Relative higher relation modules, and a property of irreducible K- representations of \(GL_ n(K)\)
The object of the paper is the persistence of free submodules for finite subgroups of \(GL_ n(K)\) in the following sense. Let K be an arbitrary field, and V be a finite-dimensional irreducible (left) module for \(GL_ n(K)\) over K. If K is infinite, assume that V affords a polynomial representation of \(GL_ n(K)\). Let E be the natural left \(GL_ n(K)\)- module and G be a nontrivial finite subgroup of \(GL_ n(K)\). Suppose that \(E\cong KG\oplus E_ 1\), as KG-modules, for some KG-module \(E_ 1\). Then the main result says that one of the following holds. (a) V contains a nonzero free KG-submodule; (b) \(\dim_ K(V)=1\); (c) \(E_ 1=0\), G is an elementary abelian 2-group of order \(>2\), the structure of V is explicitly given and V contains no nontrivial free KG-submodule. Using standad facts the proof of the main result is reduced to the case when K is infinite. The resulting special case corrects theorem 3.3 of a joint paper of the author and \textit{A. I. Lichtman} [J. Pure Appl. Algebra 22, 75-89 (1981; Zbl 0467.20009)]. Finally using results of Borel and Tits a similar theorem to the main result is proved when K is infinite but V does not necessarily afford a polynomial representation of \(GL_ n(K)\). The connection to the study of relative higher relation modules is given [cf. loc. cit.].
scientific article; zbMATH DE number 4027732