R-systems. Pavel Galashin. MIT AMS Joint Meeting, San Diego, CA, January 13, 2018 Joint work with Pavlo Pylyavskyy

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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 / 22

Part 1: Definition

A system of equations Let G = (V, E) be a strongly connected digraph. b c c(c + d) d(c + d) a d bc X X ac Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 3 / 22

A system of equations Let G = (V, E) be a strongly connected digraph. b c c(c + d) d(c + d) a d bc X X ac Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 3 / 22

A system of equations Let G = (V, E) be a strongly connected digraph. b c c(c b + d) d(c + c d) a d a bc X X ac d Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 3 / 22

A system of equations Let G = (V, E) be a strongly connected digraph. b c c(c b + d) d(c + c d) a d a bc X X ac d ( ) ( v V, X v X v = X w v w u v 1 X u ) 1 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 3 / 22

A system of equations Let G = (V, E) be a strongly connected digraph. b c c(c + d) d(c + d) a d bc X X ac ( ) ( v V, X v X v = X w v w u v 1 X u ) 1 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 3 / 22

A system of equations Let G = (V, E) be a strongly connected digraph. b c c(c + d) d(c + d) a d bc X X ac v V, X v X v = ( ) ( X w v w u v ( 1 d ac = (a) c(c + d) + 1 d(c + d) 1 X u ) 1 ) 1 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 3 / 22

Solution Theorem (G.-Pylyavskyy, 2017) Let G = (V, E) be a strongly connected digraph. Then there exists a birational map φ : P V P V such that X, X P V give a solution X = φ(x ). Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 4 / 22

Periodic examples (exercise) c b b a d a c f g e f e d Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 5 / 22

Arborescence formula a b c d a b c d a b c d wt = acd wt = ad 2 wt = abd a b c d a b c d a b c d wt = abc wt = bd 2 wt = bcd Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 6 / 22

Map Cluster algebras LP algebras Integrable systems Zamolodchikov periodicity R-systems Superpotential & Mirror symmetry Birational rowmotion Birational toggling Geometric RSK Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 7 / 22

Map Cluster algebras LP algebras Integrable systems Zamolodchikov periodicity R-systems Superpotential & Mirror symmetry Birational rowmotion Birational toggling Geometric RSK Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 8 / 22

Birational rowmotion R-systems ˆ1 ˆ1 ˆ1 ˆ0 P ˆ0 ˆ0 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 9 / 22

Birational rowmotion R-systems ˆ1 ˆ1 ˆ1 ˆ0 P ˆ0 ˆP ˆ0 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 9 / 22

Birational rowmotion R-systems ˆ1 ˆ1 ˆ1 ˆ0 ˆ0 ˆ0 P ˆP G(P) Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 9 / 22

Birational rowmotion R-systems ˆ1 ˆ1 ˆ1 ˆ0 ˆ0 P ˆP G(P) ˆ0 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 9 / 22

Part 2: Singularity confinement

Map Cluster algebras LP algebras Integrable systems Zamolodchikov periodicity R-systems Superpotential & Mirror symmetry Birational rowmotion Birational toggling Geometric RSK Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 11 / 22

The Laurent phenomenon Somos-4 sequence: τ n+4 = ατ n+1τ n+3 +βτ 2 n+2 τ n. Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 12 / 22

The Laurent phenomenon Somos-4 sequence: τ n+4 = ατ n+1τ n+3 +βτ 2 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 San Diego, CA, 01/13/2018 12 / 22

The Laurent phenomenon Somos-4 sequence: τ n+4 = ατ n+1τ n+3 +βτ 2 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 San Diego, CA, 01/13/2018 12 / 22

Singularity confinement Consider a mapping of the plane (x n 1, x n ) (x n, x n+1 ) given by x n+1 = αx n+β x n 1. substitute x xn 2 n = τ n+1τ n 1 τn 2 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 13 / 22

Singularity confinement Consider a mapping of the plane (x n 1, x n ) (x n, x n+1 ) given by x n+1 = αx n+β x n 1. substitute x xn 2 n = τ n+1τ n 1 τn 2 x 3 = αx 2+β x 1 x 2 2 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 13 / 22

Singularity confinement Consider a mapping of the plane (x n 1, x n ) (x n, x n+1 ) given by x n+1 = αx n+β x n 1. substitute x xn 2 n = τ n+1τ n 1 τn 2 x 3 = αx 2+β x 1 x 2 2 x 4 = (βx 1x 2 2 +α2 x 2 +αβ)x 1 x 2 (αx 2 +β) 2 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 13 / 22

Singularity confinement Consider a mapping of the plane (x n 1, x n ) (x n, x n+1 ) given by x n+1 = αx n+β x n 1. substitute x xn 2 n = τ n+1τ n 1 τn 2 x 3 = αx 2+β x 1 x 2 2 x 4 = (βx 1x 2 2 +α2 x 2 +αβ)x 1 x 2 (αx 2 +β) 2 x 5 = (αβx2 1 x3 2 + +β3 )(αx 2 +β) (βx 1 x 2 2 +α2 x 2 +αβ) 2 x 1 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 13 / 22

Singularity confinement Consider a mapping of the plane (x n 1, x n ) (x n, x n+1 ) given by x n+1 = αx n+β x n 1. substitute x xn 2 n = τ n+1τ n 1 τn 2 x 3 = αx 2+β x 1 x 2 2 x 4 = (βx 1x 2 2 +α2 x 2 +αβ)x 1 x 2 (αx 2 +β) 2 x 5 = (αβx2 1 x3 2 + +β3 )(αx 2 +β) (βx 1 x 2 2 +α2 x 2 +αβ) 2 x 1 x 6 = (α3 βx 2 1 x4 2 + +αβ4 )(βx 1 x 2 2 +α2 x 2 +αβ) (αβx 2 1 x3 2 + +β3 ) 2 x 2 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 13 / 22

Singularity confinement Consider a mapping of the plane (x n 1, x n ) (x n, x n+1 ) given by x n+1 = αx n+β x n 1. substitute x xn 2 n = τ n+1τ n 1 τn 2 x 3 = αx 2+β x 1 x 2 2 x 4 = (βx 1x 2 2 +α2 x 2 +αβ)x 1 x 2 (αx 2 +β) 2 x 5 = (αβx2 1 x3 2 + +β3 )(αx 2 +β) (βx 1 x 2 2 +α2 x 2 +αβ) 2 x 1 x 6 = (α3 βx 2 1 x4 2 + +αβ4 )(βx 1 x 2 2 +α2 x 2 +αβ) (αβx 2 1 x3 2 + +β3 ) 2 x 2 x 7 = (αβ3 x 4 1 x6 2 + +β6 x 2 )(αβx 2 1 x3 2 + +β3 )x 1 x 2 (α 3 βx 2 1 x4 2 + +αβ4 ) 2 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 13 / 22

Singularity confinement Consider a mapping of the plane (x n 1, x n ) (x n, x n+1 ) given by x n+1 = αx n+β x n 1. substitute x xn 2 n = τ n+1τ n 1 τn 2 x 3 = αx 2+β x 1 x 2 2 x 4 = (βx 1x 2 2 +α2 x 2 +αβ)x 1 x 2 (αx 2 +β) 2 x 5 = (αβx2 1 x3 2 + +β3 )(αx 2 +β) (βx 1 x 2 2 +α2 x 2 +αβ) 2 x 1 x 6 = (α3 βx 2 1 x4 2 + +αβ4 )(βx 1 x 2 2 +α2 x 2 +αβ) (αβx 2 1 x3 2 + +β3 ) 2 x 2 x 7 = (αβ3 x 4 1 x6 2 + +β6 x 2 )(αβx 2 1 x3 2 + +β3 )x 1 x 2 (α 3 βx 2 1 x4 2 + +αβ4 ) 2 x 8 = (α3 β 3 x 6 1 x8 2 + +αβ8 )(α 3 βx 2 1 x4 2 + +αβ4 ) (αβ 3 x 4 1 x6 2 + +β6 x 2 ) 2 x 2 1 x 2 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 13 / 22

Singularity confinement Consider a mapping of the plane (x n 1, x n ) (x n, x n+1 ) given by x n+1 = αx n+β x n 1. substitute x xn 2 n = τ n+1τ n 1 τn 2 x 3 = αx 2+β x 1 x 2 2 x 4 = (βx 1x 2 2 +α2 x 2 +αβ)x 1 x 2 (αx 2 +β) 2 x 5 = (αβx2 1 x3 2 + +β3 )(αx 2 +β) (βx 1 x 2 2 +α2 x 2 +αβ) 2 x 1 x 6 = (α3 βx 2 1 x4 2 + +αβ4 )(βx 1 x 2 2 +α2 x 2 +αβ) (αβx 2 1 x3 2 + +β3 ) 2 x 2 x 7 = (αβ3 x 4 1 x6 2 + +β6 x 2 )(αβx 2 1 x3 2 + +β3 )x 1 x 2 (α 3 βx 2 1 x4 2 + +αβ4 ) 2 x 8 = (α3 β 3 x 6 1 x8 2 + +αβ8 )(α 3 βx 2 1 x4 2 + +αβ4 ) (αβ 3 x 4 1 x6 2 + +β6 x 2 ) 2 x 2 1 x 2 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 13 / 22

Singularity confinement Consider a mapping of the plane (x n 1, x n ) (x n, x n+1 ) given by x n+1 = αx n+β x n 1 x 2 n. substitute x n = τ n+1τ n 1 τ n 2 x 3 = αx 2+β x 1 x 2 2 x 4 = (βx 1x 2 2 +α2 x 2 +αβ)x 1 x 2 (αx 2 +β) 2 x 5 = (αβx2 1 x3 2 + +β3 )(αx 2 +β) (βx 1 x 2 2 +α2 x 2 +αβ) 2 x 1 x 6 = (α3 βx 2 1 x4 2 + +αβ4 )(βx 1 x 2 2 +α2 x 2 +αβ) (αβx 2 1 x3 2 + +β3 ) 2 x 2 τ 4 = αx 2 + β τ 5 = βx 1 x2 2 + α2 x 2 + αβ τ 6 = αβx1 2x 2 3 + + β3 τ 7 = α 3 βx1 2x 2 4 + + αβ4 x 7 = (αβ3 x 4 1 x6 2 + +β6 x 2 )(αβx 2 1 x3 2 + +β3 )x 1 x 2 (α 3 βx 2 1 x4 2 + +αβ4 ) 2 τ 8 = αβ 3 x 4 1 x 6 2 + + β6 x 2 x 8 = (α3 β 3 x 6 1 x8 2 + +αβ8 )(α 3 βx 2 1 x4 2 + +αβ4 ) (αβ 3 x 4 1 x6 2 + +β6 x 2 ) 2 x 2 1 x 2 τ 9 = α 3 β 3 x 6 1 x 8 2 + + αβ8 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 13 / 22

Examples: Somos and Gale-Robinson sequences Somos-4 Somos-5 Somos-5 Somos-6 = GR(1 + 2 + 3) Somos-7 = GR(1 + 2 + 4) dp 3 Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 14 / 22

Examples: subgraphs of a bidirected cycle Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 15 / 22

Examples: subgraphs of a bidirected cycle Controlled by a cluster algebra Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 15 / 22

Examples: rectangle posets (Grinberg-Roby) Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 16 / 22

Examples: rectangle posets (Grinberg-Roby) Controlled by a Y -system Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 16 / 22

Examples: cylindric posets Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 17 / 22

Examples: cylindric posets Controlled by an LP algebra Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 17 / 22

Examples: toric digraphs.................. D C... A B C D A B...... C D A B C D... A B... A B C D A B..................... Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 18 / 22

Examples: toric digraphs.................. D C... A B C D A B...... C D A B C D... A B... A B C D A B..................... Controlled by??? Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 18 / 22

Examples: toric digraphs.................. D C... A B C D A B...... C D A B C D... A B... A B C D A B... Controlled by???.................. R v (t) = τv (t 1) τ v (t) ; Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 18 / 22

Examples: toric digraphs.................. D C... A B C D A B...... C D A B C D... A B... A B C D A B... Controlled by???.................. R v (t) = τv (t 1) τ v (t) ; Conjecture (G.-Pylyavskyy, 2017) τ v (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 18 / 22

Examples: toric digraphs.................. D C... A B C D A B...... C D A B C D... A B... A B C D A B... Controlled by???.................. R v (t) = τv (t 1) τ v (t) ; Conjecture (G.-Pylyavskyy, 2017) τ v (t) is an irreducible polynomial with κ (t+2 2 ) monomials [κ = # Arb(G; u)] τ v (t + 1) = T Arb(G;v) some product of τ u(t)-s and τ w (t 1)-s. some other product of τ u (t)-s and τ w (t 1)-s Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 18 / 22

Example: the universal R-system C B A D E Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 19 / 22

Example: the universal R-system C B A D Controlled by??? E Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 19 / 22

Example: the universal R-system C B A D Controlled by??? E R v (t) = τv (t 1) τ v (t) ; Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 19 / 22

Example: the universal R-system C B A D Controlled by??? E R v (t) = τv (t 1) τ v (t) ; Conjecture (G.-Pylyavskyy, 2017) τ v (t) is an irreducible polynomial with κ θ(t) monomials [κ = # Arb(G; u)] Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 19 / 22

Example: the universal R-system C B A D Controlled by??? E R v (t) = τv (t 1) τ v (t) ; Conjecture (G.-Pylyavskyy, 2017) τ v (t) is an irreducible polynomial with κ θ(t) monomials [κ = # Arb(G; u)] τ v (t + 1) = T Arb(G;v) some product of τ u(t)-s and τ w (t 1)-s. some other product of τ u (t)-s and τ w (t 1)-s Pavel Galashin (MIT) R-systems San Diego, CA, 01/13/2018 19 / 22

Bibliography Slides: Sage code: http://math.mit.edu/~galashin/slides/san_diego.pdf 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 Painlevé equations. Phys. Lett. A, 262(2-3):152 157, 1999.

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

Bibliography Slides: Sage code: http://math.mit.edu/~galashin/slides/san_diego.pdf 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 Painlevé equations. Phys. Lett. A, 262(2-3):152 157, 1999.

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