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Zbl 1161.16034
Glasby, S.P.
Writing representations over proper division subrings. (English)
[J] J. Algebra 319, No. 1, 77-92 (2008). ISSN 0021-8693

Summary: Let $\bbfE$ be a division ring, and $G$ a finite group of automorphisms of $\bbfE$ whose elements are distinct modulo inner automorphisms of $\bbfE$. Let $\bbfF=\bbfE^G$ be the division subring of elements of $\bbfE$ fixed by $G$. Given a representation $\rho\colon\germ A\to\bbfE^{d\times d}$ of an $\bbfF$-algebra $\germ A$, we give necessary and sufficient conditions for $\rho$ to be writable over $\bbfF$. (Here $\bbfE^{d\times d}$ denotes the algebra of $d\times d$ matrices over $\bbfE$, and a matrix $A$ writes $\rho$ over $\bbfF$ if $A^{-1}\rho(\germ A)A\subseteq\bbfF^{d\times d}$.) We give an algorithm for constructing an $A$, or proving that no $A$ exists. The case of particular interest to us is when $\bbfE$ 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 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 $\cal C_5$ in Aschbacher's matrix group classification scheme [{\it M. Aschbacher}, Invent. Math. 76, 469-514 (1984; Zbl 0537.20023); {\it S.-Z. Li}, Math. Appl., Dordr. 365, 70-90 (1996; Zbl 0879.20026)].

MSC 2000:: *16Z05 Computational aspects of associative rings
16S50 Endomorphism rings: matrix rings
16K40 Infinite dimensional and general division rings
16W22 Actions of groups and semigroups
20C40 Computational methods (representations of groups)

Keywords: division rings; finite groups of automorphisms; algebras of matrices; algorithms; absolutely irreducible group representations; Hilbert Theorem 90; division subrings

Citations: Zbl 0537.20023; Zbl 0879.20026

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