algebras LP algebras Integrable systems R-systems Birational rowmotion Cluster algebras Integrable systems

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 8 / 23

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Map

R-systems Superpotential & Mirror symmetry Birational toggling Geometric RSK Birational rowmotion Zamolodchikov periodicity Cluster algebras LP algebras Integrable systems Cluster algebras Integrable systems R-systems Birational rowmotion Zamolodchikov periodicity Birational toggling Geometric RSK

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 9 / 23

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Birational rowmotion ⊆ R-systems

ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ 1 P

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 10 / 23

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Birational rowmotion ⊆ R-systems

ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ 1 P ˆ P

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 10 / 23

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Birational rowmotion ⊆ R-systems

ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ 1 P ˆ P G(P)

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 10 / 23

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Birational rowmotion ⊆ R-systems

ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ 1 P ˆ P G(P)

Proposition (G.-Pylyavskyy, 2017)

Birational rowmotion on P = R-system associated with G(P).

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 10 / 23

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Part 2: Singularity confinement

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Map

R-systems Superpotential & Mirror symmetry Birational toggling Geometric RSK Birational rowmotion Zamolodchikov periodicity Cluster algebras LP algebras Integrable systems Cluster algebras Integrable systems Birational rowmotion Cluster algebras LP algebras Integrable systems R-systems

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 12 / 23

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The Laurent phenomenon

Somos-4 sequence: τn+4 =

ατn+1τn+3+βτ 2

n+2

τn

.

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 13 / 23

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The Laurent phenomenon

Somos-4 sequence: τn+4 =

ατn+1τn+3+βτ 2

n+2

τn

.

Theorem (Fomin-Zelevinsky, 2002)

For each n > 4, τn is a Laurent polynomial in α, β, τ1, τ2, τ3, τ4.

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 13 / 23

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The Laurent phenomenon

Somos-4 sequence: τn+4 =

ατn+1τn+3+βτ 2

n+2

τn

.

Theorem (Fomin-Zelevinsky, 2002)

For each n > 4, τn is a Laurent polynomial in α, β, τ1, τ2, τ3, τ4.

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 13 / 23

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Singularity confinement

Consider a mapping of the plane (xn−1, xn) → (xn, xn+1) given by

xn+1 = αxn+β

xn−1x2

n .

substitute xn = τn+1τn−1

τ 2

n

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 14 / 23

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Singularity confinement

Consider a mapping of the plane (xn−1, xn) → (xn, xn+1) given by

xn+1 = αxn+β

xn−1x2

n .

substitute xn = τn+1τn−1

τ 2

n

x3 = αx2+β

x1x2

2 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 14 / 23

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Singularity confinement

Consider a mapping of the plane (xn−1, xn) → (xn, xn+1) given by

xn+1 = αxn+β

xn−1x2

n .

substitute xn = τn+1τn−1

τ 2

n

x3 = αx2+β

x1x2

2

x4 = (βx1x2

2 +α2x2+αβ)x1x2

(αx2+β)2

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 14 / 23

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Singularity confinement

Consider a mapping of the plane (xn−1, xn) → (xn, xn+1) given by

xn+1 = αxn+β

xn−1x2

n .

substitute xn = τn+1τn−1

τ 2

n

x3 = αx2+β

x1x2

2

x4 = (βx1x2

2 +α2x2+αβ)x1x2

(αx2+β)2

x5 = (αβx2

1 x3 2 +···+β3)(αx2+β)

(βx1x2

2 +α2x2+αβ)2x1 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 14 / 23

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SLIDE 28

Singularity confinement

Consider a mapping of the plane (xn−1, xn) → (xn, xn+1) given by

xn+1 = αxn+β

xn−1x2

n .

substitute xn = τn+1τn−1

τ 2

n

x3 = αx2+β

x1x2

2

x4 = (βx1x2

2 +α2x2+αβ)x1x2

(αx2+β)2

x5 = (αβx2

1 x3 2 +···+β3)(αx2+β)

(βx1x2

2 +α2x2+αβ)2x1

x6 = (α3βx2

1 x4 2 +···+αβ4)(βx1x2 2 +α2x2+αβ)

(αβx2

1 x3 2 +···+β3)2x2 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 14 / 23

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Singularity confinement

Consider a mapping of the plane (xn−1, xn) → (xn, xn+1) given by

xn+1 = αxn+β

xn−1x2

n .

substitute xn = τn+1τn−1

τ 2

n

x3 = αx2+β

x1x2

2

x4 = (βx1x2

2 +α2x2+αβ)x1x2

(αx2+β)2

x5 = (αβx2

1 x3 2 +···+β3)(αx2+β)

(βx1x2

2 +α2x2+αβ)2x1

x6 = (α3βx2

1 x4 2 +···+αβ4)(βx1x2 2 +α2x2+αβ)

(αβx2

1 x3 2 +···+β3)2x2

x7 = (αβ3x4

1 x6 2 +···+β6x2)(αβx2 1 x3 2 +···+β3)x1x2

(α3βx2

1 x4 2 +···+αβ4)2 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 14 / 23

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SLIDE 30

Singularity confinement

Consider a mapping of the plane (xn−1, xn) → (xn, xn+1) given by

xn+1 = αxn+β

xn−1x2

n .

substitute xn = τn+1τn−1

τ 2

n

x3 = αx2+β

x1x2

2

x4 = (βx1x2

2 +α2x2+αβ)x1x2

(αx2+β)2

x5 = (αβx2

1 x3 2 +···+β3)(αx2+β)

(βx1x2

2 +α2x2+αβ)2x1

x6 = (α3βx2

1 x4 2 +···+αβ4)(βx1x2 2 +α2x2+αβ)

(αβx2

1 x3 2 +···+β3)2x2

x7 = (αβ3x4

1 x6 2 +···+β6x2)(αβx2 1 x3 2 +···+β3)x1x2

(α3βx2

1 x4 2 +···+αβ4)2

x8 = (α3β3x6

1 x8 2 +···+αβ8)(α3βx2 1 x4 2 +···+αβ4)

(αβ3x4

1 x6 2 +···+β6x2)2x2 1 x2 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 14 / 23

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SLIDE 31

Singularity confinement

Consider a mapping of the plane (xn−1, xn) → (xn, xn+1) given by

xn+1 = αxn+β

xn−1x2

n .

substitute xn = τn+1τn−1

τ 2

n

x3 = αx2+β

x1x2

2

x4 = (βx1x2

2 +α2x2+αβ)x1x2

(αx2+β)2

x5 = (αβx2

1 x3 2 +···+β3)(αx2+β)

(βx1x2

2 +α2x2+αβ)2x1

x6 = (α3βx2

1 x4 2 +···+αβ4)(βx1x2 2 +α2x2+αβ)

(αβx2

1 x3 2 +···+β3)2x2

x7 = (αβ3x4

1 x6 2 +···+β6x2)(αβx2 1 x3 2 +···+β3)x1x2

(α3βx2

1 x4 2 +···+αβ4)2

x8 = (α3β3x6

1 x8 2 +···+αβ8)(α3βx2 1 x4 2 +···+αβ4)

(αβ3x4

1 x6 2 +···+β6x2)2x2 1 x2 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 14 / 23

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SLIDE 32

Singularity confinement

Consider a mapping of the plane (xn−1, xn) → (xn, xn+1) given by

xn+1 = αxn+β

xn−1x2

n .

substitute xn = τn+1τn−1

τ 2

n

substitute xn = τn+1τn−1

τ 2

n

x3 = αx2+β

x1x2

2

τ4 = αx2 + β x4 = (βx1x2

2 +α2x2+αβ)x1x2

(αx2+β)2

τ5 = βx1x2

2 + α2x2 + αβ

x5 = (αβx2

1 x3 2 +···+β3)(αx2+β)

(βx1x2

2 +α2x2+αβ)2x1

τ6 = αβx2

1x3 2 + · · · + β3

x6 = (α3βx2

1 x4 2 +···+αβ4)(βx1x2 2 +α2x2+αβ)

(αβx2

1 x3 2 +···+β3)2x2

τ7 = α3βx2

1x4 2 + · · · + αβ4

x7 = (αβ3x4

1 x6 2 +···+β6x2)(αβx2 1 x3 2 +···+β3)x1x2

(α3βx2

1 x4 2 +···+αβ4)2

τ8 = αβ3x4

1x6 2 + · · · + β6x2

x8 = (α3β3x6

1 x8 2 +···+αβ8)(α3βx2 1 x4 2 +···+αβ4)

(αβ3x4

1 x6 2 +···+β6x2)2x2 1 x2

τ9 = α3β3x6

1x8 2 + · · · + αβ8

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 14 / 23

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Examples: Somos and Gale-Robinson sequences

Somos-4 Somos-5 Somos-5 Somos-6 = GR(1 + 2 + 3) Somos-7 = GR(1 + 2 + 4) dP3

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 15 / 23

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Examples: subgraphs of a bidirected cycle

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 16 / 23

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Examples: subgraphs of a bidirected cycle

Controlled by a cluster algebra

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 16 / 23

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Examples: rectangle posets (Grinberg-Roby)

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 17 / 23

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Examples: rectangle posets (Grinberg-Roby)

Controlled by a Y -system

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 17 / 23

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SLIDE 38

Examples: cylindric posets

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 18 / 23

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Examples: cylindric posets

Controlled by an LP algebra

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 18 / 23

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Examples: toric digraphs

A B C D

. . . A B C D A B . . . . . . C D A B C D . . . . . . A B C D A B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 19 / 23

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Examples: toric digraphs

A B C D

. . . A B C D A B . . . . . . C D A B C D . . . . . . A B C D A B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Controlled by ???

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 19 / 23

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Examples: toric digraphs

A B C D

. . . A B C D A B . . . . . . C D A B C D . . . . . . A B C D A B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Controlled by ??? Rv(t) = τv(t−1)

τv(t) ;

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 19 / 23

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SLIDE 43

Examples: toric digraphs

A B C D

. . . A B C D A B . . . . . . C D A B C D . . . . . . A B C D A B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Controlled by ??? Rv(t) = τv(t−1)

τv(t) ;

Conjecture (G.-Pylyavskyy, 2017)

τv(t) is an irreducible polynomial with κ(t+2

2 ) monomials [κ = # Arb(G; u)] Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 19 / 23

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SLIDE 44

Examples: toric digraphs

A B C D

. . . A B C D A B . . . . . . C D A B C D . . . . . . A B C D A B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Controlled by ??? Rv(t) = τv(t−1)

τv(t) ;

Conjecture (G.-Pylyavskyy, 2017)

τv(t) is an irreducible polynomial with κ(t+2

2 ) monomials [κ = # Arb(G; u)]

τv(t + 1) =

  • T∈Arb(G;v) some product of τu(t)-s and τw(t − 1)-s

some other product of τu(t)-s and τw(t − 1)-s .

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 19 / 23

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Example: the universal R-system

A B C D E

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 20 / 23

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SLIDE 46

Example: the universal R-system

A B C D E

Controlled by ???

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 20 / 23

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SLIDE 47

Example: the universal R-system

A B C D E

Controlled by ??? Rv(t) = τv(t−1)

τv(t) ;

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 20 / 23

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SLIDE 48

Example: the universal R-system

A B C D E

Controlled by ??? Rv(t) = τv(t−1)

τv(t) ;

Conjecture (G.-Pylyavskyy, 2017)

τv(t) is an irreducible polynomial with κθ(t) monomials [κ = # Arb(G; u)]

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 20 / 23

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SLIDE 49

Example: the universal R-system

A B C D E

Controlled by ??? Rv(t) = τv(t−1)

τv(t) ;

Conjecture (G.-Pylyavskyy, 2017)

τv(t) is an irreducible polynomial with κθ(t) monomials [κ = # Arb(G; u)]

τv(t + 1) =

  • T∈Arb(G;v) some product of τu(t)-s and τw(t − 1)-s

some other product of τu(t)-s and τw(t − 1)-s .

Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 20 / 23

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SLIDE 50

Bibliography

Slides: http://math.mit.edu/~galashin/slides/san_diego.pdf Sage code: http://math.mit.edu/~galashin/applets.html Pavel Galashin and Pavlo Pylyavskyy. R-systems arXiv preprint arXiv:1709.00578, 2017. David Einstein and James Propp. Combinatorial, piecewise-linear, and birational homomesy for products of two chains. arXiv preprint arXiv:1310.5294, 2013. Sergey Fomin and Andrei Zelevinsky. Cluster algebras. I. Foundations.

  • J. Amer. Math. Soc., 15(2):497–529 (electronic), 2002.

Darij Grinberg and Tom Roby. Iterative properties of birational rowmotion II: rectangles and triangles.

  • Electron. J. Combin., 22(3):Paper 3.40, 49, 2015.

Andrew N. W. Hone. Laurent polynomials and superintegrable maps. SIGMA Symmetry Integrability Geom. Methods Appl., 3:Paper 022, 18, 2007.

  • Y. Ohta, K. M. Tamizhmani, B. Grammaticos, and A. Ramani.

Singularity confinement and algebraic entropy: the case of the discrete Painlev´ e equations.

  • Phys. Lett. A, 262(2-3):152–157, 1999.
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SLIDE 51

Thanks!

Superpotential & Mirror symmetry Birational toggling Geometric RSK Birational rowmotion Zamolodchikov periodicity Cluster algebras LP algebras Integrable systems

Thanks!

Birational rowmotion Cluster algebras Integrable systems

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SLIDE 52

Bibliography

Slides: http://math.mit.edu/~galashin/slides/san_diego.pdf Sage code: http://math.mit.edu/~galashin/applets.html Pavel Galashin and Pavlo Pylyavskyy. R-systems arXiv preprint arXiv:1709.00578, 2017. David Einstein and James Propp. Combinatorial, piecewise-linear, and birational homomesy for products of two chains. arXiv preprint arXiv:1310.5294, 2013. Sergey Fomin and Andrei Zelevinsky. Cluster algebras. I. Foundations.

  • J. Amer. Math. Soc., 15(2):497–529 (electronic), 2002.

Darij Grinberg and Tom Roby. Iterative properties of birational rowmotion II: rectangles and triangles.

  • Electron. J. Combin., 22(3):Paper 3.40, 49, 2015.

Andrew N. W. Hone. Laurent polynomials and superintegrable maps. SIGMA Symmetry Integrability Geom. Methods Appl., 3:Paper 022, 18, 2007.

  • Y. Ohta, K. M. Tamizhmani, B. Grammaticos, and A. Ramani.

Singularity confinement and algebraic entropy: the case of the discrete Painlev´ e equations.

  • Phys. Lett. A, 262(2-3):152–157, 1999.