problem with orthogonal groups in dimension 2?

[ThomasBreuer]

When I wrote tests for the extension of GAP functionality based on the Forms package (see pull request #4), I found the following strange behaviour with some orthogonal groups in dimension 2.

I start with the group GO(-1,2,3) in GAP.

gap> d:= 2;;  q:= 3;;  g:= GO(-1,d,q);;
gap> mat1:= InvariantQuadraticForm( g ).matrix;;
gap> form1:= QuadraticFormByMatrix( mat1, GF(q) );;
gap> Display( form1 );
Quadratic form
Gram Matrix:
 1 1
 . 2
gap> bas1:= BaseChangeToCanonical( form1 );;
gap> can1:= bas1 * mat1 * TransposedMat( bas1 );
[ [ Z(3)^0, Z(3) ], [ Z(3)^0, Z(3)^0 ] ]
gap> canform1:= QuadraticFormByMatrix( can1, GF(q) );;
gap> Display( canform1 );
Quadratic form
Gram Matrix:
 1 .
 . 1

Now I conjugate the group with a permutation matrix.
A Gram matrix of the invariant quadratic form of the image is given by conjugating the Gram matrix of the form for the original group.
However, I get a different canonical form for it.

gap> pi:= PermutationMat( (1,2), d, GF(q) );;
gap> mat2:= pi * mat1 * TransposedMat( pi );;
gap> g2:= ConjugateGroup( g, pi );;
gap> form2:= QuadraticFormByMatrix( mat2, GF(q) );;
gap> bas2:= BaseChangeToCanonical( form2 );;
gap> can2:= bas2 * mat2 * TransposedMat( bas2 );
[ [ Z(3), Z(3)^0 ], [ Z(3), Z(3) ] ]
gap> canform2:= QuadraticFormByMatrix( can2, GF(q) );;
gap> Display( canform2 );
Quadratic form
Gram Matrix:
 2 .
 . 2
gap> canform1 = canform2;
false

(With the base change bas1 * pi^-1 instead of bas2, we would get can1 from mat2, and everything would work as expected.)
What am I missing?

P.S.:
PreservedQuadraticForms( GO(-1,2,q) ) seems to find nothing if q is odd. As far as I see, this is not related to the problem sketched above.

[fingolfin]

This seems like a bug to me. It seems to contradict what https://www.gap-system.org/Manuals/pkg/forms/doc/chap5.html says is the canonical form of a elliptic bilinear form.

[fingolfin]

This doesn't seem to be limited to dimension 2. The manual contains this example:

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

But when I run this in GAP 4.11 or master, I get

 1 3 1 2 2
 3 3 3 2 2
 1 3 2 4 1
 2 2 4 3 1
 2 2 1 1 2

and in GAP 4.10, I get

 2 1 2 4 4
 1 1 1 4 4
 2 1 4 3 2
 4 4 3 1 2
 4 4 2 2 4

All of these just differ by a scalar.

[ThomasBreuer]

When I ask for the forms that are preserved by a given group then the natural result is a vector space of forms, and one has to be aware of scalar multiples. However, if a function promises to compute a (base change to a) canonical form from a given form then the result should be unique.

[fingolfin]

@ThomasBreuer is right, and I was wrong: of course nothing anywhere claims that PreservedSesquilinearForms returns specific forms, just that it returns "all" forms "up to scalar". Which now leaves me wondering: what if the space of preserved forms has more than one dimension? Is then a list of forms returned which are a basis of the space of all forms? Perhaps this could be clarified?

(And in the case of a 1-dimensional form space, one could make them "canonical" by e.g. forcing the first entry of the first non-zero row to be 1 -- whether that's necessary or useful beyond making it a tad easier to make the rests reproducible is another question, though ;-) )