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### 2 Examples

Here we give some simple examples that display some of the functionality of Forms.

#### 2.1 A conic of $$\mathrm{PG}(2,8)$$

Consider the three-dimensional vector space $$V$$ over the finite field $$\mathrm{GF}(8)$$ , and consider the following quadratic polynomial in 3 variables:

$x_1^2+x_2x_3.$

Then this polynomial defines a quadratic form on $$V$$ and the zeros form a conic of the associated projective plane. So in particular, our quadratic form defines a degenerate parabolic quadric of Witt Index 1. We will see now how we can use Forms to view this example.

gap> gf := GF(8);
GF(2^3)
gap> vec := gf^3;
( GF(2^3)^3 )
gap> r := PolynomialRing( gf, 3);
PolynomialRing(..., [ x_1, x_2, x_3 ])
gap> poly := r.1^2 + r.2 * r.3;
x_1^2+x_2*x_3
gap> form := QuadraticFormByPolynomial( poly, r );
gap> Display( form );
Gram Matrix:
1 . .
. . 1
. . .
Polynomial: x_1^2+x_2*x_3
gap> IsDegenerateForm( form );
#I  Testing degeneracy of the *associated bilinear form*
true
gap> IsSingularForm( form );
false
gap> WittIndex( form );
1
gap> IsParabolicForm( form );
true
<vector space over GF(2^3), with 0 generators>


Now our conic is stabilised by a group isomorphic to $$\mathrm{GO}(3,8)$$, but which is not identical to the group returned by the GAP command GO(3,8). However, our conic is the canonical conic given in Forms.

gap> canonical := IsometricCanonicalForm( form );
gap> form = canonical;
true


So we change forms''...

gap> go := GO(3,8);
GO(0,3,8)
gap> mat := InvariantQuadraticForm( go )!.matrix;
[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), Z(2)^0, 0*Z(2) ] ]
gap> gapform := QuadraticFormByMatrix( mat, GF(8) );
gap> b := BaseChangeToCanonical( gapform );
[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2) ],
[ 0*Z(2), 0*Z(2), Z(2)^0 ] ]
gap> hom := BaseChangeHomomorphism( b, GF(8) );
^[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2) ],
[ 0*Z(2), 0*Z(2), Z(2)^0 ] ]
gap> newgo := Image(hom, go);
Group(
[ [ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2^3), 0*Z(2) ], [ 0*Z(2), 0*Z(2),
Z(2^3)^6 ] ],
[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ Z(2)^0, Z(2)^0, Z(2)^0 ],
[ 0*Z(2), Z(2)^0, 0*Z(2) ] ] ])


Now we look at the action of our new $$\mathrm{GO}(3,8)$$ on the conic.

gap> conic := Filtered(vec, x -> IsZero( x^form ));;
gap> Size(conic);
64
gap> orbs := Orbits(newgo, conic, OnRight);;
gap> List(orbs,Size);
[ 1, 63 ]


So we see that there is a fixed point, which is actually the nucleus of the conic, or in other words, the radical of the form.

#### 2.2 A form for $$\mathrm{W}(5,3)$$

The symplectic polar space $$\mathrm{W}(5,q)$$ is defined by an alternating reflexive bilinear form on the six-dimensional vector space over the finite field $$\mathrm{GF}(q)$$. Any invertible $$6 \times 6$$ matrix $$A$$ which satisfies $$A+A^T=0$$ is a candidate for the Gram matrix of a symplectic polarity. The canonical form we adopt in Forms for an alternating form is

$f(x,y)=x_1y_2-x_2y_1+x_3y_4-x_4y_3\cdots+x_{2n-1}y_{2n}-x_{2n}y_{2n-1}.$

gap> f := GF(3);
GF(3)
gap> gram := [
> [0,0,0,1,0,0],
> [0,0,0,0,1,0],
> [0,0,0,0,0,1],
> [-1,0,0,0,0,0],
> [0,-1,0,0,0,0],
> [0,0,-1,0,0,0]] * One(f);;
gap> form := BilinearFormByMatrix( gram, f );
< bilinear form >
gap> IsSymplecticForm( form );
true
gap> Display( form );
Symplectic form
Gram Matrix:
. . . 1 . .
. . . . 1 .
. . . . . 1
2 . . . . .
. 2 . . . .
. . 2 . . .
gap> b := BaseChangeToCanonical( form );
[ [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ],
[ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ],
[ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ],
[ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ],
[ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ],
[ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ] ]
gap> Display( b );
1 . . . . .
. . . 1 . .
. 1 . . . .
. . . . 1 .
. . 1 . . .
. . . . . 1
gap> Display( b * gram * TransposedMat(b) );
. 1 . . . .
2 . . . . .
. . . 1 . .
. . 2 . . .
. . . . . 1
. . . . 2 .


#### 2.3 What is the form preserved by this group?

Here we start with a matrix group which is available in GAP, namely $$\mathrm{GO}(5,5)$$. We then conjugate this group by an element of $$\mathrm{GL}(5,5)$$, and then we find the forms left invariant by this copy of $$\mathrm{GO}(5,5)$$ (which we expect to be a symmetric bilinear form).

gap> go := GO(5, 5);
GO(0,5,5)
gap> x :=
> [ [ Z(5)^0, Z(5)^3, 0*Z(5), Z(5)^3, Z(5)^3 ],
>   [ Z(5)^2, Z(5)^3, 0*Z(5), Z(5)^2, Z(5) ],
>   [ Z(5)^2, Z(5)^2, Z(5)^0, Z(5), Z(5)^3 ],
>   [ Z(5)^0, Z(5)^3, Z(5), Z(5)^0, Z(5)^3 ],
>   [ Z(5)^3, 0*Z(5), Z(5)^0, 0*Z(5), Z(5) ]
>  ];;
gap> go2 := go^x;
<matrix group of size 18720000 with 2 generators>
gap> forms := PreservedSesquilinearForms( go2 );
[ < bilinear form > ]
gap> Display( forms[1] );
Bilinear form
Gram Matrix:
4 2 4 3 3
2 2 2 3 3
4 2 3 1 4
3 3 1 2 4
3 3 4 4 3

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