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Let 𝔼 be a division ring, and G a finite group of automorphisms of E whose elements are distinct modulo inner automorphisms of 𝔼. Let 𝔽 = 𝔼G be the division subring of elements of 𝔼 fixed by G. Given a representation p : 𝔄 →𝔼d×d of an 𝔽 -algebra 𝔄, we give necessary and sufficient conditions for p to be writable over 𝔽. (Here 𝔼d×d denotes the algebra of d×d matrices over 𝔼, and a matrix A writes p over 𝔽 if A−1p(𝔄)A ⊆ Fd×d.) 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 p is absolutely irreducible. The algorithm relies on an explicit formula for A, and a generalization of Hilbert’s 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 C5 in Aschbacher’s matrix group classification scheme [1, 13].
Glasby, Stephen P., "Writing representations over proper division subrings" (2008). All Faculty Scholarship for the College of the Sciences. 249.
Journal of Algebra
Version of Record: © 2007 Elsevier Inc. All rights reserved.