Classification Scheme for Painlev´ e Equations n=1: L. Fuchs, H. Poincar´ e � dy � 2 = 4 y 3 − g 2 y − g 3 , g 2 , g 3 ∈ C Weierstrass ℘ ( t | g 2 , g 3 ) dt dy dt = a ( t ) y 2 + b ( t ) y + c ( t ), (Riccati equation) n=2: P. Painlev´ e, B. Gambier — Painlev´ e equations and Painlev´ e transcendents: (P-I) d 2 y dt 2 = 6 y 2 + t ; (P-II) d 2 y dt 2 = 2 y 3 + ty + α ; � dy (P-III) d 2 y � 2 dt 2 = 1 − 1 dt + 1 dy t ( α y 2 + β ) + γ y 3 + δ y ; y dt t � dy (P-IV) d 2 y � 2 dt 2 = 1 + 3 2 y 3 + 4 ty 2 + 2( t 2 − α ) y + β y ; 2 y dt � 1 � � dy (P-V) d 2 y dt + ( y − 1) 2 � 2 1 − 1 dy � α y + β � + γ y t + δ y ( y + 1) dt 2 = 2 y + ; y − 1 dt t t 2 y y − 1 � 1 � � dy � 1 � dy (P-VI) d 2 y � 2 dt 2 = 1 1 1 1 1 y + y − 1 + − t + t − 1 + dt + 2 y − t dt y − t y ( y − 1)( y − t ) α + β t ( y − 1) 2 + δ t ( t − 1) t − 1 � � y 2 + γ . t 2 ( t − 1) 2 ( y − t ) 2 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 4 / 41
Classification Scheme for Painlev´ e Equations n=1: L. Fuchs, H. Poincar´ e � dy � 2 = 4 y 3 − g 2 y − g 3 , g 2 , g 3 ∈ C Weierstrass ℘ ( t | g 2 , g 3 ) dt dy dt = a ( t ) y 2 + b ( t ) y + c ( t ), (Riccati equation) n=2: P. Painlev´ e, B. Gambier — Painlev´ e equations and Painlev´ e transcendents: (P-I) d 2 y dt 2 = 6 y 2 + t ; Painlev´ e equations have parameters! (P-II) d 2 y dt 2 = 2 y 3 + ty + α ; � dy (P-III) d 2 y � 2 dt 2 = 1 − 1 dy dt + 1 t ( α y 2 + β ) + γ y 3 + δ y ; y dt t � dy (P-IV) d 2 y � 2 dt 2 = 1 + 3 2 y 3 + 4 ty 2 + 2( t 2 − α ) y + β y ; 2 y dt � 1 � � dy (P-V) d 2 y dt + ( y − 1) 2 � 2 1 − 1 dy � α y + β � + γ y t + δ y ( y + 1) dt 2 = 2 y + ; y − 1 dt t t 2 y y − 1 � 1 � � dy � 1 � dy (P-VI) d 2 y � 2 dt 2 = 1 1 1 1 1 y + y − 1 + − t + t − 1 + dt + 2 y − t dt y − t y ( y − 1)( y − t ) α + β t ( y − 1) 2 + δ t ( t − 1) t − 1 � � y 2 + γ . t 2 ( t − 1) 2 ( y − t ) 2 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 4 / 41
Classification Scheme for Painlev´ e Equations n=1: L. Fuchs, H. Poincar´ e � dy � 2 = 4 y 3 − g 2 y − g 3 , g 2 , g 3 ∈ C Weierstrass ℘ ( t | g 2 , g 3 ) dt dy dt = a ( t ) y 2 + b ( t ) y + c ( t ), (Riccati equation) n=2: P. Painlev´ e, B. Gambier — Painlev´ e equations and Painlev´ e transcendents: (P-I) d 2 y dt 2 = 6 y 2 + t ; P IV Painlev´ e equations have parameters! P VI P V P II P I (P-II) d 2 y dt 2 = 2 y 3 + ty + α ; P III � dy (P-III) d 2 y � 2 dt 2 = 1 − 1 dy dt + 1 t ( α y 2 + β ) + γ y 3 + δ y ; y dt t � dy (P-IV) d 2 y � 2 dt 2 = 1 + 3 2 y 3 + 4 ty 2 + 2( t 2 − α ) y + β y ; 2 y dt � 1 � � dy (P-V) d 2 y dt + ( y − 1) 2 � 2 1 − 1 dy � α y + β � + γ y t + δ y ( y + 1) dt 2 = 2 y + ; y − 1 dt t t 2 y y − 1 � 1 � � dy � 1 � dy (P-VI) d 2 y � 2 dt 2 = 1 1 1 1 1 y + y − 1 + − t + t − 1 + dt + 2 y − t dt y − t y ( y − 1)( y − t ) α + β t ( y − 1) 2 + δ t ( t − 1) t − 1 � � y 2 + γ . t 2 ( t − 1) 2 ( y − t ) 2 n ≥ 3: Still open. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 4 / 41
Classification Scheme for Painlev´ e Equations n=1: L. Fuchs, H. Poincar´ e � dy � 2 = 4 y 3 − g 2 y − g 3 , g 2 , g 3 ∈ C Weierstrass ℘ ( t | g 2 , g 3 ) dt dy dt = a ( t ) y 2 + b ( t ) y + c ( t ), (Riccati equation) n=2: P. Painlev´ e, B. Gambier — Painlev´ e equations and Painlev´ e transcendents: (P-I) d 2 y dt 2 = 6 y 2 + t ; P IV Painlev´ e equations have parameters! P VI P V P II P I (P-II) d 2 y dt 2 = 2 y 3 + ty + α ; P III � dy (P-III) d 2 y � 2 dt 2 = 1 − 1 dy dt + 1 t ( α y 2 + β ) + γ y 3 + δ y ; y dt t Hermite � dy (P-IV) d 2 y � 2 dt 2 = 1 + 3 2 y 3 + 4 ty 2 + 2( t 2 − α ) y + β Airy Gauss Kummer y ; Bessel 2 y dt � 1 � � dy (P-V) d 2 y dt + ( y − 1) 2 � 2 1 − 1 dy � α y + β � + γ y t + δ y ( y + 1) dt 2 = 2 y + ; y − 1 dt t t 2 y y − 1 � 1 � � dy � 1 � dy (P-VI) d 2 y � 2 dt 2 = 1 1 1 1 1 y + y − 1 + − t + t − 1 + dt + 2 y − t dt y − t y ( y − 1)( y − t ) α + β t ( y − 1) 2 + δ t ( t − 1) t − 1 � � y 2 + γ . t 2 ( t − 1) 2 ( y − t ) 2 n ≥ 3: Still open. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 4 / 41
Classification Scheme for Discrete Painlev´ e Equations In 2001 H. Sakai, developing the ideas of K. Okamoto in the differential case, proposed a classification scheme for Painlev´ e equations based on algebraic geometry. To each equation corresponds a pair of orthogonal sub-lattices (Π( R ) , Π( R ⊥ )) — the surface and the symmetry sub-lattice in the E (1) lattice, and a translation element in ˜ W ( R ⊥ ). 8 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 5 / 41
Classification Scheme for Discrete Painlev´ e Equations In 2001 H. Sakai, developing the ideas of K. Okamoto in the differential case, proposed a classification scheme for Painlev´ e equations based on algebraic geometry. To each equation corresponds a pair of orthogonal sub-lattices (Π( R ) , Π( R ⊥ )) — the surface and the symmetry sub-lattice in the E (1) lattice, and a translation element in ˜ W ( R ⊥ ). 8 q- P I � e � q � A (1) � E (1) 1 8 | α | 2=8 q- P VI q- P V q- P IV, q- P III � q � q � q � q � q � q � q � E (1) � E (1) � E (1) � D (1) � A (1) ( A 2 + A 1)(1) � q ( A 1 + A 1)(1) � q � A (1) � A (1) � � 8 7 6 5 4 1 0 | α | 2 = 14 | α | 2 = 4 � δ � δ � δ � c ,δ � c ,δ � c ,δ � c ,δ � c � � � � � � � � A (1) E (1) E (1) E (1) D (1) A (1) 2 A (1) A (1) 8 7 6 4 3 1 1 0 P VI, d- P V P V, d- P IV P III P III P III d- P III alt. d- P II � c ,δ � c ,δ � c � � � A (1) 2 A (1) A (1) 1 1 0 P IV, d- P II P II, alt.d- P I P I Symmetry-type classification scheme for Painlev´ e equations Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 5 / 41
Classification Scheme for Discrete Painlev´ e Equations In 2001 H. Sakai, developing the ideas of K. Okamoto in the differential case, proposed a classification scheme for Painlev´ e equations based on algebraic geometry. To each equation corresponds a pair of orthogonal sub-lattices (Π( R ) , Π( R ⊥ )) — the surface and the symmetry sub-lattice in the E (1) lattice, and a translation element in ˜ W ( R ⊥ ). 8 q- P I � e � q � A (1) � E (1) 1 8 | α | 2=8 q- P VI q- P V q- P IV, q- P III � q � q � q � q � q � q � q � E (1) � E (1) � E (1) � D (1) � A (1) ( A 2 + A 1)(1) � q ( A 1 + A 1)(1) � q � A (1) � A (1) � � 8 7 6 5 4 1 0 | α | 2 = 14 | α | 2 = 4 � δ � δ � δ � c ,δ � c ,δ � c ,δ � c ,δ � c � � � � � � � � A (1) E (1) E (1) E (1) D (1) A (1) 2 A (1) A (1) 8 7 6 4 3 1 1 0 P VI, d- P V P V, d- P IV P III P III P III d- P III alt. d- P II � c ,δ � c ,δ � c � � � A (1) 2 A (1) A (1) 1 1 0 P IV, d- P II P II, alt.d- P I P I Symmetry-type classification scheme for Painlev´ e equations So we see that in the discrete case the classification scheme is very rich, there are twenty-two different cases, and moreover, there are no generic expressions for equations of each type, the classification scheme is very algebraic. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 5 / 41
Classification Scheme for Discrete Painlev´ e Equations In 2001 H. Sakai, developing the ideas of K. Okamoto in the differential case, proposed a classification scheme for Painlev´ e equations based on algebraic geometry. To each equation corresponds a pair of orthogonal sub-lattices (Π( R ) , Π( R ⊥ )) — the surface and the symmetry sub-lattice in the E (1) lattice, and a translation element in ˜ W ( R ⊥ ). 8 q- P I � e � q � A (1) � E (1) 1 8 | α | 2=8 q- P VI q- P V q- P IV, q- P III � q � q � q � q � q � q � q � E (1) � E (1) � E (1) � D (1) � A (1) ( A 2 + A 1)(1) � q ( A 1 + A 1)(1) � q � A (1) � A (1) � � 8 7 6 5 4 1 0 | α | 2 = 14 | α | 2 = 4 � δ � δ � δ � c ,δ � c ,δ � c ,δ � c ,δ � c � � � � � � � � A (1) E (1) E (1) E (1) D (1) A (1) 2 A (1) A (1) 8 7 6 4 3 1 1 0 P VI, d- P V P V, d- P IV P III P III P III d- P III alt. d- P II � c ,δ � c ,δ � c � � � A (1) 2 A (1) A (1) 1 1 0 P IV, d- P II P II, alt.d- P I P I Symmetry-type classification scheme for Painlev´ e equations So we see that in the discrete case the classification scheme is very rich, there are twenty-two different cases, and moreover, there are no generic expressions for equations of each type, the classification scheme is very algebraic. One of my goals for today is to explain the main ideas behind this scheme. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 5 / 41
Classification Scheme for Discrete Painlev´ e Equations First, we note that although the previous classification scheme according to symmetries is more traditional, for the geometric approach, it is more natural to look at the classification based on the point-configuration or surface type : Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 6 / 41
Classification Scheme for Discrete Painlev´ e Equations First, we note that although the previous classification scheme according to symmetries is more traditional, for the geometric approach, it is more natural to look at the classification based on the point-configuration or surface type : A (1) A (1) 0 7 A (1) ∗ A (1) A (1) A (1) A (1) A (1) A (1) A (1) A (1) 0 1 2 3 4 5 6 7 8 A (1) ∗∗ A (1) ∗ A (1) ∗ D (1) D (1) D (1) D (1) D (1) 0 1 2 4 5 6 7 8 E (1) E (1) E (1) 6 7 8 Surface-type classification scheme for Painlev´ e equations Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 6 / 41
Classification Scheme for Discrete Painlev´ e Equations First, we note that although the previous classification scheme according to symmetries is more traditional, for the geometric approach, it is more natural to look at the classification based on the point-configuration or surface type : A (1) A (1) 0 7 A (1) ∗ A (1) A (1) A (1) A (1) A (1) A (1) A (1) A (1) 0 1 2 3 4 5 6 7 8 A (1) ∗∗ A (1) ∗ A (1) ∗ D (1) D (1) D (1) D (1) D (1) 0 1 2 4 5 6 7 8 E (1) E (1) E (1) 6 7 8 Surface-type classification scheme for Painlev´ e equations In this description each letter stands for a Dynkin diagram describing intersection configuration of the irreducible components of an anti-canonical divisor of a certain algebraic surface, known as the generalized Halphen surface , that is obtained by blowing up P 1 × P 1 at eight (possibly infinitely-close) points. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 6 / 41
Classification Scheme for Discrete Painlev´ e Equations First, we note that although the previous classification scheme according to symmetries is more traditional, for the geometric approach, it is more natural to look at the classification based on the point-configuration or surface type : A (1) A (1) 0 7 A (1) ∗ A (1) A (1) A (1) A (1) A (1) A (1) A (1) A (1) 0 1 2 3 4 5 6 7 8 A (1) ∗∗ A (1) ∗ A (1) ∗ D (1) D (1) D (1) D (1) D (1) 0 1 2 4 5 6 7 8 E (1) E (1) E (1) 6 7 8 Surface-type classification scheme for Painlev´ e equations In this description each letter stands for a Dynkin diagram describing intersection configuration of the irreducible components of an anti-canonical divisor of a certain algebraic surface, known as the generalized Halphen surface , that is obtained by blowing up P 1 × P 1 at eight (possibly infinitely-close) points. The very top A (1) node corresponds to the the points lying on a smooth elliptic curves, A (1) ∗ is a 0 0 nodal curve (hence the multiplicative dynamic on the parameters), A (1) ∗∗ is a cusp curve (hence 0 the additive dynamic on the parameters), and others are various degenerations into rational components. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 6 / 41
Origins of Discrete Painlev´ e Equations Discrete Painlev´ e Equations appear in a variety of contexts: Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 7 / 41
Origins of Discrete Painlev´ e Equations Discrete Painlev´ e Equations appear in a variety of contexts: As certain second -order (or two -dimensional) non-autonomous nonlinear recurrence relations that has the usual differential Painlev´ e equations as some continuous limits (Shohat, Br´ ezin-Kazakov, Gross-Migdal). Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 7 / 41
Origins of Discrete Painlev´ e Equations Discrete Painlev´ e Equations appear in a variety of contexts: As certain second -order (or two -dimensional) non-autonomous nonlinear recurrence relations that has the usual differential Painlev´ e equations as some continuous limits (Shohat, Br´ ezin-Kazakov, Gross-Migdal). As symmetries (B¨ acklund transformations) of continous Painlev´ e equations. Geometric theory of B¨ acklund transformations of continous Painlev´ e equations was developed in the works of Okamoto, who also introduced the important notion of the space of initial conditions , which eventually led to Sakai’s theory. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 7 / 41
Origins of Discrete Painlev´ e Equations Discrete Painlev´ e Equations appear in a variety of contexts: As certain second -order (or two -dimensional) non-autonomous nonlinear recurrence relations that has the usual differential Painlev´ e equations as some continuous limits (Shohat, Br´ ezin-Kazakov, Gross-Migdal). As symmetries (B¨ acklund transformations) of continous Painlev´ e equations. Geometric theory of B¨ acklund transformations of continous Painlev´ e equations was developed in the works of Okamoto, who also introduced the important notion of the space of initial conditions , which eventually led to Sakai’s theory. As deautonomizations of different autonomous systems, such as QRT maps, together with requiring the singularity confinement condition (Grammaticos, Ramani, Ohta, Papageorgiu, Nijhoff). Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 7 / 41
Origins of Discrete Painlev´ e Equations Discrete Painlev´ e Equations appear in a variety of contexts: As certain second -order (or two -dimensional) non-autonomous nonlinear recurrence relations that has the usual differential Painlev´ e equations as some continuous limits (Shohat, Br´ ezin-Kazakov, Gross-Migdal). As symmetries (B¨ acklund transformations) of continous Painlev´ e equations. Geometric theory of B¨ acklund transformations of continous Painlev´ e equations was developed in the works of Okamoto, who also introduced the important notion of the space of initial conditions , which eventually led to Sakai’s theory. As deautonomizations of different autonomous systems, such as QRT maps, together with requiring the singularity confinement condition (Grammaticos, Ramani, Ohta, Papageorgiu, Nijhoff). As birational representations of affine Weyl groups (Noumi, Yamada, Kajiwara, Ohta). Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 7 / 41
Origins of Discrete Painlev´ e Equations Discrete Painlev´ e Equations appear in a variety of contexts: As certain second -order (or two -dimensional) non-autonomous nonlinear recurrence relations that has the usual differential Painlev´ e equations as some continuous limits (Shohat, Br´ ezin-Kazakov, Gross-Migdal). As symmetries (B¨ acklund transformations) of continous Painlev´ e equations. Geometric theory of B¨ acklund transformations of continous Painlev´ e equations was developed in the works of Okamoto, who also introduced the important notion of the space of initial conditions , which eventually led to Sakai’s theory. As deautonomizations of different autonomous systems, such as QRT maps, together with requiring the singularity confinement condition (Grammaticos, Ramani, Ohta, Papageorgiu, Nijhoff). As birational representations of affine Weyl groups (Noumi, Yamada, Kajiwara, Ohta). As reductions of Schlesinger transformations of Fuchsian systems (Jimbo, Sakai, Grammaticos, Ramani) Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 7 / 41
Origins of Discrete Painlev´ e Equations Discrete Painlev´ e Equations appear in a variety of contexts: As certain second -order (or two -dimensional) non-autonomous nonlinear recurrence relations that has the usual differential Painlev´ e equations as some continuous limits (Shohat, Br´ ezin-Kazakov, Gross-Migdal). As symmetries (B¨ acklund transformations) of continous Painlev´ e equations. Geometric theory of B¨ acklund transformations of continous Painlev´ e equations was developed in the works of Okamoto, who also introduced the important notion of the space of initial conditions , which eventually led to Sakai’s theory. As deautonomizations of different autonomous systems, such as QRT maps, together with requiring the singularity confinement condition (Grammaticos, Ramani, Ohta, Papageorgiu, Nijhoff). As birational representations of affine Weyl groups (Noumi, Yamada, Kajiwara, Ohta). As reductions of Schlesinger transformations of Fuchsian systems (Jimbo, Sakai, Grammaticos, Ramani) (the list of names above is, of course, far from complete). Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 7 / 41
Origins of Discrete Painlev´ e Equations Discrete Painlev´ e Equations appear in a variety of contexts: As certain second -order (or two -dimensional) non-autonomous nonlinear recurrence relations that has the usual differential Painlev´ e equations as some continuous limits (Shohat, Br´ ezin-Kazakov, Gross-Migdal). As symmetries (B¨ acklund transformations) of continous Painlev´ e equations. Geometric theory of B¨ acklund transformations of continous Painlev´ e equations was developed in the works of Okamoto, who also introduced the important notion of the space of initial conditions , which eventually led to Sakai’s theory. As deautonomizations of different autonomous systems, such as QRT maps, together with requiring the singularity confinement condition (Grammaticos, Ramani, Ohta, Papageorgiu, Nijhoff). As birational representations of affine Weyl groups (Noumi, Yamada, Kajiwara, Ohta). As reductions of Schlesinger transformations of Fuchsian systems (Jimbo, Sakai, Grammaticos, Ramani) (the list of names above is, of course, far from complete). For the geometric description, the most natural approach is via the deautonomization of QRT maps, so we briefly recall this construction. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 7 / 41
Geometry of a QRT Map The Quispel-Roberts-Thompson (or QRT ) mapping can be describing geometrically as follows (following T. Tsuda). Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 8 / 41
Geometry of a QRT Map The Quispel-Roberts-Thompson (or QRT ) mapping can be describing geometrically as follows (following T. Tsuda). Consider a bi-quadractic curve Γ on P 1 × P 1 . In an affine C 2 -chart Γ is given by a bi-degree (2 , 2) polynomial equation a 00 x 2 y 2 + a 01 x 2 y + a 02 x 2 + a 10 xy 2 + a 11 xy + a 12 x + a 20 y 2 + a 21 y + a 22 = 0 . Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 8 / 41
Geometry of a QRT Map The Quispel-Roberts-Thompson (or QRT ) mapping can be describing geometrically as follows (following T. Tsuda). Consider a bi-quadractic curve Γ on P 1 × P 1 . In an affine C 2 -chart Γ is given by a bi-degree (2 , 2) polynomial equation a 00 x 2 y 2 + a 01 x 2 y + a 02 x 2 + a 10 xy 2 + a 11 xy + a 12 x + a 20 y 2 + a 21 y + a 22 = 0 . This equation can be written as y 2 a 00 a 01 a 02 2 � � x 2 1 � x T Ay = a ij x 2 − i y 2 − j = 0 , = x a 10 a 11 a 12 y a 20 a 21 a 22 1 i , j =0 where x = � x 2 , x , 1 � , y = � y 2 , y , 1 � , A ∈ Mat 3 × 3 ( C ) . Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 8 / 41
Geometry of a QRT Map The Quispel-Roberts-Thompson (or QRT ) mapping can be describing geometrically as follows (following T. Tsuda). Consider a bi-quadractic curve Γ on P 1 × P 1 . In an affine C 2 -chart Γ is given by a bi-degree (2 , 2) polynomial equation a 00 x 2 y 2 + a 01 x 2 y + a 02 x 2 + a 10 xy 2 + a 11 xy + a 12 x + a 20 y 2 + a 21 y + a 22 = 0 . This equation can be written as y 2 a 00 a 01 a 02 2 � � x 2 1 � x T Ay = a ij x 2 − i y 2 − j = 0 , = x a 10 a 11 a 12 y a 20 a 21 a 22 1 i , j =0 where x = � x 2 , x , 1 � , y = � y 2 , y , 1 � , A ∈ Mat 3 × 3 ( C ) . In general, Γ is an elliptic curve that can be rewritten in a Weierstrass normal form Y 2 = 4 X 3 − g 2 X − g 3 . Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 8 / 41
Geometry of a QRT Map Since Γ has bi-degree (2 , 2), we can define two involutions , r x : ( x , y ) → ( x ′ , y ) and r y : ( x , y ) → ( x , y ′ ) ( x, y 0 ) (¯ x, ¯ y ) r x r y r x ( x 0 , y ) ( x, y ) Γ as well as their composition r x ◦ r y : ( x , y ) → (¯ x , ¯ y ). Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 9 / 41
Geometry of a QRT Map Since Γ has bi-degree (2 , 2), we can define two involutions , r x : ( x , y ) → ( x ′ , y ) and r y : ( x , y ) → ( x , y ′ ) ( x, y 0 ) (¯ x, ¯ y ) r x r y r x ( x 0 , y ) ( x, y ) Γ as well as their composition r x ◦ r y : ( x , y ) → (¯ x , ¯ y ). The main idea of the QRT map is to extend r x ◦ r y to all of the P 1 × P 1 . Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 9 / 41
Geometry of a QRT Map For that, take two matrices A , B ∈ Mat 3 × 3 ( C ) and consider a pencil (i.e., a one-dimensional family) of such curves α x T Ay + β x T By = 0 , [ α : β ] ∈ P 1 Γ [ α : β ] : Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 10 / 41
Geometry of a QRT Map For that, take two matrices A , B ∈ Mat 3 × 3 ( C ) and consider a pencil (i.e., a one-dimensional family) of such curves α x T Ay + β x T By = 0 , [ α : β ] ∈ P 1 Γ [ α : β ] : Γ [ α : β ] x T Ay = 0 x T By = 0 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 10 / 41
Geometry of a QRT Map For that, take two matrices A , B ∈ Mat 3 × 3 ( C ) and consider a pencil (i.e., a one-dimensional family) of such curves α x T Ay + β x T By = 0 , [ α : β ] ∈ P 1 Γ [ α : β ] : Γ [ α : β ] x T Ay = 0 x T By = 0 Then, given a point ( x ∗ , y ∗ ), there is only one curve from a family with the parameter [ α : β ] = [ − x T ∗ By ∗ , x T ∗ Ay ∗ ], except for the eight base points x T ∗ Ay ∗ = x T ∗ By ∗ = 0. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 10 / 41
Geometry of a QRT Map For that, take two matrices A , B ∈ Mat 3 × 3 ( C ) and consider a pencil (i.e., a one-dimensional family) of such curves α x T Ay + β x T By = 0 , [ α : β ] ∈ P 1 Γ [ α : β ] : Γ [ α : β ] x T Ay = 0 x T By = 0 Then, given a point ( x ∗ , y ∗ ), there is only one curve from a family with the parameter [ α : β ] = [ − x T ∗ By ∗ , x T ∗ Ay ∗ ], except for the eight base points x T ∗ Ay ∗ = x T ∗ By ∗ = 0. Resolving these points using the blowup, we get a rational elliptic surface X with the QRT automorphism r x ◦ r y preserving the elliptic fibration π : X → P 1 , and π − 1 ([ α : β ]) is an elliptic curve except for 12 points corresponding to singular fibers (classified by K. Kodaira into 22 types). Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 10 / 41
Geometry of a QRT Map For that, take two matrices A , B ∈ Mat 3 × 3 ( C ) and consider a pencil (i.e., a one-dimensional family) of such curves α x T Ay + β x T By = 0 , [ α : β ] ∈ P 1 Γ [ α : β ] : Γ [ α : β ] x T Ay = 0 x T By = 0 Then, given a point ( x ∗ , y ∗ ), there is only one curve from a family with the parameter [ α : β ] = [ − x T ∗ By ∗ , x T ∗ Ay ∗ ], except for the eight base points x T ∗ Ay ∗ = x T ∗ By ∗ = 0. Resolving these points using the blowup, we get a rational elliptic surface X with the QRT automorphism r x ◦ r y preserving the elliptic fibration π : X → P 1 , and π − 1 ([ α : β ]) is an elliptic curve except for 12 points corresponding to singular fibers (classified by K. Kodaira into 22 types). Deautonomization is performed with different choices of a fiber on which the blowup points lie (that fiber is exactly the anti-canonical divisor, − K X that is the key object of the geometric theory). Allowing the points move along a particular fiber, either smooth or singular, breaks down the elliptic surface structure and the dynamic becomes non-autonomus. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 10 / 41
The A (1) ∗ Deautonomization Example 2 For the example that we are interested in, we can take the matrices A , B as − ( a + a − 1 ) 0 0 0 1 1 , − ( a + a − 1 ) ( b + b − 1 ) 2 , A = 0 0 1 B = 0 0 1 0 1 0 1 where a � = b � = 0 , ± 1. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 11 / 41
The A (1) ∗ Deautonomization Example 2 For the example that we are interested in, we can take the matrices A , B as − ( a + a − 1 ) 0 0 0 1 1 , − ( a + a − 1 ) ( b + b − 1 ) 2 , A = 0 0 1 B = 0 0 1 0 1 0 1 where a � = b � = 0 , ± 1. Here is an example of two fibers in this family: a smooth elliptic A (1) 0 -fiber and a singular A (1) ∗ -fiber: 2 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 11 / 41
The A (1) ∗ Deautonomization Example 2 For the example that we are interested in, we can take the matrices A , B as − ( a + a − 1 ) 0 0 0 1 1 , − ( a + a − 1 ) ( b + b − 1 ) 2 , A = 0 0 1 B = 0 0 1 0 1 0 1 where a � = b � = 0 , ± 1. Here is an example of two fibers in this family: a smooth elliptic A (1) 0 -fiber and a singular A (1) ∗ -fiber: 2 The base points of the map are shown in red, allowing them to move along the A (1) ∗ -fiber (points 2 shown in blue) resulted in the deautonomization example of Grammaticos-Ramani-Ohta. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 11 / 41
The A (1) ∗ Deautonomization Example 2 For the example that we are interested in, we can take the matrices A , B as − ( a + a − 1 ) 0 0 0 1 1 , − ( a + a − 1 ) ( b + b − 1 ) 2 , A = 0 0 1 B = 0 0 1 0 1 0 1 where a � = b � = 0 , ± 1. Here is an example of two fibers in this family: a smooth elliptic A (1) 0 -fiber and a singular A (1) ∗ -fiber: 2 The base points of the map are shown in red, allowing them to move along the A (1) ∗ -fiber (points 2 shown in blue) resulted in the deautonomization example of Grammaticos-Ramani-Ohta. However, we can in fact create the mapping starting just from the Dynkin diagrams and a choice of a translation element. We explain how to do that next. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 11 / 41
Canonical Model of the Okamoto Surface of Type A (1) ∗ 2 Let us start by understanding the structure of a generalized Halphen surface of type A (1) ∗ . 2 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 12 / 41
Canonical Model of the Okamoto Surface of Type A (1) ∗ 2 Let us start by understanding the structure of a generalized Halphen surface of type A (1) ∗ . 2 Such a surface X is obtained by blowing up P 1 × P 1 at 8 points, and the type corresponds to the intersection structure of the irreducible components of the anti-canonical divisor −K X in the � � 8 Picard Lattice Pic( X ) = Z H f Z H g i =1 Z E i , ⊕ Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 12 / 41
Canonical Model of the Okamoto Surface of Type A (1) ∗ 2 Let us start by understanding the structure of a generalized Halphen surface of type A (1) ∗ . 2 Such a surface X is obtained by blowing up P 1 × P 1 at 8 points, and the type corresponds to the intersection structure of the irreducible components of the anti-canonical divisor −K X in the � � 8 Picard Lattice Pic( X ) = Z H f Z H g i =1 Z E i , ⊕ � −K X = 2 H f + 2 H g − E 1 − · · · − E 8 = m i D i . i Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 12 / 41
Canonical Model of the Okamoto Surface of Type A (1) ∗ 2 Let us start by understanding the structure of a generalized Halphen surface of type A (1) ∗ . 2 Such a surface X is obtained by blowing up P 1 × P 1 at 8 points, and the type corresponds to the intersection structure of the irreducible components of the anti-canonical divisor −K X in the � � 8 Picard Lattice Pic( X ) = Z H f Z H g i =1 Z E i , ⊕ � −K X = 2 H f + 2 H g − E 1 − · · · − E 8 = m i D i . i Dynkin diagram A (1) and the anti-canonical divisor decomposition 2 D 2 − 2 1 1 1 − 2 1 −K X = D 0 + D 1 + D 2 1 1 − 2 D 0 D 1 Dynkin diagram A (1) its Cartan matrix −K X decomposition 2 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 12 / 41
Canonical Model of the Okamoto Surface of Type A (1) ∗ 2 Let us start by understanding the structure of a generalized Halphen surface of type A (1) ∗ . 2 Such a surface X is obtained by blowing up P 1 × P 1 at 8 points, and the type corresponds to the intersection structure of the irreducible components of the anti-canonical divisor −K X in the � � 8 Picard Lattice Pic( X ) = Z H f Z H g i =1 Z E i , ⊕ � −K X = 2 H f + 2 H g − E 1 − · · · − E 8 = m i D i . i Dynkin diagram A (1) and the anti-canonical divisor decomposition 2 D 2 − 2 1 1 1 − 2 1 −K X = D 0 + D 1 + D 2 1 1 − 2 D 0 D 1 Dynkin diagram A (1) its Cartan matrix −K X decomposition 2 Without loss of generality, we can put Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 12 / 41
Canonical Model of the Okamoto Surface of Type A (1) ∗ 2 Let us start by understanding the structure of a generalized Halphen surface of type A (1) ∗ . 2 Such a surface X is obtained by blowing up P 1 × P 1 at 8 points, and the type corresponds to the intersection structure of the irreducible components of the anti-canonical divisor −K X in the � � 8 Picard Lattice Pic( X ) = Z H f Z H g i =1 Z E i , ⊕ � −K X = 2 H f + 2 H g − E 1 − · · · − E 8 = m i D i . i Dynkin diagram A (1) and the anti-canonical divisor decomposition 2 D 2 − 2 1 1 1 − 2 1 −K X = D 0 + D 1 + D 2 1 1 − 2 D 0 D 1 Dynkin diagram A (1) its Cartan matrix −K X decomposition 2 Without loss of generality, we can put D 0 = H f + H g − E 1 − E 2 − E 3 − E 4 D 1 = H f − E 5 − E 6 D 2 = H g − E 7 − E 8 . Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 12 / 41
Canonical Model of the Okamoto Surface of Type A (1) ∗ 2 There are, however, two different geometric configurations related to the algebraic intersection structure given by this Dynkin diagram: Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 13 / 41
Canonical Model of the Okamoto Surface of Type A (1) ∗ 2 There are, however, two different geometric configurations related to the algebraic intersection structure given by this Dynkin diagram: D 2 D 0 D 1 Dynkin diagram A (1) A (1) A (1) ∗ surface (multiplicative) surface (additive) 2 2 2 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 13 / 41
Canonical Model of the Okamoto Surface of Type A (1) ∗ 2 There are, however, two different geometric configurations related to the algebraic intersection structure given by this Dynkin diagram: D 2 D 0 D 1 Dynkin diagram A (1) A (1) A (1) ∗ surface (multiplicative) surface (additive) 2 2 2 We are interested in the additive dynamic given by A (1) ∗ , so we want all of the irreducible 2 components of the anti-canonical divisor to intersect at one point. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 13 / 41
Canonical Model of the Okamoto Surface of Type A (1) ∗ 2 There are, however, two different geometric configurations related to the algebraic intersection structure given by this Dynkin diagram: D 2 D 0 D 1 Dynkin diagram A (1) A (1) A (1) ∗ surface (multiplicative) surface (additive) 2 2 2 We are interested in the additive dynamic given by A (1) ∗ , so we want all of the irreducible 2 components of the anti-canonical divisor to intersect at one point. Again, without the loss of generality (i.e., acting by affine transformations on each of the two P 1 factors) we can assume that the component D 1 = H f − E 5 − E 6 under the blowing down map projects to the line f = ∞ (and so there are two blowup points p 5 ( ∞ , b 5 ) and p 6 ( ∞ , b 6 ) on that line), the component D 2 = H g − E 7 − E 8 projects to the line g = ∞ with points p 7 ( − b 6 , ∞ ) and p 8 ( − b 8 , ∞ ), and the component D 0 = H f + H g − E 1 − E 2 − E 3 − E 4 projects to the line f + g = 0. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 13 / 41
Canonical Model of the Okamoto Surface of Type A (1) ∗ 2 Thus, we get the following geometric realization of a (family of) surface(s) X b of type A (1) ∗ : 2 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 14 / 41
Canonical Model of the Okamoto Surface of Type A (1) ∗ 2 Thus, we get the following geometric realization of a (family of) surface(s) X b of type A (1) ∗ : 2 f = 0 f = ∞ h = f + g = 0 p 7 ( − b 7 , ∞ ) p 8 ( − b 8 , ∞ ) E 7 E 8 g = ∞ H g H g − E 7 − E 8 E 4 p 4 ( b 4 , − b 4 ) p 6 ( ∞ , b 6 ) E 3 E 6 p 3 ( b 3 , − b 3 ) E 2 p 2 ( b 2 , − b 2 ) p 5 ( ∞ , b 5 ) E 5 E 1 p 1 ( b 1 , − b 1 ) g = 0 H g H f + H g H f + H g − E 1 − E 2 − E 3 − E 4 H f − E 5 − E 6 H f H f Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 14 / 41
Canonical Model of the Okamoto Surface of Type A (1) ∗ 2 Thus, we get the following geometric realization of a (family of) surface(s) X b of type A (1) ∗ : 2 f = 0 f = ∞ h = f + g = 0 p 7 ( − b 7 , ∞ ) p 8 ( − b 8 , ∞ ) E 7 E 8 g = ∞ H g H g − E 7 − E 8 E 4 p 4 ( b 4 , − b 4 ) p 6 ( ∞ , b 6 ) E 3 E 6 p 3 ( b 3 , − b 3 ) E 2 p 2 ( b 2 , − b 2 ) p 5 ( ∞ , b 5 ) E 5 E 1 p 1 ( b 1 , − b 1 ) g = 0 H g H f + H g H f + H g − E 1 − E 2 − E 3 − E 4 H f − E 5 − E 6 H f H f Note that the lines in the above configuration form a pole divisor of the symplectic form ω = df ∧ dg ( f + g ) = − dF ∧ dg F (1 + Fg ) = − df ∧ dG G ( fG + 1) = dF ∧ dG ( F + G ) = dh ∧ dg = · · · h Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 14 / 41
Canonical Model of the Okamoto Surface of Type A (1) ∗ 2 Thus, we get the following geometric realization of a (family of) surface(s) X b of type A (1) ∗ : 2 f = 0 f = ∞ h = f + g = 0 p 7 ( − b 7 , ∞ ) p 8 ( − b 8 , ∞ ) E 7 E 8 g = ∞ H g H g − E 7 − E 8 E 4 p 4 ( b 4 , − b 4 ) p 6 ( ∞ , b 6 ) E 3 E 6 p 3 ( b 3 , − b 3 ) E 2 p 2 ( b 2 , − b 2 ) p 5 ( ∞ , b 5 ) E 5 E 1 p 1 ( b 1 , − b 1 ) g = 0 H g H f + H g H f + H g − E 1 − E 2 − E 3 − E 4 H f − E 5 − E 6 H f H f Note that the lines in the above configuration form a pole divisor of the symplectic form ω = df ∧ dg ( f + g ) = − dF ∧ dg F (1 + Fg ) = − df ∧ dG G ( fG + 1) = dF ∧ dG ( F + G ) = dh ∧ dg = · · · h However, there is still a two-parameter family of transformations preserving this configuration: � b 1 � � α b 1 + β � b 2 b 3 b 4 α b 2 + β α b 3 + β α b 4 + β b 8 ; f , g ∼ α b 8 − β ; α f + β, α g − β , α � = 0 . b 5 b 6 b 7 α b 5 − β α b 6 − β α b 7 − β Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 14 / 41
The Symmetry Group and the Symmetry Sub-Lattice A more invariant way to parameterize the surface is to use the so-called Period Map . For that we first need to define the symmetry sublattice. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 15 / 41
The Symmetry Group and the Symmetry Sub-Lattice A more invariant way to parameterize the surface is to use the so-called Period Map . For that we first need to define the symmetry sublattice. Symmetry sublattice Q ⊳ Pic( X ) � 2 ) ⊥ � � � Q = (Span Z {D 0 , D 1 , D 2 } ) ⊥ = Q ( A (1) E (1) = Span Z { α 0 , α 1 , α 2 , α 3 , α 4 , α 5 , α 6 } = Q , 6 where the simple roots α i are given by α 6 α 0 = E 3 − E 4 , α 4 = E 7 − E 8 , α 1 = E 2 − E 3 , α 5 = H g − E 1 − E 5 , α 5 α 2 = E 1 − E 2 , α 6 = E 5 − E 6 . α 3 = H f − E 1 − E 7 , α 0 α 1 α 2 α 3 α 4 Note also that δ = −K X = α 0 + 2 α 1 + 3 α 2 + 2 α 3 + α 4 + 2 α 5 + α 6 . Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 15 / 41
The Symmetry Group and the Symmetry Sub-Lattice A more invariant way to parameterize the surface is to use the so-called Period Map . For that we first need to define the symmetry sublattice. Symmetry sublattice Q ⊳ Pic( X ) � 2 ) ⊥ � � � Q = (Span Z {D 0 , D 1 , D 2 } ) ⊥ = Q ( A (1) E (1) = Span Z { α 0 , α 1 , α 2 , α 3 , α 4 , α 5 , α 6 } = Q , 6 where the simple roots α i are given by α 6 α 0 = E 3 − E 4 , α 4 = E 7 − E 8 , α 1 = E 2 − E 3 , α 5 = H g − E 1 − E 5 , α 5 α 2 = E 1 − E 2 , α 6 = E 5 − E 6 . α 3 = H f − E 1 − E 7 , α 0 α 1 α 2 α 3 α 4 Note also that δ = −K X = α 0 + 2 α 1 + 3 α 2 + 2 α 3 + α 4 + 2 α 5 + α 6 . The period mapping is the map χ : Q → C , χ ( α i ) = a i defined on the simple roots and then extended by the linearity. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 15 / 41
The Period Map � Q i � � � 1 P i Q i [ C 1 i ] − [ C 0 χ ( α i ) = χ 1 ] = ω D k 2 π i P i D k � Q i ω = df ∧ dg = res D k ω, C 0 C 1 f + g P i i i Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 16 / 41
The Period Map � Q i � � � 1 P i Q i [ C 1 i ] − [ C 0 χ ( α i ) = χ 1 ] = ω D k 2 π i P i D k � Q i ω = df ∧ dg = res D k ω, C 0 C 1 f + g P i i i Examples of the Period Map computations p 7 ( − b 7 , ∞ ) p 8 ( − b 8 , ∞ ) p 4 ( b 4 , − b 4 ) p 3 ( b 3 , − b 3 ) p 5 ( ∞ , b 5 ) p 2 ( b 2 , − b 2 ) p 1 ( b 1 , − b 1 ) p 6 ( ∞ , b 6 ) Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 16 / 41
The Period Map � Q i � � � 1 P i Q i [ C 1 i ] − [ C 0 χ ( α i ) = χ 1 ] = ω D k 2 π i P i D k � Q i ω = df ∧ dg = res D k ω, C 0 C 1 f + g P i i i Examples of the Period Map computations • α 0 = E 3 − E 4 = [ E 3 ] − [ E 4 ] , D k = D 0 = { h = f + g = 0 } ω = df ∧ dg = dh ∧ dg , res h =0 ω = dg p 7 ( − b 7 , ∞ ) p 8 ( − b 8 , ∞ ) f + g h � − b 3 p 4 ( b 4 , − b 4 ) χ ( α 0 ) = dg = b 4 − b 3 = a 0 − b 4 p 3 ( b 3 , − b 3 ) p 5 ( ∞ , b 5 ) p 2 ( b 2 , − b 2 ) p 1 ( b 1 , − b 1 ) p 6 ( ∞ , b 6 ) Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 16 / 41
The Period Map � Q i � � � 1 P i Q i [ C 1 i ] − [ C 0 χ ( α i ) = χ 1 ] = ω D k 2 π i P i D k � Q i ω = df ∧ dg = res D k ω, C 0 C 1 f + g P i i i Examples of the Period Map computations • α 0 = E 3 − E 4 = [ E 3 ] − [ E 4 ] , D k = D 0 = { h = f + g = 0 } H f − E 1 ω = df ∧ dg = dh ∧ dg , res h =0 ω = dg p 7 ( − b 7 , ∞ ) p 8 ( − b 8 , ∞ ) f + g h � − b 3 p 4 ( b 4 , − b 4 ) χ ( α 0 ) = dg = b 4 − b 3 = a 0 − b 4 p 3 ( b 3 , − b 3 ) p 5 ( ∞ , b 5 ) • α 3 = H f − E 1 − E 7 = [ H f − E 1 ] − [ E 7 ] , p 2 ( b 2 , − b 2 ) D k = D 2 = { g = ∞} = { G = 0 } p 1 ( b 1 , − b 1 ) p 6 ( ∞ , b 6 ) ω = df ∧ dg = − df ∧ dG G ( fG + 1) , res G =0 ω = df f + g f = b 1 � b 1 χ ( α 3 ) = df = b 1 + b 7 = a 3 − b 7 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 16 / 41
The Period Map The Period Map, a i = χ ( α i ) are the root variables a 0 = b 4 − b 3 , a 3 = b 1 + b 7 , a 6 = b 6 − b 5 , a 1 = b 3 − b 2 , a 4 = b 8 − b 7 , a 2 = b 2 − b 1 , a 5 = b 1 + b 5 . Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 17 / 41
The Period Map The Period Map, a i = χ ( α i ) are the root variables a 0 = b 4 − b 3 , a 3 = b 1 + b 7 , a 6 = b 6 − b 5 , a 1 = b 3 − b 2 , a 4 = b 8 − b 7 , a 2 = b 2 − b 1 , a 5 = b 1 + b 5 . Parameterization by the root variables a i � b 1 � � � b 2 b 3 b 4 b 1 b 1 + a 2 b 1 + a 1 + a 2 b 1 + a 0 + a 1 + a 2 b 8 ; f , g = ; f , g , b 5 b 6 b 7 a 5 − b 1 a 5 + a 6 − b 1 a 3 − b 1 a 3 + a 4 − b 1 and so we see that b 1 is one free parameter (translation of the origin). To fix the global scaling parameter we usually normalize χ ( δ ) = χ ( −K X ) = χ ( a 0 + 2 a 1 + 3 a 2 + 2 a 3 + a 4 + 2 a 5 + a 6 ) = b 1 + b 2 + b 3 + b 4 + b 5 + b 6 + b 7 + b 8 . Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 17 / 41
The Period Map The Period Map, a i = χ ( α i ) are the root variables a 0 = b 4 − b 3 , a 3 = b 1 + b 7 , a 6 = b 6 − b 5 , a 1 = b 3 − b 2 , a 4 = b 8 − b 7 , a 2 = b 2 − b 1 , a 5 = b 1 + b 5 . Parameterization by the root variables a i � b 1 � � � b 2 b 3 b 4 b 1 b 1 + a 2 b 1 + a 1 + a 2 b 1 + a 0 + a 1 + a 2 b 8 ; f , g = ; f , g , b 5 b 6 b 7 a 5 − b 1 a 5 + a 6 − b 1 a 3 − b 1 a 3 + a 4 − b 1 and so we see that b 1 is one free parameter (translation of the origin). To fix the global scaling parameter we usually normalize χ ( δ ) = χ ( −K X ) = χ ( a 0 + 2 a 1 + 3 a 2 + 2 a 3 + a 4 + 2 a 5 + a 6 ) = b 1 + b 2 + b 3 + b 4 + b 5 + b 6 + b 7 + b 8 . The usual normalization is to put χ ( δ ) = 1, and one can also ask the same for b 1 . We will not do that, but we will require that, when resolving the normalization ambiguity, both χ ( δ ) and b 1 are fixed — this ensures the group structure on the level of elementary birational maps . Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 17 / 41
� � E (1) The Extended Affine Weyl Symmetry Group � W 6 � � A (1) ∗ The next step in understanding the structure of difference Painlev´ e equations of type d- P 2 is to describe the realization of the symmetry group in terms of elementary bilinear maps. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 18 / 41
� � E (1) The Extended Affine Weyl Symmetry Group � W 6 � � A (1) ∗ The next step in understanding the structure of difference Painlev´ e equations of type d- P 2 is to describe the realization of the symmetry group in terms of elementary bilinear maps. � � E (1) = Aut( E (1) ) ⋉ W ( E (1) � W ) 6 6 6 � � E (1) The full extended Weyl symmetry group � W is a semi-direct product of 6 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 18 / 41
� � E (1) The Extended Affine Weyl Symmetry Group � W 6 � � A (1) ∗ The next step in understanding the structure of difference Painlev´ e equations of type d- P 2 is to describe the realization of the symmetry group in terms of elementary bilinear maps. � � E (1) = Aut( E (1) ) ⋉ W ( E (1) � W ) 6 6 6 � � E (1) The full extended Weyl symmetry group � W is a semi-direct product of 6 The affine Weyl symmetry group of reflections w i = w α i � � w 2 i = e � α 6 � � � � w i ◦ w j = w j ◦ w i when α i W ( E (1) � α j α 5 ) = w 0 , . . . , w 6 � 6 � � w i ◦ w j ◦ w i = w j ◦ w i ◦ w j when α i � α j α 0 α 1 α 2 α 3 α 4 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 18 / 41
� � E (1) The Extended Affine Weyl Symmetry Group � W 6 � � A (1) ∗ The next step in understanding the structure of difference Painlev´ e equations of type d- P 2 is to describe the realization of the symmetry group in terms of elementary bilinear maps. � � E (1) = Aut( E (1) ) ⋉ W ( E (1) � W ) 6 6 6 � � E (1) The full extended Weyl symmetry group � W is a semi-direct product of 6 The affine Weyl symmetry group of reflections w i = w α i � � w 2 i = e � α 6 � � � � w i ◦ w j = w j ◦ w i when α i W ( E (1) � α j α 5 ) = w 0 , . . . , w 6 � 6 � � w i ◦ w j ◦ w i = w j ◦ w i ◦ w j when α i � α j α 0 α 1 α 2 α 3 α 4 The finite group of Dynkin diagram automorphisms � � � � E (1) A (1) Aut ≃ Aut ≃ D 3 , 6 2 0 = r 3 = e , m 0 r = r 2 m 0 � is the usual dihedral where D 3 = { e , m 0 , m 1 , m 2 , r , r 2 } = � m 0 , r | m 2 group of the symmetries of a triangle. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 18 / 41
The Affine Weyl Group W ( E (1) 6 ) Theorem Reflections w i are induced by the following elementary birational mappings (also denoted by w i ) on the family X b fixing b 1 and χ ( δ ) (we put b i ··· k = b i + · · · + b k , e.g., b 12 = b 1 + b 2 and so on) Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 19 / 41
The Affine Weyl Group W ( E (1) 6 ) Theorem Reflections w i are induced by the following elementary birational mappings (also denoted by w i ) on the family X b fixing b 1 and χ ( δ ) (we put b i ··· k = b i + · · · + b k , e.g., b 12 = b 1 + b 2 and so on) � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 2 b 4 b 8 ; f b 3 w 0 �− → , b 5 b 6 b 7 g b 5 b 6 b 7 g Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 19 / 41
The Affine Weyl Group W ( E (1) 6 ) Theorem Reflections w i are induced by the following elementary birational mappings (also denoted by w i ) on the family X b fixing b 1 and χ ( δ ) (we put b i ··· k = b i + · · · + b k , e.g., b 12 = b 1 + b 2 and so on) � b 1 � � b 1 � b 2 b 3 b 8 ; f b 4 b 2 b 4 b 8 ; f b 3 w 0 �− → , b 5 b 6 b 7 g b 5 b 6 b 7 g � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 3 b 2 b 8 ; f b 4 w 1 �− → , b 5 b 6 b 7 g b 5 b 6 b 7 g Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 19 / 41
The Affine Weyl Group W ( E (1) 6 ) Theorem Reflections w i are induced by the following elementary birational mappings (also denoted by w i ) on the family X b fixing b 1 and χ ( δ ) (we put b i ··· k = b i + · · · + b k , e.g., b 12 = b 1 + b 2 and so on) � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 2 b 4 b 8 ; f b 3 w 0 �− → , b 5 b 6 b 7 g b 5 b 6 b 7 g � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 3 b 2 b 8 ; f b 4 w 1 �− → , b 5 b 6 b 7 g b 5 b 6 b 7 g � b 1 � � � b 2 b 3 b 8 ; f b 4 b 1 b 11 − b 2 b 13 − b 2 b 14 − b 2 b 82 − b 1 ; f + b 1 − b 2 w 2 �− → g − b 1 + b 2 , , b 5 b 6 b 7 g b 52 − b 1 b 62 − b 1 b 72 − b 1 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 19 / 41
The Affine Weyl Group W ( E (1) 6 ) Theorem Reflections w i are induced by the following elementary birational mappings (also denoted by w i ) on the family X b fixing b 1 and χ ( δ ) (we put b i ··· k = b i + · · · + b k , e.g., b 12 = b 1 + b 2 and so on) � b 1 � � b 1 � b 2 b 3 b 8 ; f b 4 b 2 b 4 b 8 ; f b 3 w 0 �− → , b 5 b 6 b 7 g b 5 b 6 b 7 g � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 3 b 2 b 8 ; f b 4 w 1 �− → , b 5 b 6 b 7 g b 5 b 6 b 7 g � b 1 � � � b 2 b 3 b 4 b 8 ; f b 1 b 11 − b 2 b 13 − b 2 b 82 − b 1 ; f + b 1 − b 2 b 14 − b 2 w 2 �− → g − b 1 + b 2 , , b 5 b 6 b 7 g b 52 − b 1 b 62 − b 1 b 72 − b 1 � � � b 1 � f + b 17 b 2 b 3 b 8 ; f b 4 b 1 b 217 b 317 b 417 w 3 �− → b 8 − b 17 ; , ( g + b 1 )( f + b 7 ) b 5 b 6 b 7 g b 5 b 6 − b 117 − b 1 f − b 1 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 19 / 41
The Affine Weyl Group W ( E (1) 6 ) Theorem Reflections w i are induced by the following elementary birational mappings (also denoted by w i ) on the family X b fixing b 1 and χ ( δ ) (we put b i ··· k = b i + · · · + b k , e.g., b 12 = b 1 + b 2 and so on) � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 2 b 4 b 8 ; f b 3 w 0 �− → , b 5 b 6 b 7 g b 5 b 6 b 7 g � b 1 � � b 1 � b 2 b 3 b 8 ; f b 4 b 3 b 2 b 4 b 8 ; f w 1 �− → , b 5 b 6 b 7 g b 5 b 6 b 7 g � b 1 � � � b 2 b 3 b 4 b 8 ; f b 1 b 11 − b 2 b 13 − b 2 b 82 − b 1 ; f + b 1 − b 2 b 14 − b 2 w 2 �− → g − b 1 + b 2 , , b 5 b 6 b 7 g b 52 − b 1 b 62 − b 1 b 72 − b 1 � � � b 1 � f + b 17 b 2 b 3 b 4 b 8 ; f b 1 b 217 b 317 b 417 w 3 �− → b 8 − b 17 ; , ( g + b 1 )( f + b 7 ) b 5 b 6 b 7 g b 5 b 6 − b 117 − b 1 f − b 1 � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 2 b 3 b 7 ; f b 4 w 4 �− → , b 5 b 6 b 7 g b 5 b 6 b 8 g Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 19 / 41
The Affine Weyl Group W ( E (1) 6 ) Theorem Reflections w i are induced by the following elementary birational mappings (also denoted by w i ) on the family X b fixing b 1 and χ ( δ ) (we put b i ··· k = b i + · · · + b k , e.g., b 12 = b 1 + b 2 and so on) � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 2 b 4 b 8 ; f b 3 w 0 �− → , b 5 b 6 b 7 g b 5 b 6 b 7 g � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 3 b 2 b 8 ; f b 4 w 1 �− → , b 5 b 6 b 7 g b 5 b 6 b 7 g � b 1 � � � b 2 b 3 b 8 ; f b 4 b 1 b 11 − b 2 b 13 − b 2 b 82 − b 1 ; f + b 1 − b 2 b 14 − b 2 w 2 �− → g − b 1 + b 2 , , b 5 b 6 b 7 g b 52 − b 1 b 62 − b 1 b 72 − b 1 � � � b 1 � f + b 17 b 2 b 3 b 4 b 8 ; f b 1 b 217 b 317 b 417 w 3 �− → b 8 − b 17 ; , ( g + b 1 )( f + b 7 ) b 5 b 6 b 7 g b 5 b 6 − b 117 − b 1 f − b 1 � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 2 b 3 b 4 b 7 ; f w 4 �− → , b 5 b 6 b 7 g b 5 b 6 b 8 g � � � b 1 � ( f − b 1 )( g − b 5 ) b 2 b 3 b 4 b 8 ; f b 1 b 215 b 315 b 415 + b 1 w 5 �− → b 8 ; g + b 1 , b 5 b 6 b 7 g − b 115 b 6 − b 15 b 7 g − b 15 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 19 / 41
The Affine Weyl Group W ( E (1) 6 ) Theorem Reflections w i are induced by the following elementary birational mappings (also denoted by w i ) on the family X b fixing b 1 and χ ( δ ) (we put b i ··· k = b i + · · · + b k , e.g., b 12 = b 1 + b 2 and so on) � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 2 b 4 b 3 b 8 ; f w 0 �− → , b 5 b 6 b 7 g b 5 b 6 b 7 g � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 3 b 2 b 4 b 8 ; f w 1 �− → , b 5 b 6 b 7 g b 5 b 6 b 7 g � b 1 � � � b 2 b 3 b 8 ; f b 4 b 1 b 11 − b 2 b 13 − b 2 b 14 − b 2 b 82 − b 1 ; f + b 1 − b 2 w 2 �− → g − b 1 + b 2 , , b 5 b 6 b 7 g b 52 − b 1 b 62 − b 1 b 72 − b 1 � � � b 1 � f + b 17 b 2 b 3 b 4 b 8 ; f b 1 b 217 b 317 b 417 w 3 �− → b 8 − b 17 ; , ( g + b 1 )( f + b 7 ) b 5 b 6 b 7 g b 5 b 6 − b 117 − b 1 f − b 1 � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 2 b 3 b 4 b 7 ; f w 4 �− → , b 5 b 6 b 7 g b 5 b 6 b 8 g � � � b 1 � ( f − b 1 )( g − b 5 ) b 2 b 3 b 8 ; f b 4 b 1 b 215 b 315 b 415 + b 1 w 5 �− → b 8 ; g + b 1 , b 5 b 6 b 7 g − b 115 b 6 − b 15 b 7 g − b 15 � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 2 b 3 b 4 b 8 ; f w 6 �− → . b 5 b 6 b 7 g b 6 b 5 b 7 g Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 19 / 41
Sketch of the proof Since α 0 = E 2 − E 3 , w 0 : E 2 ↔ E 3 , which just swaps the parameters b 2 and b 3 . Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 20 / 41
Sketch of the proof Since α 0 = E 2 − E 3 , w 0 : E 2 ↔ E 3 , which just swaps the parameters b 2 and b 3 . Similarly, since α 2 = E 1 − E 2 , w 2 : E 1 ↔ E 2 , which swaps b 1 and b 2 , but then we need to use the normalization freedom to ensure that b 1 is fixed, � b 2 � � � b 1 b 4 b 3 b 8 ; f b 1 b 11 − b 2 b 13 − b 2 b 82 − b 1 ; f + b 1 − b 2 b 14 − b 2 ∼ g − b 1 + b 2 , . b 5 b 6 b 7 g b 52 − b 1 b 62 − b 1 b 72 − b 1 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 20 / 41
Sketch of the proof Since α 0 = E 2 − E 3 , w 0 : E 2 ↔ E 3 , which just swaps the parameters b 2 and b 3 . Similarly, since α 2 = E 1 − E 2 , w 2 : E 1 ↔ E 2 , which swaps b 1 and b 2 , but then we need to use the normalization freedom to ensure that b 1 is fixed, � b 2 � � � b 1 b 4 b 3 b 8 ; f b 1 b 11 − b 2 b 13 − b 2 b 14 − b 2 b 82 − b 1 ; f + b 1 − b 2 ∼ g − b 1 + b 2 , . b 5 b 6 b 7 g b 52 − b 1 b 62 − b 1 b 72 − b 1 Consider now α 3 = H f − E 1 − E 7 . Then w 3 ( H f ) = H f , w 3 ( H g ) = H f + H g − E 1 − E 7 , w 3 ( E 1 ) = H f − E 7 , w 3 ( E 7 ) = H f − E 1 , and w 3 ( E i ) = E i otherwise. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 20 / 41
Sketch of the proof Since α 0 = E 2 − E 3 , w 0 : E 2 ↔ E 3 , which just swaps the parameters b 2 and b 3 . Similarly, since α 2 = E 1 − E 2 , w 2 : E 1 ↔ E 2 , which swaps b 1 and b 2 , but then we need to use the normalization freedom to ensure that b 1 is fixed, � b 2 � � � b 1 b 4 b 3 b 8 ; f b 1 b 11 − b 2 b 13 − b 2 b 14 − b 2 b 82 − b 1 ; f + b 1 − b 2 ∼ g − b 1 + b 2 , . b 5 b 6 b 7 g b 52 − b 1 b 62 − b 1 b 72 − b 1 Consider now α 3 = H f − E 1 − E 7 . Then w 3 ( H f ) = H f , w 3 ( H g ) = H f + H g − E 1 − E 7 , w 3 ( E 1 ) = H f − E 7 , w 3 ( E 7 ) = H f − E 1 , and w 3 ( E i ) = E i otherwise. Thus, w − 1 ( H ¯ g ) = H f + H g − E 1 − E 7 , i.e., ¯ g is a coordinate on a pencil of (1 , 1) curves passing 3 through p 1 ( b 1 , − b 1 ) and p 7 ( − b 7 , ∞ ): g | = { Afg + Bf + Cg + D = 0 | − Ab 2 | H ¯ 1 + ( B − C ) b 1 + D = − Ab 7 + C = 0 } = { A ( fg + b 7 g + b 2 1 + b 1 b 7 ) + b ( f − b 1 = 0 } = ⇒ g = P ( fg + b 7 g + b 2 1 + b 1 b 7 ) + Q ( f − b 1 ) f = Lf + M ¯ ¯ 1 + b 1 b 7 ) + S ( f − b 1 ) , Nf + T . R ( fg + b 7 g + b 2 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 20 / 41
Sketch of the proof Since α 0 = E 2 − E 3 , w 0 : E 2 ↔ E 3 , which just swaps the parameters b 2 and b 3 . Similarly, since α 2 = E 1 − E 2 , w 2 : E 1 ↔ E 2 , which swaps b 1 and b 2 , but then we need to use the normalization freedom to ensure that b 1 is fixed, � b 2 � � � b 1 b 4 b 8 ; f b 3 b 1 b 11 − b 2 b 13 − b 2 b 82 − b 1 ; f + b 1 − b 2 b 14 − b 2 ∼ g − b 1 + b 2 , . b 5 b 6 b 7 g b 52 − b 1 b 62 − b 1 b 72 − b 1 Consider now α 3 = H f − E 1 − E 7 . Then w 3 ( H f ) = H f , w 3 ( H g ) = H f + H g − E 1 − E 7 , w 3 ( E 1 ) = H f − E 7 , w 3 ( E 7 ) = H f − E 1 , and w 3 ( E i ) = E i otherwise. Thus, w − 1 ( H ¯ g ) = H f + H g − E 1 − E 7 , i.e., ¯ g is a coordinate on a pencil of (1 , 1) curves passing 3 through p 1 ( b 1 , − b 1 ) and p 7 ( − b 7 , ∞ ): g | = { Afg + Bf + Cg + D = 0 | − Ab 2 | H ¯ 1 + ( B − C ) b 1 + D = − Ab 7 + C = 0 } = { A ( fg + b 7 g + b 2 1 + b 1 b 7 ) + b ( f − b 1 = 0 } = ⇒ g = P ( fg + b 7 g + b 2 1 + b 1 b 7 ) + Q ( f − b 1 ) f = Lf + M ¯ ¯ 1 + b 1 b 7 ) + S ( f − b 1 ) , Nf + T . R ( fg + b 7 g + b 2 g ( − b 8 , ∞ ) = ∞ , R = 0, and since ¯ Since ¯ f ( ∞ , b 5 ) = ∞ , N = 0. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 20 / 41
Sketch of the proof (cont.) So we have g = P fg + b 7 g + b 2 1 + b 1 b 7 ¯ f = Lf + M , ¯ + Q . f − b 1 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 21 / 41
Sketch of the proof (cont.) So we have g = P fg + b 7 g + b 2 1 + b 1 b 7 ¯ f = Lf + M , ¯ + Q . f − b 1 Then we have (¯ b 1 , − ¯ b 1 ) = (¯ f , ¯ g )( f = − b 7 ) = ( − Lb 7 + M , − Pb 1 + Q ) , (¯ b 2 , − ¯ b 2 ) = (¯ f , ¯ g )( b 2 , − b 2 ) = ( Lb 2 + M , − P ( b 1 + b 2 + b 7 ) + Q ) , (¯ b 3 , − ¯ b 3 ) = (¯ f , ¯ g )( b 3 , − b 3 ) = ( Lb 3 + M , − P ( b 1 + b 3 + b 7 ) + Q ) , (¯ b 4 , − ¯ b 4 ) = (¯ f , ¯ g )( b 4 , − b 4 ) = ( Lb 4 + M , − P ( b 1 + b 4 + b 7 ) + Q ) , ( ∞ , ¯ b 5 ) = (¯ f , ¯ g )( ∞ , b 5 ) = ( ∞ , Pb 5 + Q ) , ( ∞ , ¯ b 6 ) = (¯ f , ¯ g )( ∞ , b 6 ) = ( ∞ , Pb 6 + Q ) , ( − ¯ b 7 , ∞ ) = (¯ f , ¯ g )( f = b 1 ) = ( Lb 1 + M , ∞ ) , ( − ¯ b 8 , ∞ ) = (¯ f , ¯ g )( − b 8 , ∞ ) = ( − Lb 8 + M , ∞ ) . Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 21 / 41
Sketch of the proof (cont.) So we have g = P fg + b 7 g + b 2 1 + b 1 b 7 ¯ f = Lf + M , ¯ + Q . f − b 1 Then we have (¯ b 1 , − ¯ b 1 ) = (¯ f , ¯ g )( f = − b 7 ) = ( − Lb 7 + M , − Pb 1 + Q ) , (¯ b 2 , − ¯ b 2 ) = (¯ f , ¯ g )( b 2 , − b 2 ) = ( Lb 2 + M , − P ( b 1 + b 2 + b 7 ) + Q ) , (¯ b 3 , − ¯ b 3 ) = (¯ f , ¯ g )( b 3 , − b 3 ) = ( Lb 3 + M , − P ( b 1 + b 3 + b 7 ) + Q ) , (¯ b 4 , − ¯ b 4 ) = (¯ f , ¯ g )( b 4 , − b 4 ) = ( Lb 4 + M , − P ( b 1 + b 4 + b 7 ) + Q ) , ( ∞ , ¯ b 5 ) = (¯ f , ¯ g )( ∞ , b 5 ) = ( ∞ , Pb 5 + Q ) , ( ∞ , ¯ b 6 ) = (¯ f , ¯ g )( ∞ , b 6 ) = ( ∞ , Pb 6 + Q ) , ( − ¯ b 7 , ∞ ) = (¯ f , ¯ g )( f = b 1 ) = ( Lb 1 + M , ∞ ) , ( − ¯ b 8 , ∞ ) = (¯ f , ¯ g )( − b 8 , ∞ ) = ( − Lb 8 + M , ∞ ) . The first four equations give L = P and then to preserve χ ( δ ) we must have L = P = 1. Then, to fix b 1 , M = b 1 + b 7 and Q = 0, which gives the required map. Other cases are similar to the ones considered. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 21 / 41
The Automorphism Group Aut( A (1) 2 ) ≃ Aut( E (1) 6 ) ≃ D 3 Theorem The acton of the automorphisms on the Picard lattice Pic( X ) , the symmetry sub-lattice Span Z { α i } and the surface sub-lattice Span Z {D i } is given by: m 0 = ( D 1 D 2 ) = ( α 3 α 5 )( α 4 α 6 ) , H f → H g , E 1 → E 1 , E 3 → E 3 , E 5 → E 7 , E 7 → E 5 , H g → H f , E 2 → E 2 , E 4 → E 4 , E 6 → E 8 , E 8 → E 6 ; m 1 = ( D 0 D 2 ) = ( α 0 α 4 )( α 1 α 3 ) , H f → H f , E 1 → H f − E 2 , E 3 → E 7 , E 5 → E 5 , E 7 → E 3 , H g → H f + H g − E 1 − E 2 , E 2 → H f − E 1 , E 4 → E 8 , E 6 → E 6 , E 8 → E 4 ; m 2 = ( D 0 D 1 ) = ( α 0 α 6 )( α 1 α 5 ) , H f → H f + H g − E 1 − E 2 , E 1 → H g − E 2 , E 3 → E 5 , E 5 → E 3 , E 7 → E 7 , H g → H g , E 2 → H g − E 1 , E 4 → E 6 , E 6 → E 4 , E 8 → E 8 ; r = ( D 0 D 1 D 2 ) = ( α 0 α 6 α 4 )( α 1 α 5 α 3 ) , H f → H g , E 1 → H g − E 2 , E 3 → E 5 , E 5 → E 7 , E 7 → E 3 , H g → H f + H g − E 1 − E 2 , E 2 → H g − E 1 , E 4 → E 6 , E 6 → E 8 , E 8 → E 4 ; r 2 = ( D 0 D 2 D 1 ) = ( α 0 α 4 α 6 )( α 1 α 3 α 5 ) , H f → H f + H g − E 1 − E 2 , E 1 → H f − E 2 , E 3 → E 7 , E 5 → E 3 , E 7 → E 5 , H g → H f , E 2 → H f − E 1 , E 4 → E 8 , E 6 → E 4 , E 8 → E 6 . Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 22 / 41
Sketch of the proof This is almost obvious from looking at the diagrams. For example, for m 2 we have Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 23 / 41
Sketch of the proof This is almost obvious from looking at the diagrams. For example, for m 2 we have D 2 = H g − E 7 − E 8 D 0 = H f + H g − E 1 − E 2 − E 3 − E 4 D 1 = H f − E 5 − E 6 α 4 = E 7 − E 8 α 3 = H f − E 1 − E 7 α 2 = E 1 − E 2 α 5 = H g − E 1 − E 5 α 1 = E 2 − E 3 α 6 = E 5 − E 6 α 0 = E 3 − E 4 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 23 / 41
Sketch of the proof This is almost obvious from looking at the diagrams. For example, for m 2 we have D 2 = H g − E 7 − E 8 D 0 = H f + H g − E 1 − E 2 − E 3 − E 4 D 1 = H f − E 5 − E 6 α 4 = E 7 − E 8 α 3 = H f − E 1 − E 7 α 2 = E 1 − E 2 α 5 = H g − E 1 − E 5 α 1 = E 2 − E 3 α 6 = E 5 − E 6 α 0 = E 3 − E 4 Hence, m 2 is given by H f → H f + H g − E 1 − E 2 , E 1 → H g − E 2 , E 3 → E 5 , E 5 → E 3 , E 7 → E 7 , H g → H g , E 2 → H g − E 1 , E 4 → E 6 , E 6 → E 4 , E 8 → E 8 ; Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 23 / 41
The Automorphism Group Aut( A (1) 2 ) ≃ Aut( E (1) 6 ) ≃ D 3 Theorem The automorphisms are given by the following elementary birational maps on the family X b fixing b 1 and χ ( δ ) � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 2 b 4 b 6 ; − f b 3 m 0 �− → , b 5 b 6 b 7 g b 7 b 8 b 5 − g � � � b 1 � b 12 − f b 2 b 3 b 4 b 8 ; f b 1 b 2 b 127 b 128 m 1 �− → b 4 − b 12 ; , g ( f − b 12 ) − b 1 b 2 b 5 b 6 b 7 g b 5 b 6 b 3 − b 12 f + g � � � b 1 � f ( g + b 12 ) − b 1 b 2 b 2 b 3 b 4 b 8 ; f b 1 b 2 b 125 b 126 m 2 �− → b 8 ; f + g , b 5 b 6 b 7 g b 3 − b 12 b 4 − b 12 b 7 − g − b 12 � � � b 1 � b 6 ; − g ( f − b 12 ) − b 1 b 2 b 2 b 3 b 4 b 8 ; f b 1 b 2 b 127 b 128 r �− → f + g , b 5 b 6 b 7 g b 3 − b 12 b 4 − b 12 b 5 f − b 12 � � � b 1 � g + b 12 b 2 b 3 b 4 b 8 ; f r 2 b 1 b 2 b 125 b 126 �− → b 4 − b 12 ; . − f ( g + b 12 ) − b 1 b 2 b 5 b 6 b 7 g b 7 b 8 b 3 − b 12 f + g Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 24 / 41
The Automorphism Group Aut( A (1) 2 ) ≃ Aut( E (1) 6 ) ≃ D 3 Theorem The automorphisms are given by the following elementary birational maps on the family X b fixing b 1 and χ ( δ ) � b 1 � � b 1 � b 2 b 3 b 4 b 8 ; f b 2 b 4 b 6 ; − f b 3 m 0 �− → , b 5 b 6 b 7 g b 7 b 8 b 5 − g � � � b 1 � b 12 − f b 2 b 3 b 4 b 8 ; f b 1 b 2 b 127 b 128 m 1 �− → b 4 − b 12 ; , g ( f − b 12 ) − b 1 b 2 b 5 b 6 b 7 g b 5 b 6 b 3 − b 12 f + g � � � b 1 � f ( g + b 12 ) − b 1 b 2 b 2 b 3 b 4 b 8 ; f b 1 b 2 b 125 b 126 m 2 �− → b 8 ; f + g , b 5 b 6 b 7 g b 3 − b 12 b 4 − b 12 b 7 − g − b 12 � � � b 1 � b 6 ; − g ( f − b 12 ) − b 1 b 2 b 2 b 3 b 4 b 8 ; f b 1 b 2 b 127 b 128 r �− → f + g , b 5 b 6 b 7 g b 3 − b 12 b 4 − b 12 b 5 f − b 12 � � � b 1 � g + b 12 b 2 b 3 b 4 b 8 ; f r 2 b 1 b 2 b 125 b 126 �− → b 4 − b 12 ; . − f ( g + b 12 ) − b 1 b 2 b 5 b 6 b 7 g b 7 b 8 b 3 − b 12 f + g Proof is similar to the previous theorem. Notice that the group structure is preserved on the level of the maps. Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 24 / 41
The Semi-Direct Product Structure W ( E (1) The extended affine Weyl group � ) is a semi-direct product of its normal subgroup 6 W ( E (1) W ( E (1) ) and the subgroup of the diagram automorphisms Aut( E (1) ) ⊳ � ), 6 6 6 � W ( E (1) ) = Aut( D (1) 6 ) ⋉ W ( D (1) 6 ) . 6 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 25 / 41
The Semi-Direct Product Structure W ( E (1) The extended affine Weyl group � ) is a semi-direct product of its normal subgroup 6 W ( E (1) W ( E (1) ) and the subgroup of the diagram automorphisms Aut( E (1) ) ⊳ � ), 6 6 6 � W ( E (1) ) = Aut( D (1) 6 ) ⋉ W ( D (1) 6 ) . 6 We have just described the group structure of W ( E (1) ) and Aut( E (1) ) using generators and 6 6 relations, so it remains to give the action of Aut( E (1) ) on W ( E (1) ). 6 6 Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 25 / 41
The Semi-Direct Product Structure W ( E (1) The extended affine Weyl group � ) is a semi-direct product of its normal subgroup 6 W ( E (1) W ( E (1) ) and the subgroup of the diagram automorphisms Aut( E (1) ) ⊳ � ), 6 6 6 � W ( E (1) ) = Aut( D (1) 6 ) ⋉ W ( D (1) 6 ) . 6 We have just described the group structure of W ( E (1) ) and Aut( E (1) ) using generators and 6 6 relations, so it remains to give the action of Aut( E (1) ) on W ( E (1) ). 6 6 But elements of Aut( E (1) ) act as permutations of the simple roots α i , and so the action is just 6 the corresponding permutation of the corresponding reflections, σ t w α i σ − 1 = w t ( α i ) , where t is t the permutation of α i ’s corresponding to σ t . Anton Dzhamay (UNC) Geometry of Discrete Painlev´ e Equations August 25, 2017 25 / 41
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