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examples.xml Forms package documentation John Bamberg and Jan De Beule Copyright (C) 2021, The University of Western Australia Copyright (C) 2021, Vrije Universiteit Brussel This chapter gives examples for the usage of this package. --> <Chapter Label="examples"> <Heading>Examples</Heading> Here we give some simple examples that display some of the functionality of <Package>Forms</Package>. <Section> <Heading>A conic of <Alt Only="Text"><M>PG(2,8)</M></Alt><Alt Only="LaTeX"><M>&pg;(2,8)</M></Alt> <Alt Only="HTML MathJax"><M>&pg;(2,8)</M></Alt><Alt Only="HTML noMathJax">PG(2,8)</Alt> </Heading> Consider the three-dimensional vector space <M>V</M> over the finite field <Alt Only="Text"><M>GF(8)</M></Alt><Alt Only="LaTeX"><M>&gf;(8)</M></Alt><Alt Only="HTML MathJax"><M>&gf;(8)</M></ <Alt Only="HTML noMathJax">GF(8)</Alt>, and consider the following quadratic polynomial in 3 variables: <Alt Only="HTML MathJax"><Display>x_1^2+x_2x_3.</Display></Alt> <Alt Only="LaTeX"><Display>x_1^2+x_2x_3.</Display></Alt> <Alt Only="Text"><Display>x_1^2+x_2x_3.</Display></Alt> <Alt Only="HTML noMathJax"><center>x<sub>1</sub><sup>2</sup>+x<sub>2</sub>x<sub>3</sub>.</center></Alt> Then this polynomial defines a quadratic form on <M>V</M> and the zeros form a <E>conic</E> 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 <Package>Forms</Package> to view this example. <Example><![CDATA[ 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> ]]></Example> Now our conic is stabilised by a group isomorphic to <Alt Only="Text"><M>GO(3,8)</M></Alt><Alt Only="LaTeX"><M>&go;(3,8)</M></Alt> <Alt Only="HTML MathJax"><M>&go;(3,8)</M></Alt><Alt Only="HTML noMathJax">GO(3,8)</Alt>, but which is not identical to the group returned by the GAP command <C>GO(3,8)</C>. However, our conic is the canonical conic given in <Package>Forms</Package>. <Example><![CDATA[ gap> canonical := IsometricCanonicalForm( form ); < parabolic quadratic form > gap> form = canonical; true ]]></Example> So we ``change forms''... <Example><![CDATA[ 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) ] ] ]) ]]></Example> Now we look at the action of our new <Alt Only="Text"><M>GO(3,8)</M></Alt><Alt Only="LaTeX"><M>&go;(3,8)</M></Alt> <Alt Only="HTML MathJax"><M>&go;(3,8)</M></Alt><Alt Only="HTML noMathJax">GO(3,8)</Alt> on the conic. <Example><![CDATA[ gap> conic := Filtered(vec, x -> IsZero( x^form ));; gap> Size(conic); 64 gap> orbs := Orbits(newgo, conic, OnRight);; gap> List(orbs,Size); [ 1, 63 ] ]]></Example> So we see that there is a fixed point, which is actually the <E>nucleus</E> of the conic, or in other words, the radical of the form. </Section> <Section> <Heading>A form for <Alt Only="Text"><M>W(5,3)</M></Alt><Alt Only="LaTeX"><M>\mathrm{W}(5,3)</M></Alt> <Alt Only="HTML MathJax"><M>\mathrm{W}(5,3)</M></Alt><Alt Only="HTML noMathJax">W(5,3)</Alt></H The symplectic polar space <Alt Only="Text"><M>W(5,q)</M></Alt><Alt Only="LaTeX"><M>\mathrm{W}(5,q <Alt Only="HTML MathJax"><M>\mathrm{W}(5,q)</M></Alt><Alt Only="HTML noMathJax">W(5,q)</Alt> is alternating reflexive bilinear form on the six-dimensional vector space over the finite field &gfq;. Any invertible <M>6 \times 6</M> matrix <M>A</M> which satisfies <Alt Only="Text"><M>A+A^T=0</M></Alt><Alt Only="LaTeX"><M>A+A^T=0</M></Alt> <Alt Only="HTML MathJax"><M>A+A^T=0</M></Alt><Alt Only="HTML noMathJax">A+A<sup>T</sup>=0</Alt> is a candidate for the Gram matrix of a symplectic polarity. The canonical form we adopt in <Package>Forms</Package> for an alternating form is <Alt Only="Text"><Display>f(x,y)=x_1y_2-x_2y_1+x_3y_4-x_4y_3\cdots+x_{2n-1}y_{2n}-x_{ <Alt Only="LaTeX"><Display>f(x,y)=x_1y_2-x_2y_1+x_3y_4-x_4y_3\cdots+x_{2n-1}y_{2n}-x_ <Alt Only="HTML MathJax"><Display>f(x,y)=x_1y_2-x_2y_1+x_3y_4-x_4y_3\cdots+x_{2n-1}y_ <Alt Only="HTML noMathJax"><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></Alt> <Example> <#Include SYSTEM "../examples/include/w53.include"> </Example> </Section> <Section> <Heading>What is the form preserved by this group?</Heading> Here we start with a matrix group which is available in GAP, namely <Alt Only="Text"><M>GO(5,5)</M></Alt><Alt Only="LaTeX"><M>\mathrm{GO}(5,5)</M></Alt> <Alt Only="HTML MathJax"><M>\mathrm{GO}(5,5)</M></Alt><Alt Only="HTML noMathJax"><M>GO(5,5)</M></ We then conjugate this group by an element of <Alt Only="Text"><M>GL(5,5)</M></Alt><Alt Only="LaTeX"><M>\mathrm{GL}(5,5)</M></Alt> <Alt Only="HTML MathJax"><M>\mathrm{GL}(5,5)</M></Alt><Alt Only="HTML noMathJax"><M>GL(5,5)</M></ and then we find the forms left invariant by this copy of <Alt Only="Text"><M>GO(5,5)</M></Alt><Alt Only="LaTeX"><M>\mathrm{GO}(5,5)</M></Alt> <Alt Only="HTML MathJax"><M>\mathrm{GO}(5,5)</M></Alt><Alt Only="HTML noMathJax"><M>GO(5,5)</M></ (which we expect to be a symmetric bilinear form). <Example> <#Include SYSTEM "../examples/include/preservedform.include"> </Example> </Section> </Chapter> ¤ Dauer der Verarbeitung: 0.38 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28