Here we give some simple examples that display some of the functionality of Forms.
Consider the three-dimensional vector space V over the finite field GF(8), and consider the following quadratic polynomial in 3 variables:
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>
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 GO(3,8). However, our conic is the canonical conic given in Forms.
gap> canonical := IsometricCanonicalForm( form ); < parabolic quadratic form > gap> form = canonical; true
So we ``change forms''...
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) ] ] ])
Now we look at the action of our new GO(3,8) on the conic.
gap> conic := Filtered(vec, x -> IsZero( x^form ));; gap> Size(conic); 64 gap> orbs := Orbits(newgo, conic, OnRight);; gap> List(orbs,Size); [ 1, 63 ]
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.
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 6 × 6 matrix A which satisfies A+AT=0 is a candidate for the Gram matrix of a symplectic polarity. The canonical form we adopt in Forms for an alternating form is
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 ); [ [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ] ] gap> Display( b ); 1 . . . . . . . . 1 . . . 1 . . . . . . . . 1 . . . 1 . . . . . . . . 1 gap> Display( b * gram * TransposedMat(b) ); . 1 . . . . 2 . . . . . . . . 1 . . . . 2 . . . . . . . . 1 . . . . 2 .
Here we start with a matrix group which is available in GAP, namely GO(5,5). We then conjugate this group by an element of GL(5,5), and then we find the forms left invariant by this copy of GO(5,5) (which we expect to be a symmetric bilinear form).
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: 4 2 4 3 3 2 2 2 3 3 4 2 3 1 4 3 3 1 2 4 3 3 4 4 3
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