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<p id="mathjaxlink" class="pcenter"><a href="chap2_mj.html">[MathJax on]</a></p>
<p><a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a></p>
<div class="ChapSects"><a href="chap2.html#X7A489A5D79DA9E5C">2 <span class="Heading">Examples</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X83510E647FBB2475">2.1 <span class="Heading">A conic of 
PG(2,8)
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X781F69578636E8C5">2.2 <span class="Heading">A form for 
W(5,3)</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X78638D21797AC9A0">2.3 <span class="Heading">What is the form preserved by this group?</span></a>
</span>
</div>
</div>

<h3>2 <span class="Heading">Examples</span></h3>

<p>Here we give some simple examples that display some of the functionality of <strong class="pkg">Forms</strong>.</p>

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

<h4>2.1 <span class="Heading">A conic of 
PG(2,8)
</span></h4>

<p>Consider the three-dimensional vector space <span class="SimpleMath">V</span> over the finite field GF(8), and consider the following quadratic polynomial in 3 variables:</p>

<p><center>x<sub>1</sub><sup>2</sup>+x<sub>2</sub>x<sub>3</sub>.</center> Then this polynomial defines a quadratic form on <span class="SimpleMath">V</span> 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>


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

<p>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 <strong class="pkg">Forms</strong>.</p>


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

<p>So we ``change forms''...</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">go := GO(3,8);</span>
GO(0,3,8)
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := InvariantQuadraticForm( go )!.matrix;</span>
[ [ 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) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">gapform := QuadraticFormByMatrix( mat, GF(8) );</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">b := BaseChangeToCanonical( gapform );</span>
[ [ 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 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">hom := BaseChangeHomomorphism( b, GF(8) );</span>
^[ [ 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 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">newgo := Image(hom, go);</span>
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>

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


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

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

<h4>2.2 <span class="Heading">A form for 
W(5,3)</span></h4>

<p>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 <span class="SimpleMath">6 × 6</span> matrix <span class="SimpleMath">A</span> which satisfies A+A<sup>T</sup>=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><center> f(x,y)=x<sub>1</sub>y<sub>2</sub><font face="symbol">-</font >x<sub>2</sub>y<sub>1</sub>+x<sub>3</sub>y<sub>4</sub><font face="symbol">-</font >x<sub>4</sub>y<sub>3</sub>+ ... +x<sub>2n<font face="symbol">-</font >1</sub>y<sub>2n</sub><font face="symbol">-</font >x<sub>2n</sub>y<sub>2n<font face="symbol">-</font >1</sub></center></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := GF(3);</span>
GF(3)
<span class="GAPprompt">gap></span> <span class="GAPinput">gram := [</span>
<span class="GAPprompt">></span> <span class="GAPinput">[0,0,0,1,0,0], </span>
<span class="GAPprompt">></span> <span class="GAPinput">[0,0,0,0,1,0],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[0,0,0,0,0,1],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[-1,0,0,0,0,0],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[0,-1,0,0,0,0],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[0,0,-1,0,0,0]] * One(f);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByMatrix( gram, f );</span>
< bilinear form >
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSymplecticForm( form );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( form );</span>
Symplectic form
Gram Matrix:
 . . . 1 . .
 . . . . 1 .
 . . . . . 1
 2 . . . . .
 . 2 . . . .
 . . 2 . . .
<span class="GAPprompt">gap></span> <span class="GAPinput">b := BaseChangeToCanonical( form );</span>
< immutable compressed matrix 6x6 over GF(3) >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( b );</span>
 1 . . . . .
 . . . 1 . .
 . . 1 . . .
 . . . . . 1
 . . . . 1 .
 . 2 . . . .
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( b * gram * TransposedMat(b) );</span>
 . 1 . . . .
 2 . . . . .
 . . . 1 . .
 . . 2 . . .
 . . . . . 1
 . . . . 2 .
 
</pre></div>

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

<h4>2.3 <span class="Heading">What is the form preserved by this group?</span></h4>

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


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">go := GO(5, 5);</span>
GO(0,5,5)
<span class="GAPprompt">gap></span> <span class="GAPinput">x := </span>
<span class="GAPprompt">></span> <span class="GAPinput">[ [ Z(5)^0, Z(5)^3, 0*Z(5), Z(5)^3, Z(5)^3 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput">  [ Z(5)^2, Z(5)^3, 0*Z(5), Z(5)^2, Z(5) ], </span>
<span class="GAPprompt">></span> <span class="GAPinput">  [ Z(5)^2, Z(5)^2, Z(5)^0, Z(5), Z(5)^3 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">  [ Z(5)^0, Z(5)^3, Z(5), Z(5)^0, Z(5)^3 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput">  [ Z(5)^3, 0*Z(5), Z(5)^0, 0*Z(5), Z(5) ] </span>
<span class="GAPprompt">></span> <span class="GAPinput"> ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">go2 := go^x;</span>
<matrix group of size 18720000 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">forms := PreservedSesquilinearForms( go2 );</span>
[ < bilinear form > ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( forms[1] );</span>
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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