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On the field of definition of a complex reflection group. (Sur le corps de définition d’un groupe de réflexions complexe.) (French) Zbl 0946.20024
Let V be a finite-dimensional vector space over the complex number field , G be a complex reflection group regarded as a finite subgroup of GL(V). The inclusion GGL(V) is called the natural representation of G. The extension field of the rational number field generated by the values of the characters of the natural representation of G is called the field of definition of G. M. Benard [J. Algebra 38, 318-342 (1976; Zbl 0327.20004)] announced the following theorem: Let K be the field of definition of a complex reflection group G, then all the complex representations of G are rational over K. The proof by Benard had some errors. Another proof is given in this paper. First, the conclusion is proved for the infinite families of complex reflection groups. Then for the exceptional groups, the computer is used to do the calculations.
20G20Linear algebraic groups over , ,
20F55Reflection groups; Coxeter groups
51F15Reflection groups, reflection geometries
20C15Ordinary representations and characters of groups
20G05Representation theory of linear algebraic groups
20H15Other geometric groups, including crystallographic groups