[1m[4m[31m2. Examples[0m Here we give some simple examples that display some of the functionality of [1mForms[0m. [1m[4m[31m2.1 A conic of PG(2,8)[0m Consider the three-dimensional vector space V=GF(8)^3 over GF(8), and consider the following quadratic polynomial in 3 variables: \[ x_1^2+x_2x_3. \] Then this polynomial defines a quadratic form in V and the zeros form a [22m[36mconic[0m 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 [1mForms[0m to view this example. [22m[35m--------------------------- Example ----------------------------[0m [22m[35mgap> gf := GF(8);[0m [22m[35mGF(2^3)[0m [22m[35mgap> vec := gf^3;[0m [22m[35m( GF(2^3)^3 )[0m [22m[35mgap> r := PolynomialRing( gf, 3 );[0m [22m[35mGF(2^3)[x_1,x_2,x_3][0m [22m[35mgap> poly := r.1^2 + r.2 * r.3;[0m [22m[35mx_1^2+x_2*x_3[0m [22m[35mgap> form := QuadraticFormByPolynomial( poly, r );[0m [22m[35m< quadratic form >[0m [22m[35mgap> Display( form );[0m [22m[35mQuadratic form[0m [22m[35mGram Matrix:[0m [22m[35m 1 . .[0m [22m[35m . . 1[0m [22m[35m . . .[0m [22m[35mPolynomial: x_1^2+x_2*x_3[0m [22m[35mgap> IsDegenerateForm( form );[0m [22m[35mtrue[0m [22m[35mgap> WittIndex( form );[0m [22m[35m1[0m [22m[35mgap> IsParabolicForm( form );[0m [22m[35mtrue[0m [22m[35mgap> RadicalOfForm( form );[0m [22m[35m<vector space of dimension 1 over GF(2^3)>[0m [22m[35m------------------------------------------------------------------[0m 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 [1mForms[0m. [22m[35m--------------------------- Example ----------------------------[0m [22m[35mgap> canonical := IsometricCanonicalForm( form );[0m [22m[35m< quadratic form >[0m [22m[35mgap> form = canonical;[0m [22m[35mtrue[0m [22m[35m------------------------------------------------------------------[0m So we ``change forms''... [22m[35m--------------------------- Example ----------------------------[0m [22m[35mgap> go := GO(3,8);[0m [22m[35mGO(0,3,8)[0m [22m[35mgap> mat := InvariantQuadraticForm( go )!.matrix;[0m [22m[35m[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2) ], [0m [22m[35m[ 0*Z(2), Z(2)^0, 0*Z(2) ] ][0m [22m[35mgap> gapform := QuadraticFormByMatrix( mat, GF(8) );[0m [22m[35m< quadratic form >[0m [22m[35mgap> b := BaseChangeToCanonical( gapform );[0m [22m[35m[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2) ], [0m [22m[35m[ 0*Z(2), 0*Z(2), Z(2)^0 ] ][0m [22m[35mgap> hom := BaseChangeHomomorphism( b, GF(8) );[0m [22m[35m^[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2) ], [0m [22m[35m[ 0*Z(2), 0*Z(2), Z(2)^0 ] ][0m [22m[35mgap> newgo := Image(hom, go);[0m [22m[35mGroup([ [ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2^3), 0*Z(2) ], [0m [22m[35m[ 0*Z(2), 0*Z(2), Z(2^3)^6 ] ], [ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [0m [22m[35m[ Z(2)^0, Z(2)^0, Z(2)^0 ], [ 0*Z(2), Z(2)^0, 0*Z(2) ] ] ])[0m [22m[35m------------------------------------------------------------------[0m Now we look at the action of our new GO(3,8) on the conic. [22m[35m--------------------------- Example ----------------------------[0m [22m[35mgap> conic := Filtered(vec, x -> IsZero( x^form ));;[0m [22m[35mgap> Size( conic );[0m [22m[35m64[0m [22m[35mgap> orbs := Orbits(newgo, conic, OnRight);;[0m [22m[35mgap> List(orbs, Size);[0m [22m[35m[ 1, 63 ][0m [22m[35m------------------------------------------------------------------[0m So we see that there is a fixed point, which is actually the [22m[36mnucleus[0m of the conic, or in other words, the radical of the form. [1m[4m[31m2.2 A form for W(5,3)[0m 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 [1mForms[0m for an alternating form is \[ f(x,y)=x_1y_2-x_2y_1+x_3y_4-x_4y_3\cdots+x_{2n-1}y_{2n}-x_{2n}y_{2n-1}. \] [22m[35m--------------------------- Example ----------------------------[0m [22m[35mgap> f := GF(3);[0m [22m[35mGF(3)[0m [22m[35mgap> gram := [[0m [22m[35m[0,0,0,1,0,0], [0m [22m[35m[0,0,0,0,1,0],[0m [22m[35m[0,0,0,0,0,1],[0m [22m[35m[-1,0,0,0,0,0],[0m [22m[35m[0,-1,0,0,0,0],[0m [22m[35m[0,0,-1,0,0,0]] * One(f);;[0m [22m[35mgap> form := BilinearFormByMatrix( gram, f );[0m [22m[35m< bilinear form >[0m [22m[35mgap> IsSymplecticForm( form );[0m [22m[35mtrue[0m [22m[35mgap> Display( form );[0m [22m[35mBilinear form[0m [22m[35mGram Matrix:[0m [22m[35m . . . 1 . .[0m [22m[35m . . . . 1 .[0m [22m[35m . . . . . 1[0m [22m[35m 2 . . . . .[0m [22m[35m . 2 . . . .[0m [22m[35m . . 2 . . .[0m [22m[35mgap> b := BaseChangeToCanonical( form );;[0m [22m[35mgap> Display( b );[0m [22m[35m . . . . . 1[0m [22m[35m . . 2 . . .[0m [22m[35m . . . . 1 .[0m [22m[35m . 2 . . . .[0m [22m[35m . . . 1 . .[0m [22m[35m 2 . . . . .[0m [22m[35mgap> Display( b * gram * TransposedMat(b) );[0m [22m[35m . 1 . . . .[0m [22m[35m 2 . . . . .[0m [22m[35m . . . 1 . .[0m [22m[35m . . 2 . . .[0m [22m[35m . . . . . 1[0m [22m[35m . . . . 2 .[0m [22m[35m------------------------------------------------------------------[0m