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<title>GAP (Forms) - Chapter 2: Examples</title>
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<p><a id="s0ss0" name="s0ss0"></a></p>
<h3>2. Examples</h3>
<p>Here we give some simple examples that display some of the functionality of <strong class="pkg">Forms</strong>.</p>
<p><a id="s1ss0" name="s1ss0"></a></p>
<h4>2.1 A conic of PG(2,8)</h4>
<p>Consider the three-dimensional vector space V=GF(8)^3 over GF(8), and consider the following quadratic polynomial in 3 variables:</p>
<p class="pcenter">\[
x_1^2+x_2x_3.
\]</p>
<p>Then this polynomial defines a quadratic form in V and the zeros form a <em>conic</em> 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 <strong class="pkg">Forms</strong> to view this example.</p>
<table class="example">
<tr><td><pre>
gap> gf := GF(8);
GF(2^3)
gap> vec := gf^3;
( GF(2^3)^3 )
gap> r := PolynomialRing( gf, 3 );
GF(2^3)[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 );
true
gap> WittIndex( form );
1
gap> IsParabolicForm( form );
true
gap> RadicalOfForm( form );
<vector space of dimension 1 over GF(2^3)>
</pre></td></tr></table>
<p>Now our conic is stabilised by GO(3,8), but not the same GO(3,8) that is installed in GAP. However, our conic is the canonical conic given in <strong class="pkg">Forms</strong>.</p>
<table class="example">
<tr><td><pre>
gap> canonical := IsometricCanonicalForm( form );
< quadratic form >
gap> form = canonical;
true
</pre></td></tr></table>
<p>So we ``change forms''...</p>
<table class="example">
<tr><td><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></td></tr></table>
<p>Now we look at the action of our new GO(3,8) on the conic.</p>
<table class="example">
<tr><td><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></td></tr></table>
<p>So we see that there is a fixed point, which is actually the <em>nucleus</em> of the conic, or in other words, the radical of the form.</p>
<p><a id="s2ss0" name="s2ss0"></a></p>
<h4>2.2 A form for W(5,3)</h4>
<p>The symplectic polar space W(5,q) is defined by an alternating reflexive bilinear form on the six-dimensional vector space GF(q)^6. Any invertible 6times 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 <strong class="pkg">Forms</strong> for an alternating form is</p>
<p class="pcenter">\[f(x,y)=x_1y_2-x_2y_1+x_3y_4-x_4y_3\cdots+x_{2n-1}y_{2n}-x_{2n}y_{2n-1}. \]</p>
<table class="example">
<tr><td><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 );
Bilinear form
Gram Matrix:
. . . 1 . .
. . . . 1 .
. . . . . 1
2 . . . . .
. 2 . . . .
. . 2 . . .
gap> b := BaseChangeToCanonical( form );;
gap> Display( b );
. . . . . 1
. . 2 . . .
. . . . 1 .
. 2 . . . .
. . . 1 . .
2 . . . . .
gap> Display( b * gram * TransposedMat(b) );
. 1 . . . .
2 . . . . .
. . . 1 . .
. . 2 . . .
. . . . . 1
. . . . 2 .
</pre></td></tr></table>
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