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

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 connected digraph . c ( c + d ) d ( c + d ) b c a ac d bc X X ′ Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 3 / 31

A system of equations Let G = ( V , E ) be a strongly connected digraph . c ( c + d ) d ( c + d ) b c a ac d bc X X ′ Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 3 / 31

A system of equations Let G = ( V , E ) be a strongly connected digraph . c ( c + d ) d ( c + d ) b c a ac d bc X X ′ Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 3 / 31

Solution Theorem (G.-Pylyavskyy, 2017) Let G = ( V , E ) be a strongly connected digraph. Then there exists a birational map φ : P V ( K ) ��� P V ( K ) such that X , X ′ ∈ P V ( K ) give a solution if and only if X ′ = φ ( X ) . Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 4 / 31

Arborescence formula b c b c b c a a a d d d wt = ad 2 wt = acd wt = abd b c b c b c a a a d d d wt = bd 2 wt = abc wt = bcd Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 5 / 31

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 P V ( K ) that consists of iterating the map φ . Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 6 / 31

Boring examples: a directed cycle c c b d a e d b φ e a f f Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 7 / 31

Boring examples: a directed cycle c c b d a e d b φ e a f f Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 7 / 31

Boring examples: a bidirected graph 1 1 b a b a 1 1 1 g c f g c f φ e 1 1 d e d Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 8 / 31

Boring examples: a bidirected graph 1 1 b a b a 1 1 1 g c f g c f φ e 1 1 d e d Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 8 / 31

Map Cluster Cluster Integrable Integrable LP algebras algebras algebras systems systems Zamolodchikov Superpotential & R-systems R-systems periodicity Mirror symmetry Birational Birational Birational Geometric rowmotion rowmotion toggling RSK Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 9 / 31

Part 2: Toggle!

Map Cluster Cluster Integrable Integrable LP algebras algebras algebras systems systems Zamolodchikov Zamolodchikov Superpotential & R-systems R-systems periodicity periodicity Mirror symmetry Birational Birational Birational Birational Geometric Geometric rowmotion rowmotion toggling toggling RSK RSK Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 11 / 31

Birational toggling Let ( P , ≤ ) be a poset and X = ( X v ) v ∈ P . Add ˆ 0 and ˆ 1 to P and set X ˆ 0 = X ˆ 1 = 1. Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 12 / 31

Birational toggling Let ( P , ≤ ) be a poset and X = ( X v ) v ∈ P . Add ˆ 0 and ˆ 1 to P and set X ˆ 0 = X ˆ 1 = 1. Definition (Kirillov (2001), Einstein-Propp (2013)) Birational toggle operation : � − 1 �� � �� 1 X ′ v X v = X w . X u v ⋖ w u ⋖ v Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 12 / 31

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

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 A n − 1 ⊗ A k − 1 is periodic with period n + k. Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 13 / 31

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 A n − 1 ⊗ A k − 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

Birational rowmotion ⊆ R -systems ˆ ˆ ˆ 1 1 1 ˆ ˆ ˆ 0 0 0 P Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 14 / 31

Birational rowmotion ⊆ R -systems ˆ ˆ ˆ ˆ 1 1 1 1 ˆ ˆ ˆ ˆ 0 0 0 0 ˆ P P Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 14 / 31

Birational rowmotion ⊆ R -systems ˆ ˆ ˆ ˆ 1 1 1 1 ˆ ˆ ˆ ˆ 0 0 0 0 ˆ P P G ( P ) Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 14 / 31

Birational rowmotion ⊆ R -systems ˆ ˆ ˆ 1 1 1 ˆ ˆ ˆ 0 0 0 ˆ P 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

Birational rowmotion ⊆ R -systems 1 1 c + d c + d c d ad + bc + bd ac c + d d a b a b 1 1 Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 15 / 31

Birational rowmotion ⊆ R -systems 1 1 1 c + d c + d c ac c d d ad + bc + bd c + d d a a a b b b 1 1 1 Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 15 / 31

Birational rowmotion ⊆ R -systems 1 1 c + d c + d c ac c d d ad + bc + bd c + d c + d d a b b a a b 1 1 Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 15 / 31

Birational rowmotion ⊆ R -systems 1 1 c + d c + d c ac c d d ad + bc + bd c + d d d a b a b b 1 1 Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 15 / 31

Birational rowmotion ⊆ R -systems 1 1 c + d c + d c + d c d d ad + bc + bd ac ac c + d d a b a b 1 1 Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 15 / 31

Birational rowmotion ⊆ R -systems 1 1 c + d c + d c + d c d ad + bc + bd ad + bc + bd ac c + d d a b a b 1 1 Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 15 / 31

Birational rowmotion ⊆ R -systems 1 c + d c + d c d ad + bc + bd ac φ c + d d a b a b 1 1 Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 16 / 31

Map Cluster Cluster Integrable Integrable LP algebras algebras algebras systems systems Zamolodchikov Superpotential & R-systems R-systems periodicity Mirror symmetry Birational Birational Birational Geometric rowmotion rowmotion toggling RSK Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 17 / 31

Part 3: Singularity confinement

Map Cluster Cluster Cluster Integrable Integrable Integrable LP algebras LP algebras algebras algebras algebras systems systems systems Zamolodchikov Superpotential & R-systems R-systems periodicity Mirror symmetry Birational Birational Birational Geometric rowmotion rowmotion toggling RSK Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 19 / 31

The Laurent phenomenon ατ n +1 τ n +3 + βτ 2 Somos-4 sequence: τ n +4 = n +2 . τ n Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 20 / 31

The Laurent phenomenon ατ n +1 τ n +3 + βτ 2 Somos-4 sequence: τ n +4 = n +2 . τ n Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 20 / 31

The Laurent phenomenon ατ n +1 τ n +3 + βτ 2 Somos-4 sequence: τ n +4 = n +2 . τ n Theorem (Fomin-Zelevinsky, 2002) For each n > 4 , τ n is a Laurent polynomial in α, β, τ 1 , τ 2 , τ 3 , τ 4 . Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 20 / 31

Singularity confinement Consider a mapping of the plane ( x n − 1 , x n ) �→ ( x n , x n +1 ) given by substitute x n = τ n +1 τ n − 1 x n + β x n +1 x n − 1 = α n . τ 2 x 2 n Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 21 / 31

Singularity confinement Consider a mapping of the plane ( x n − 1 , x n ) �→ ( x n , x n +1 ) given by substitute x n = τ n +1 τ n − 1 x n + β x n +1 x n − 1 = α n . τ 2 x 2 n x 3 = α x 2 + β x 1 x 2 2 Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 21 / 31

Singularity confinement Consider a mapping of the plane ( x n − 1 , x n ) �→ ( x n , x n +1 ) given by substitute x n = τ n +1 τ n − 1 x n + β x n +1 x n − 1 = α n . τ 2 x 2 n x 3 = α x 2 + β x 1 x 2 2 x 4 = ( β x 1 x 2 2 + α 2 x 2 + αβ ) x 1 x 2 ( α x 2 + β ) 2 Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 21 / 31

Singularity confinement Consider a mapping of the plane ( x n − 1 , x n ) �→ ( x n , x n +1 ) given by substitute x n = τ n +1 τ n − 1 x n + β x n +1 x n − 1 = α n . τ 2 x 2 n x 3 = α x 2 + β x 1 x 2 2 x 4 = ( β x 1 x 2 2 + α 2 x 2 + αβ ) x 1 x 2 ( α x 2 + β ) 2 x 5 = ( αβ x 2 1 x 3 2 + ··· + β 3 )( α x 2 + β ) ( β x 1 x 2 2 + α 2 x 2 + αβ ) 2 x 1 Pavel Galashin (MIT) R -systems UQAM, 11/24/2017 21 / 31