Identification of matrix generators of a Chevalley group.
The problem of computing effectively (and efficiently) with matrix groups over finite fields is one which has received a great deal of attention over the last two decades, and various approaches have been suggested and adopted. Most of these involve at some stage working with simple or quasisimple groups in some, usually irreducible but otherwise arbitrary, representation. In order to identify the group with certainty, it is necessary to set up an isomorphism with some standard copy of the group in question. The paper under review proposes a new approach to this latter problem, in the case where the quasisimple group is an untwisted group of Lie type over a field of odd order. The `standard copy' in this case is the adjoint representation written with respect to a Chevalley basis. The `arbitrary' representation is assumed to be in the defining characteristic, as in non-defining characteristic the group must be `small' and permutation group methods are more appropriate. The method relies on the fact that the tensor product of any such representation with its dual contains a copy of the adjoint representation. Hence the adjoint can be constructed by well-known (so-called `Meataxe') methods. The next step is to construct the Lie bracket -- here lies the real novelty of the author's approach -- and thence compute a Chevalley basis. Thus the standard representation of the group is obtained, on the original (user-defined) generators. In order to complete the construction of the group isomorphism it is then necessary to obtain `standard generators' as well, but that is a separate problem addressed elsewhere.
scientific article; zbMATH DE number 5141726