Matrix generators for the Ree groups

For the purposes of [K] and [KM] it became necessary to have 7× 7 matrix generators for a Sylow-3-subgroup of the Ree groups G2(q) and its normalizer. For example in [K] we wanted to show that in a seven dimensional representation the Jordan canonical form of any element of order nine is a single Jordan block of size 7. In [KM] we develop group recognition algorithms. At some stage we need to identify a certain subset of group elements with the Sylow-3-subgroup of G2(q). This is most easily done using a faithful matrix representation of small dimension. In this note we provide matrix generators for two distinct Sylow-3subgroups of G2(q), thereby providing a matrix generating set for the whole group. Starting with the Steinberg generators for a seven dimensional representation of G2(q) we construct our matrices following Carter [C], chapters 12 and 13. The matrices for the Steinberg generators of G2(q) were computed with the help of a computer program developed by the second author. For our setup we let G = G2(K), where K is the algebraic closure of a finite field of characteristic 3. Let F a Frobenius endomorphism of G whose set of fixed points G is a Ree group of type G2(q), q = 3 . Let T be an F -invariant maximal torus of G, and let B and B be F -invariant Borel subgroups intersecting in T with unipotent radicals U respectively U. By N we denote the normalizer NG(T ). Let Φ be the root system of G with respect to T and {a, b} its base given by B, where a is a short and b a long root. Now U respectively U is generated by subgroups Xr respectively X−r, where r ∈ Φ + (the set of positive roots). The groups Xr are isomorphic to the additive group of the field K. We denote the elements of Xr by Xr(t) where t ∈ K. The reductive group G has an irreducible 7-dimensional representation over K (with highest weight (1, 0)) which can be found as follows: In characteristic 3, the 14-dimensional adjoint representation V of G has a 7-dimensional irreducible submodule. This submodule is spanned by those elements of the Chevalley basis of V which are labeled by short roots. The restriction of this representation to G2(q) remains irreducible. The second author has implemented a computer program which computes for an arbitrary Chevalley group explicit matrices for the root elements Xr(t) in its adjoint representation with respect to a Chevalley basis. Using this for type G2, we obtain the images of the Xr(t) in the 7-dimensional representation with high weight (1, 0) by cutting out the appropriate 7 × 7-blocks of the Xr(t). (By abuse of notation we also denote these images by Xr(t) in the sequel.)