[GAP Forum] Linear representation of G2(q), q even

[Juergen Mueller]

Dear GAP-Forum, dear Gabor,

here, in addition to Frank's comment, is an idea how to show that G_2(q)
acts transitively and primitively on PG(5,q) whenever q is even:

It is known that in any characteristic the Lie type graphs associated 
to the maximal parabolic subgroups of G_2(q) are isomorphic and primitive
distance-transitive; see e.g. the book by Brouwer-Cohen-Neumaier. 
In particular, the maximal parabolic subgroups of G_2(q) are maximal 
subgroups of index (q^6-1)/(q-1).

Hence it remains to show that, in characteristic 2, one of the maximal
parabolic subgroups fixes a one-dimensional subspace of the fundamental
representation of degree 6; this should be fairly easy using the given
matrices, as the generators are probably sufficiently generic to derive
generators of the maximal parabolic subgroups therefrom.

Best wishes, Jürgen Müller

On Mon, Dec 22, 2008 at 01:53:42PM +0100, Frank Lübeck wrote:
> On Sat, Dec 20, 2008 at 04:01:37PM +0100, Nagy Gábor wrote:
> > It is "well known" that for even prime power q, the exceptional Lie group 
> > G2(q) has two linear representations of dimension 6, both are transitive on 
> > the set of non-zero vectors.
> >
> > Can you suggest me some literature about this construction?
> >
> > I am escpecially interested in the question if the induced transitive 
> > representations on the 5-dimensional projective spaces PG(5,q) are 
> > primitive?
> 
> Dear Gabor, dear Forum,
> 
> I don't have a reference, but maybe the following is of some help.
> 
> G2(q), q=2^f, has f irreducible representations in characteristic 2,
> one of the fundamental representations and its Frobenius twists 
> (i.e., combinations with the field automorphism of GF(q)/GF(2)).
> 
> I print below "generic" generators of G2(2^f), substitute the variable T 
> by Z(2^f) to get generators for some specific f.
> 
> With this you can check for q = 2, 4, 8 in GAP that the action on non-zero 
> vectors is transitive and the action on PG(5, q) is primitive. I'm not sure 
> if one could somehow use these generators also to prove these properties in 
> general. 
> 
> Best regards,
> 
>     Frank
> 
> And here are the matrices, these are generators of root subgroups:
> 
> G2char2gens := 
> [ [ [ 1, T, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0 ], [ 0, 0, 1, T^2, 0, 0 ], 
>       [ 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1, T ], [ 0, 0, 0, 0, 0, 1 ] ], 
>   [ [ 1, 0, 0, 0, 0, 0 ], [ 0, 1, -T, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0 ], 
>       [ 0, 0, 0, 1, T, 0 ], [ 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 1 ] ], 
>   [ [ 1, 0, 0, 0, 0, 0 ], [ T, 1, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0 ], 
>       [ 0, 0, T^2, 1, 0, 0 ], [ 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, T, 1 ] ], 
>   [ [ 1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0 ], [ 0, -T, 1, 0, 0, 0 ], 
>       [ 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, T, 1, 0 ], [ 0, 0, 0, 0, 0, 1 ] ] ];
> 
> -- 
> ///  Dr. Frank Lübeck, Lehrstuhl D für Mathematik, Templergraben 64,  ///
> \\\                    52062 Aachen, Germany                          \\\
> ///  E-mail: Frank.Luebeck at Math.RWTH-Aachen.De                        ///
> \\\  WWW:    http://www.math.rwth-aachen.de/~Frank.Luebeck/           \\\
> 
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