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Zbl 0946.20024
Bessis, David
On the field of definition of a complex reflection group. (Sur le corps de définition d'un groupe de réflexions complexe.) (French)
[J] Commun. Algebra 25, No.8, 2703-2716 (1997). ISSN 0092-7872; ISSN 1532-4125/e

Let $V$ be a finite-dimensional vector space over the complex number field $\bbfC$, $G$ be a complex reflection group regarded as a finite subgroup of $\text{GL}(V)$. The inclusion $G\hookrightarrow\text{GL}(V)$ is called the natural representation of $G$. The extension field of the rational number field $\bbfQ$ generated by the values of the characters of the natural representation of $G$ is called the field of definition of $G$. {\it 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.
[Chen Zhijie (Shanghai)]

MSC 2000:: *20G20 Linear algebraic groups over the reals
20F55 Coxeter groups
51F15 Reflection groups and geometries
20C15 Ordinary representations and characters of groups
20G05 Representation theory of linear algebraic groups
20H15 Other geometric groups, including crystallographic groups

Keywords: complex reflection groups; fields of definition; natural representations; exceptional groups

Citations: Zbl 0327.20004

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