Writing representations over proper sub-division rings

Abstract

Let E be a division ring and G a finite group of automorphisms of E whose elements are distinct modulo inner automorphisms of E. Given a representation \rho: B-> GL(d,E) of an F-algebra B, we give necessary and sufficient conditions for \rho to be {\it writable} over F=E^G, i.e. whether or not there exists a matrix A in GL(d,E) that conjugates \rho(B) into GL(d,F). We give an algorithm for constructing an A, or proving that no A exists. The case of particular interest to us is when E is a field, and \rho is absolutely irreducible. The algorithm relies on an explicit formula for A, and a generalization of Hilbert's Theorem 90 (Theorem~3) 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 one of the classes in Aschbacher's matrix group classification scheme.