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Writing representations over minimal fields. (English) Zbl 0888.20013
The paper describes a procedure which determines whether an absolutely irreducible representation $$\rho\colon G\to\text{GL}(d,E)$$ of a finite group $$G$$ over a finite field $$E$$ is equivalent to a representation $$G\to\text{GL}(d,F)$$, where $$F$$ is a subfield of $$E$$. If so, the algorithm produces an $$A\in\text{GL}(d,E)$$ such that $$A^{-1}\rho(g)A\in\text{GL}(d,F)$$ for each $$g\in G$$. The algorithm relies on a matrix version of Hilbert’s Theorem 90 and is probabilistic with expected running time $$O([E:F]d^3)$$.

MSC:
 20C40 Computational methods (representations of groups) (MSC2010)
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References:
 [1] DOI: 10.1017/S1446788700036016 · doi:10.1017/S1446788700036016 [2] Huppert B., Finite Groups II (1982) · Zbl 0477.20001
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