Bull. Korean Math. Soc. 2017; 54(5): 1757-1771
Online first article July 17, 2017 Printed September 30, 2017
https://doi.org/10.4134/BKMS.b160742
Copyright © The Korean Mathematical Society.
Marat Rovinsky
National Research University Higher School of Economics
Let $K$ be a field and $G$ be a group of its automorphisms. It follows from Speiser's generalization of Hilbert's Theorem 90, \cite{Speiser} that any $K$-{\it semilinear} representation of the group $G$ is isomorphic to a direct sum of copies of $K$, if $G$ is finite. In this note three examples of pairs $(K,G)$ are presented such that certain irreducible $K$-semilinear representations of $G$ admit a simple description: (i) with precompact $G$, (ii) $K$ is a field of rational functions and $G$ permutes the variables, (iii) $K$ is a universal domain over field of characteristic zero and $G$ its automorphism group. The example (iii) is new and it generalizes the principal result of \cite{adm}.
Keywords: non-compact groups
MSC numbers: 14C15, 14F20, 14F43, 20B27, 20C32
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