R -systems Pavel Galashin MIT galashin@mit.edu UQAM, November 24, - - PowerPoint PPT Presentation

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R -systems Pavel Galashin MIT galashin@mit.edu UQAM, November 24, - - PowerPoint PPT Presentation

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R -systems Pavel Galashin MIT galashin@mit.edu UQAM, November 24, 2017 Joint work with Pavlo Pylyavskyy Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 1 / 31 Part 1: Definition A system of equations Let G = ( V , E ) be a strongly

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R-systems

Pavel Galashin

MIT galashin@mit.edu

UQAM, November 24, 2017 Joint work with Pavlo Pylyavskyy

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 1 / 31

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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 UQAM, 11/24/2017 3 / 31

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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 UQAM, 11/24/2017 3 / 31

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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 UQAM, 11/24/2017 3 / 31

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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 (K) PV (K) such that X, X ′ ∈ PV (K) give a solution if and only if X ′ = φ(X).

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 4 / 31

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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 UQAM, 11/24/2017 5 / 31

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The R-system

Definition

Let G = (V , E) be a strongly connected digraph. Then the R-system associated with G is a discrete dynamical system on PV (K) that consists

  • f iterating the map φ.

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 6 / 31

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Boring examples: a directed cycle

a b c d e f φ a b c d e f

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 7 / 31

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Boring examples: a directed cycle

a b c d e f φ a b c d e f

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 7 / 31

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Boring examples: a bidirected graph

a b c d e f g φ

1 a 1 b 1 c 1 d 1 e 1 f 1 g

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 8 / 31

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Boring examples: a bidirected graph

a b c d e f g φ

1 a 1 b 1 c 1 d 1 e 1 f 1 g

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 8 / 31

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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 UQAM, 11/24/2017 9 / 31

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Part 2: Toggle!

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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 UQAM, 11/24/2017 11 / 31

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Birational toggling

Let (P, ≤) be a poset and X = (Xv)v∈P. Add ˆ 0 and ˆ 1 to P and set Xˆ

0 = Xˆ 1 = 1.

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 12 / 31

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Birational toggling

Let (P, ≤) be a poset and X = (Xv)v∈P. Add ˆ 0 and ˆ 1 to P and set Xˆ

0 = Xˆ 1 = 1.

Definition (Kirillov (2001), Einstein-Propp (2013))

Birational toggle operation: X ′

vXv =

  • v⋖w

Xw

u⋖v

1 Xu −1 .

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 12 / 31

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Birational rowmotion for the product of two chains

Theorem (Grinberg-Roby, 2015)

For P = [n] × [k], birational rowmotion is periodic with period n + k.

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 13 / 31

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Birational rowmotion for the product of two chains

Theorem (Grinberg-Roby, 2015)

For P = [n] × [k], birational rowmotion is periodic with period n + k. “Inspired by” Volkov’s proof of Zamolodchikov periodicity:

Theorem (Volkov, 2005)

The Y -system of Type An−1 ⊗ Ak−1 is periodic with period n + k.

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 13 / 31

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Birational rowmotion for the product of two chains

Theorem (Grinberg-Roby, 2015)

For P = [n] × [k], birational rowmotion is periodic with period n + k. “Inspired by” Volkov’s proof of Zamolodchikov periodicity:

Theorem (Volkov, 2005)

The Y -system of Type An−1 ⊗ Ak−1 is periodic with period n + k.

Proposition (Glick, 2016)

There is a simple monomial transformation that shows that the two theorems above are equivalent.

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 13 / 31

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Birational rowmotion ⊆ R-systems

ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ 1 P

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 14 / 31

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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 UQAM, 11/24/2017 14 / 31

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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 UQAM, 11/24/2017 14 / 31

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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 UQAM, 11/24/2017 14 / 31

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Birational rowmotion ⊆ R-systems 1 a b c d 1

c+d a d b c+d ac c+d ad+bc+bd

1 1

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 15 / 31

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Birational rowmotion ⊆ R-systems 1 a b c d 1

c+d a a d b

b

c+d acc c+d ad+bc+bd

d 1 1 1 1

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 15 / 31

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Birational rowmotion ⊆ R-systems 1 a b c d 1

c+d a c+d a d b

b

c+d acc c+d ad+bc+bd

d 1 1

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 15 / 31

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Birational rowmotion ⊆ R-systems 1 a b c d 1

c+d a d b d b c+d acc c+d ad+bc+bd

d 1 1

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 15 / 31

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Birational rowmotion ⊆ R-systems 1 a b c d 1

c+d a d b c+d ac c+d ac c+d ad+bc+bd

d 1 1

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 15 / 31

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Birational rowmotion ⊆ R-systems 1 a b c d 1

c+d a d b c+d ac c+d ad+bc+bd c+d ad+bc+bd

1 1

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 15 / 31

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Birational rowmotion ⊆ R-systems a b c d 1 φ

c+d a d b c+d ac c+d ad+bc+bd

1 1

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 16 / 31

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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 UQAM, 11/24/2017 17 / 31

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Part 3: 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 UQAM, 11/24/2017 19 / 31

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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 UQAM, 11/24/2017 20 / 31

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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 UQAM, 11/24/2017 20 / 31

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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 UQAM, 11/24/2017 20 / 31

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Singularity confinement

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

xn+1xn−1 = α

xn + β x2

n .

substitute xn = τn+1τn−1

τ 2

n

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 21 / 31

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Singularity confinement

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

xn+1xn−1 = α

xn + β x2

n .

substitute xn = τn+1τn−1

τ 2

n

x3 = αx2+β

x1x2

2 Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 21 / 31

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Singularity confinement

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

xn+1xn−1 = α

xn + β x2

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 UQAM, 11/24/2017 21 / 31

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Singularity confinement

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

xn+1xn−1 = α

xn + β x2

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 UQAM, 11/24/2017 21 / 31

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Singularity confinement

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

xn+1xn−1 = α

xn + β x2

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 UQAM, 11/24/2017 21 / 31

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Singularity confinement

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

xn+1xn−1 = α

xn + β x2

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 UQAM, 11/24/2017 21 / 31

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Singularity confinement

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

xn+1xn−1 = α

xn + β x2

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 UQAM, 11/24/2017 21 / 31

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Singularity confinement

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

xn+1xn−1 = α

xn + β x2

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 UQAM, 11/24/2017 21 / 31

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Singularity confinement

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

xn+1xn−1 = α

xn + β x2

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 UQAM, 11/24/2017 21 / 31

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Moral

Sequence with Laurent property Integrable mapping with singularity confinement monomial transformation

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Moral

Sequence with Laurent property Integrable mapping with singularity confinement monomial transformation Lots of R-systems exhibit singularity confinement!

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 22 / 31

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Examples: subgraphs of a bidirected cycle

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 23 / 31

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Examples: subgraphs of a bidirected cycle

Controlled by a cluster algebra

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 23 / 31

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Examples: rectangle posets (Grinberg-Roby)

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 24 / 31

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Examples: rectangle posets (Grinberg-Roby)

Controlled by a Y -system

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 24 / 31

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Examples: cylindric posets

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 25 / 31

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Examples: cylindric posets

Controlled by an LP algebra

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 25 / 31

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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 UQAM, 11/24/2017 26 / 31

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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 UQAM, 11/24/2017 26 / 31

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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 UQAM, 11/24/2017 26 / 31

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SLIDE 58

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 UQAM, 11/24/2017 26 / 31

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SLIDE 59

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 UQAM, 11/24/2017 26 / 31

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SLIDE 60

Coefficients

Definition

Coefficient-free R-system: XvX ′

v = v→w

Xw

u→v

1 X ′

u

−1 .

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 27 / 31

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SLIDE 61

Coefficients

Definition

Coefficient-free R-system: XvX ′

v = v→w

Xw

u→v

1 X ′

u

−1 . R-system with coefficients: XvX ′

v = v→w

wt(v → w)Xw

u→v

wt(u → v) 1 X ′

u

−1 .

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 27 / 31

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SLIDE 62

Example: the universal R-system

A B C D E

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 28 / 31

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SLIDE 63

Example: the universal R-system

A B C D E

Controlled by ???

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 28 / 31

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SLIDE 64

Example: the universal R-system

A B C D E

Controlled by ??? Rv(t) = τv(t−1)

τv(t) ;

Pavel Galashin (MIT) R-systems UQAM, 11/24/2017 28 / 31

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SLIDE 65

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 UQAM, 11/24/2017 28 / 31

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SLIDE 66

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 UQAM, 11/24/2017 28 / 31

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SLIDE 67

Bibliography

Slides: http://math.mit.edu/~galashin/slides/UQAM.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 68

Thanks!

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SLIDE 69

Bibliography

Slides: http://math.mit.edu/~galashin/slides/UQAM.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.