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

[Frank Lübeck]

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                        ///
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