No suitable invariant bilinear/quadratic form pairs exist for Omega(0,d,q) with d odd, q even

[fingolfin]

I recently decided to try and add the InvariantBilinearForm and InvariantQuadraticForm attributes for Omega(e, d, q).

It turns out that for e=0 and q even, this is not possible?! Consider for example Omega(0,3,4); -- an exhaustive search over GF(4)^[3,3] reveals that the only non-trivial bilinear form preserved by this group in GAP is (up to scalars) given by the Gram matrix:

 . . .
 . 1 .
 . . .

Actually, this can be verified for Omega(0,3,q) for any even q>=4 with a simple hand calculation, by inspecting the two generators of the group.

But this matrix cannot be written as Q + Q^tr for some matrix Q over GF(4), as such a matrix always has only zeros on the diagonal. But this is required by the GAP manual entry for InvariantQuadraticForm.

I could fix this by using a different representation of the group (e.g. the one obtained via DerivedSubgroup(GO(e,d,q))), but that might break compatibility in unforeseen ways? See also issue #500.

Perhaps @ThomasBreuer @frankluebeck @hulpke have some insights on this?

[fingolfin]

To clarify, this concerns the function OmegaZero which was added by @ThomasBreuer on 2011-06-14, so I am hoping he has some additional insights?

[ThomasBreuer]

As far as I see, the generators given in the Rylands/Taylor paper are wrong,
since they do not describe the claimed orthogonal group.
Thus my proposal is as follows.

• Delegate from OmegaZero to SO in the case that the dimension is 3
  and the characteristic is 2.
  This is in fact the definition of $\Omega(n,q)$ for odd $n$ and even $q$
  in the ATLAS of Finite Groups.

  This delegation happens already in the case of OmegaZero( 0, 5, 2 ),
  where the generators in Rylands/Taylor are also wrong.
  In that case, the error had been obvious because the group described by
  the generators from the paper has the wrong order.
  In the current case, the generators from the paper describe a group of the right isomorphism type
  but in a wrong class of subgroups inside the general linear group in question.

• Extend the documentation in the introduction of the section ''Classical Groups''
  (the source code for that is in grp/classic.gd) by a remark about the changed generators
  that are chosen in GAP, including the statement that the generators claimed in Rylands/Taylor
  do not work in these cases.

I can provide a pull request for these changes if you agree with them.

[fingolfin]

@ThomasBreuer Thanks for your analysis. I fully agree with it and the proposed plan, and would appreciate if you can indeed provide that pull request.

If we do this anyway, I wonder if we should also reconsider issue #500, and change Omega to always return subgroups of SO -- this is just a matter of conjugating by the right element. But of course this risks breaking code which relies on the specific output of Omega... so I guess the "right" solution would be to support an extended interface which allows specifying the desired form, or something like that... In any case, this surely goes beyond fixing the issue at hand, I merely mention it for completeness.