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 \(\mathbb{C}\), \(G\) 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 \(\mathbb{Q}\) 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.


20G20 Linear algebraic groups over the reals, the complexes, the quaternions
20F55 Reflection and Coxeter groups (group-theoretic aspects)
51F15 Reflection groups, reflection geometries
20C15 Ordinary representations and characters
20G05 Representation theory for linear algebraic groups
20H15 Other geometric groups, including crystallographic groups


Zbl 0327.20004


Full Text: DOI


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