aa r X i v : . [ m a t h . AG ] A p r SYMMETRIES IN THE SYSTEM OF TYPE A (2)5 YUSUKE SASANO
Abstract.
In this paper, we propose a 3-parameter family of coupled Painlev´e IIIsystems in dimension four with affine Weyl group symmetry of type A (2)5 . We alsopropose its symmetric form in which the A (2)5 -symmetries become clearly visible. Statement of main results
In [10, 11, 13], we presented some types of coupled Painlev´e systems with variousaffine Weyl group symmetries. In this paper, we present a 3-parameter family ofcoupled Painlev´e III systems with affine Weyl group symmetry of type A (2)5 . Thissystem is the first example with the B¨acklund transformations satisfying Noumi-Yamada’s universal description for A (2)5 root system (see [3]). At first, we propose a3-parameter family of autonomous ordinary differential systems with the invariantdivisors f i as variables: df dt = { f − f ) g − g f − α − α } f + α ( f − f ) ,df dt = { f − f ) g − g f − α − α } f + α ( f − f ) ,df dt = { ( f − f + 3 f ) g + ( f + 3 f − f ) g + 1 } f − α g g ,df dt = − (2 f g + 2 f g + α + α ) f − α ( f + f ) ,dg dt = ( f − f + f ) g + { ( f − f − f ) g + α + α + α } g + f ( f + 3 f − f ) ,dg dt = ( f − f + f ) g + { ( f − f − f ) g + α + α + α } g + f ( f + 3 f − f ) . (1)Here f , f , f , f and g , g denote unknown complex variables and α , . . . , α arethe parameters satisfying the condition α + α + 2 α + α = 12 . Proposition 0.1.
This system has the following invariant divisors :invariant divisors parameter’s relation f := 0 α = 0 f := 0 α = 0 f := 0 α = 0 f := 0 α = 0 Key words and phrases.
Affine Weyl group, birational symmetries, coupled Painlev´e systems.2000 Mathematics Subject Classification Numbers. 34M55, 34M45, 58F05, 32S65.
Theorem 0.1.
This system is invariant under the transformations s , s , s , s , π defined as follows : with the notation ( ∗ ) := ( f , f , f , f , g , g ; α , α , α , α ) ,s : ( ∗ ) → ( f , f , f + α g f , f , g + α f , g ; − α , α , α + α , α ) ,s : ( ∗ ) → ( f , f , f + α g f , f , g , g + α f ; α , − α , α + α , α ) ,s : ( ∗ ) → ( f − α g f , f − α g f , f , f − α ( g + g ) f , g , g ; α + α , α + α , − α , α + 2 α ) ,s : ( ∗ ) → ( f , f , f + α ( g + g ) f + α f , f , g + α f , g + α f ; α , α , α + α , − α ) ,π : ( ∗ ) → ( f , f , f , f , g , g ; α , α , α , α ) . (2)Here the Poisson bracket { , } is defined by { f , f } = g + g , { f , g } = { f , g } = 1 . Theorem 0.2.
This system has two first integrals : df dt = d ( f + f ) dt , d ( f − g g ) dt = f − g g . From this, we have f = f + f − , f − g g = e ( t + c ) . Here we set t + c = logT, q := g , p := f , q := g , p := f , then we obtain a 3-parameter family of coupled Painlev´e III systems in dimensionfour with affine Weyl group symmetry of type A (2)5 explicitly given by dq dT = 2 q p − q + ( α + α + α ) q T − p + 2 q q p T ,dp dT = − q p + 2 q p − ( α + α + α ) p + α T − p q p T ,dq dT = 2 q p − q + ( α + α + α ) q T − p + 2 q p q T ,dp dT = − q p + 2 q p − ( α + α + α ) p + α T − q p p T (3)with the Hamiltonian H = q p − q p + ( α + α + α ) q p − α q T − p + q p − q p + ( α + α + α ) q p − α q T − p + 4 p p + 2 q p q p T . (4)Here the Poisson bracket { , } is defined by { q , p } = { q , p } = 1 , { q , q } = { q , p } = { p , q } = { p , p } = 0 . YMMETRIES IN THE SYSTEM OF TYPE A (2)5 p p q q + T p + p − Figure 1.
The transformations s i satisfy the relations: s i = 1 ( i =0 , , , , ( s s ) = ( s s ) = ( s s ) = 1 , ( s s ) = ( s s ) =1 , ( s s ) = 1 . Theorem 0.3.
This system is invariant under the transformations s , s , s , s , π defined as follows : with the notation ( ∗ ) := ( q , p , q , p , T ; α , α , α , α ) ,s : ( ∗ ) → ( q + α p , p , q , p , T ; − α , α , α + α , α ) ,s : ( ∗ ) → ( q , p , q + α p , p , T ; α , − α , α + α , α ) ,s : ( ∗ ) → ( q , p − α q q q + T , q , p − α q q q + T , T ; α + α , α + α , − α , α + 2 α ) ,s : ( ∗ ) → ( q + α p + p − , p , q + α p + p − , p , T ; α , α , α + α , − α ) ,π : ( ∗ ) → ( q , p , q , p , T ; α , α , α , α ) . (5) References [1] K. Fuji and T. Suzuki,
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