Physical Sciences and Mathematics Commons^™

Writing Representations Over Proper Division Subrings, Stephen P. Glasby

All Faculty Scholarship for the College of the Sciences

Let �� be a division ring, and G a finite group of automorphisms of E whose elements are distinct modulo inner automorphisms of ��. Let �� = ��^G be the division subring of elements of �� fixed by G. Given a representation p : �� →��^d×d of an �� -algebra ��, we give necessary and sufficient conditions for p to be writable over ��. (Here ��^d×d denotes the algebra of d×d matrices over ��, and a matrix A writes p over �� if A⁻¹p(��)A ⊆ F^d×d.) We give an algorithm for constructing an …