<?xml version="1.0" encoding="ISO-8859-1"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (Forms) - Chapter 2: Examples</title> <meta http-equiv="content-type" content="text/html; charset=iso-8859-1" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> </head> <body> <div class="pcenter"><table class="chlink"><tr><td class="chlink1">Goto Chapter: </td><td><a href="chap0.html">Top</a></td><td><a href="chap1.html">1</a></td><td><a href="chap2.html">2</a></td><td><a href="chap3.html">3</a></td><td><a href="chap4.html">4</a></td><td><a href="chapBib.html">Bib</a></td><td><a href="chapInd.html">Ind</a></td></tr></table><br /></div> <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> <div class="pcenter"> <table class="chlink"><tr><td><a href="chap0.html">Top of Book</a></td><td><a href="chap1.html">Previous Chapter</a></td><td><a href="chap3.html">Next Chapter</a></td></tr></table> <br /> <div class="pcenter"><table class="chlink"><tr><td class="chlink1">Goto Chapter: </td><td><a href="chap0.html">Top</a></td><td><a href="chap1.html">1</a></td><td><a href="chap2.html">2</a></td><td><a href="chap3.html">3</a></td><td><a href="chap4.html">4</a></td><td><a href="chapBib.html">Bib</a></td><td><a href="chapInd.html">Ind</a></td></tr></table><br /></div> </div> <hr /> <p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p> </body> </html>