Abstract :
Let be a division ring, and G a finite group of automorphisms of whose elements are distinct modulo inner automorphisms of . Let be the division subring of elements of fixed by G. Given a representation of an -algebra , we give necessary and sufficient conditions for ρ to be writable over . (Here denotes the algebra of d×d matrices over , and a matrix A writes ρ over if .) We give an algorithm for constructing an A, or proving that no A exists. The case of particular interest to us is when is a field, and ρ is absolutely irreducible. The algorithm relies on an explicit formula for A, and a generalization of Hilbertʹs Theorem 90 that arises in galois cohomology. The algorithm has applications to the construction of absolutely irreducible group representations (especially for solvable groups), and to the recognition of class in Aschbacherʹs matrix group classification scheme [M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984) 469–514, MR0746539; Shangzhi Li, On the subgroup structure of classical groups, in: Group Theory in China, in: Math. Appl. (China Ser.), vol. 365, Kluwer Acad. Publ., Dordrecht, 1996, pp. 70–90, MR1447199. [1] and [13]].
