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<a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a>
<div class="ChapSects"><a href="chap2.html#X7A489A5D79DA9E5C">2 Examples</a>
<div class="ContSect"> <a href="chap2.html#X83510E647FBB2475">2.1 A conic of
PG(2,8)
</a>

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<div class="ContSect"> <a href="chap2.html#X781F69578636E8C5">2.2 A form for
W(5,3)</a>

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<div class="ContSect"> <a href="chap2.html#X78638D21797AC9A0">2.3 What is the form preserved by this group?</a>

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<h3>2 Examples</h3>

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

<a id="X83510E647FBB2475" name="X83510E647FBB2475"></a>

<h4>2.1 A conic of
PG(2,8)
</h4>

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

<center>x12+x2x3.</center> 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.

<div class="example"><pre>
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 );
< quadratic form >
gap> Display( form );
Quadratic 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
gap> RadicalOfForm( form );
<vector space over GF(2^3), with 0 generators>
</pre></div>

Now our conic is stabilised by a group isomorphic to GO(3,8), but which is not identical to the group returned by the GAP command <code class="code">GO(3,8)</code>. However, our conic is the canonical conic given in Forms.

<div class="example"><pre>
gap> canonical := IsometricCanonicalForm( form );
< parabolic quadratic form >
gap> form = canonical;
true
</pre></div>

So we ``change forms''...

<div class="example"><pre>
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) );
< quadratic form >
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) ] ] ])
</pre></div>

Now we look at the action of our new GO(3,8) on the conic.

<div class="example"><pre>
gap> conic := Filtered(vec, x -> IsZero( x^form ));;
gap> Size(conic);
64
gap> orbs := Orbits(newgo, conic, OnRight);;
gap> List(orbs,Size);
[ 1, 63 ]
</pre></div>

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.

<a id="X781F69578636E8C5" name="X781F69578636E8C5"></a>

<h4>2.2 A form for
W(5,3)</h4>

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

<center> f(x,y)=x1y2-x2y1+x3y4-x4y3+ ... +x2n-1y2n-x2ny2n-1</center>

<div class="example"><pre>
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 );
< immutable compressed matrix 6x6 over GF(3) >
gap> Display( b );
1 . . . . .
. . . 1 . .
. . 1 . . .
. . . . . 1
. . . . 1 .
. 2 . . . .
gap> Display( b * gram * TransposedMat(b) );
. 1 . . . .
2 . . . . .
. . . 1 . .
. . 2 . . .
. . . . . 1
. . . . 2 .

</pre></div>

<a id="X78638D21797AC9A0" name="X78638D21797AC9A0"></a>

<h4>2.3 What is the form preserved by this group?</h4>

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

<div class="example"><pre>
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:
3 4 3 1 1
4 4 4 1 1
3 4 1 2 3
1 1 2 4 3
1 1 3 3 1

</pre></div>

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