aa r X i v : . [ m a t h . AG ] A p r SYMMETRIES IN THE SYSTEM OF TYPE D (1)4 YUSUKE SASANO
Abstract.
In this paper, we propose a 4-parameter family of coupled Painlev´eIII systems in dimension four with affine Weyl group symmetry of type D (1)4 . Wealso propose its symmetric form in which the D (1)4 -symmetries become clearlyvisible. Statement of main results
In [9, 10, 11], we presented some types of coupled Painlev´e systems with variousaffine Weyl group symmetries. In this paper, we present a 4-parameter family ofcoupled Painlev´e III systems with affine Weyl group symmetry of type D (1)4 . Sincethese universal B¨acklund transformations have Lie theoretic origin, similarity reduc-tion of a Drinfeld-Sokolov hierarchy admits such a B¨acklund symmetry. The aimof this paper is to introduce the system of type D (1)4 . After our discovery of thissystem, they were studied from the viewpoint of Drinfeld-Sokolov hierarchy by K.Fuji independently (cf. [1]), and he succeeded to obtain our system by similarityreduction of the Drinfeld-Sokolov hierarchy of type D (1)4 .At first, we propose a 4-parameter family of autonomous ordinary differentialsystems with the invariant divisors f i as variables: df dt = − (2 f g + α ) f − α f ,df dt = − (2 f g + α ) f − α f ,df dt = { ( f + f ) g + ( f + f ) g + 1 } f − α g g ,df dt = − (2 f g + α ) f − α f ,df dt = − (2 f g + α ) f − α f ,dg dt = ( f + f ) g − { ( f + f ) g − α − α } g + ( f + f ) f ,dg dt = ( f + f ) g − { ( f + f ) g − α − α } g + ( f + f ) f . (1)Here f , f , . . . , f and g , g denote unknown complex variables and α , . . . , α arethe parameters satisfying the condition: α + α + 2 α + α + α = 1 . Proposition 0.1.
This system has the following invariant divisors : Key words and phrases.
Affine Weyl group, birational symmetries, coupled Painlev´e systems.2000 Mathematics Subject Classification Numbers. 34M55, 34M45, 58F05, 32S65. invariant divisors parameter’s relation f := 0 α = 0 f := 0 α = 0 f := 0 α = 0 f := 0 α = 0 f := 0 α = 0 Theorem 0.1.
This system is invariant under the transformations s , . . . , s defined as follows : with the notation ( ∗ ) := ( f , f , . . . , f , g , g ; α , α , . . . , α ) ,s : ( ∗ ) → ( f , f , f + α g f , f , f , g + α f , g ; − α , α , α + α , α , α ) ,s : ( ∗ ) → ( f , f , f + α g f , f , f , g + α f , g ; α , − α , α + α , α , α ) ,s : ( ∗ ) → ( f − α g f , f − α g f , f , f − α g f , f − α g f , g , g ; α + α , α + α , − α , α + α , α + α ) ,s : ( ∗ ) → ( f , f , f + α g f , f , f , g , g + α f ; α , α , α + α , − α , α ) ,s : ( ∗ ) → ( f , f , f + α g f , f , f , g , g + α f ; α , α , α + α , α , − α ) . (2) Theorem 0.2.
This system has two first integrals : d ( f − f ) dt = d ( f − f ) dt = 0 , d ( f − g g ) dt = f − g g . From this, we have f = f − , f = f − , f − g g = e ( t + c ) . Here we set t + c = logT, x := g , y := f , z := g , w := f , then we obtain a 4-parameter family of coupled Painlev´e III systems in dimensionfour with affine Weyl group symmetry of type D (1)4 explicitly given by dxdT = 2 x y − x + ( α + α ) xT − w,dydT = − xy + 2 xy − ( α + α ) y + α T ,dzdT = 2 z w − z + ( α + α ) zT − y,dwdT = − zw + 2 zw − ( α + α ) w + α T (3)with the Hamiltonian H = x y − x y + ( α + α ) xy − α xT − y + z w − z w + ( α + α ) zw − α zT − w + 2 yw. (4) YMMETRIES IN THE SYSTEM OF TYPE D (1)4 y − y xz + T ww −
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Figure 1.
The transformations s i satisfy the relations: s i = 1 ( i =0 , , , , , ( s s ) = ( s s ) = ( s s ) = ( s s ) = ( s s ) =( s s ) = 1 , ( s s ) = ( s s ) = ( s s ) = ( s s ) = 1 . Theorem 0.3.
This system is invariant under the transformations s , . . . , s , π ,π , π defined as follows : with the notation ( ∗ ) := ( x, y, z, w, T ; α , α , α , α , α ) ,s : ( ∗ ) → ( x + α y − , y, z, w, T ; − α , α , α + α , α , α ) ,s : ( ∗ ) → ( x + α y , y, z, w, T ; α , − α , α + α , α , α ) ,s : ( ∗ ) → ( x, y − α zxz + T , z, w − α xxz + T , T ; α + α , α + α , − α , α + α , α + α ) ,s : ( ∗ ) → ( x, y, z + α w − , w, T ; α , α , α + α , − α , α ) ,s : ( ∗ ) → ( x, y, z + α w , w, T ; α , α , α + α , α , − α ) ,π : ( ∗ ) → ( − x, − y, z, w, − T ; α , α , α , α , α ) ,π : ( ∗ ) → ( x, y, − z, − w, − T ; α , α , α , α , α ) ,π : ( ∗ ) → ( z, w, x, y, T ; α , α , α , α , α ) . (5) References [1] K. Fuji and T. Suzuki,
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