|
gap> mat1 := IdentityMat(6,GF(5));
< mutable compressed matrix 6x6 over GF(5) >
gap> form1 := BilinearFormByMatrix(mat1,GF(5));
< bilinear form >
gap> ps1 := PolarSpace(form1);
<polar space in ProjectiveSpace(
5,GF(5)): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0 >
gap> mat2 := [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],
> [0,0,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,0,0]]*Z(5)^0;
[ [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0 ],
[ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0, 0*Z(5) ],
[ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0, 0*Z(5), 0*Z(5) ],
[ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ],
[ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ],
[ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ] ]
gap> form2 := QuadraticFormByMatrix(mat2,GF(5));
< quadratic form >
gap> ps2 := PolarSpace(form2);
<polar space in ProjectiveSpace(5,GF(5)): x_1*x_6+x_2*x_5+x_3*x_4=0 >
gap> iso := IsomorphismPolarSpaces(ps1,ps2,true);
#I No intertwiner computed. One of the polar spaces must have a collineation group computed
<geometry morphism from <Elements of Q+(5,
5): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0> to <Elements of Q+(5,
5): x_1*x_6+x_2*x_5+x_3*x_4=0>>
gap> CollineationGroup(ps1);
#I Computing collineation group of canonical polar space...
<projective collineation group of size 58032000000 with 4 generators>
gap> CollineationGroup(ps2);
#I Computing collineation group of canonical polar space...
<projective collineation group of size 58032000000 with 4 generators>
gap> iso := IsomorphismPolarSpaces(ps1,ps2,true);
<geometry morphism from <Elements of Q+(5,
5): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0> to <Elements of Q+(5,
5): x_1*x_6+x_2*x_5+x_3*x_4=0>>
gap> hom := Intertwiner( iso );
MappingByFunction( <projective collineation group of size 58032000000 with
4 generators>, <projective collineation group of size 58032000000 with
4 generators>, function( y ) ... end, function( x ) ... end )
gap> ps1 := ParabolicQuadric(6,8);
Q(6, 8)
gap> ps2 := SymplecticSpace(5,8);
W(5, 8)
gap> em := IsomorphismPolarSpaces(ps1,ps2);
#I Have 36171 points.
#I Have 37381 points in new orbit.
#I Have 36171 points.
#I Have 37388 points in new orbit.
<geometry morphism from <Elements of Q(6, 8)> to <Elements of W(5, 8)>>
gap> hom := Intertwiner(em);
MappingByFunction( PGammaO(7,8), <projective collineation group of size
27231016821530296320 with
3 generators>, function( el ) ... end, function( el ) ... end )
[ Dauer der Verarbeitung: 0.16 Sekunden
(vorverarbeitet)
]
|
