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<div class="ChapSects"><a href="chap2_mj.html#X7A489A5D79DA9E5C">2 Examples</a>
<div class="ContSect"> <a href="chap2_mj.html#X83510E647FBB2475">2.1 A conic of
\(\mathrm{PG}(2,8)\)
</a>

</div>
<div class="ContSect"> <a href="chap2_mj.html#X781F69578636E8C5">2.2 A form for
\(\mathrm{W}(5,3)\)</a>

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<div class="ContSect"> <a href="chap2_mj.html#X78638D21797AC9A0">2.3 What is the form preserved by this group?</a>

</div>
</div>

<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
\(\mathrm{PG}(2,8)\)
</h4>

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.

<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 \(\mathrm{GO}(3,8)\), 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 \(\mathrm{GO}(3,8)\) 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
\(\mathrm{W}(5,3)\)</h4>

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

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

<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 \(\mathrm{GO}(5,5)\). 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).

<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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