|
gap> #isotropic subspaces. symplectic form
gap> q := 5;
5
gap> hdim := 3;
3
gap> dim := 2*hdim;
6
gap> f := GF(q);
GF(5)
gap> nzero := List([0..q-2],x->Z(q)^x);
[ Z(5)^0, Z(5), Z(5)^2, Z(5)^3 ]
gap> mat := NullMat(2*hdim,2*hdim,f);
[ [ 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), 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> for i in [1..hdim] do
> entry := Random(nzero);;
> mat[i][2*hdim-i+1] := entry;
> mat[2*hdim-i+1][i] := -entry;
> od;
gap> form := BilinearFormByMatrix(mat,f);
< bilinear form >
gap> IsAlternatingForm(form);
true
gap> v := f^dim;
( GF(5)^6 )
gap> lines := Subspaces(v,1);
Subspaces( ( GF(5)^6 ), 1 )
gap> matrices := List(lines,x->BasisVectors(Basis(x)));;
gap> vectors := List(matrices,x->x[1]);;
gap> results := Collected(List(vectors,x->EvaluateForm(form,x,x)));;
gap> [Zero(f),(q^dim-1)/(q-1)] in results;
true
gap> results := Collected(List(matrices,x->EvaluateForm(form,x,x)));;
gap> [[[Zero(f)]],(q^dim-1)/(q-1)] in results;
true
gap> Number(vectors,x->IsIsotropicVector(form,x))=(q^dim-1)/(q-1);
true
gap> Number(matrices,x->IsTotallyIsotropicSubspace(form,x))=(q^dim-1)/(q-1);
true
gap> planes := Subspaces(v,2);
Subspaces( ( GF(5)^6 ), 2 )
gap> matrices := List(planes,x->BasisVectors(Basis(x)));;
gap> results := Collected(List(matrices,x->EvaluateForm(form,x,x)));;
gap> z := Zero(f);
0*Z(5)
gap> [[[z,z],[z,z]],(q^2+1)*(q^6-1)/(q-1)] in results;
true
gap> Number(matrices,x->IsTotallyIsotropicSubspace(form,x))=(q^2+1)*(q^6-1)/(q-1);
true
gap> threespaces := Subspaces(v,3);
Subspaces( ( GF(5)^6 ), 3 )
gap> matrices := List(lines,x->BasisVectors(Basis(x)));;
gap> results := Collected(List(matrices,x->EvaluateForm(form,x,x)));;
gap> z := Zero(f);
0*Z(5)
gap> [[[Zero(f)]],(q^3+1)*(q^2+q+1)] in results;
true
gap> Number(matrices,x->IsTotallyIsotropicSubspace(form,x))=(q^3+1)*(q^2+q+1);
true
gap> quit;
[ Dauer der Verarbeitung: 0.14 Sekunden
(vorverarbeitet)
]
|
