On primitive linear representations of finite groups (Q1841843)

A representation \(\pi\) of a finite group \(G\) over a field \(F\) is called primitive if it is irreducible and it is not induced from any representation of a proper subgroup of \(G\). An application of Clifford's theorem shows that a primitive representation is always quasi-primitive, that is, its restriction to each normal subgroup of \(G\) is homogeneous. Given two representations \(\pi_1\) and \(\pi_2\) of groups \(G_1\) and \(G_2\), their tensor product \(\pi_1\otimes\pi_2\) is defined, and is a representation of the direct product \(G_1\times G_2\). The two main results of the paper, in the special case where \(F\) is the field of complex numbers, show that if \(\pi_1\) and \(\pi_2\) are primitive (resp.~irreducible and quasi-primitive), then \(\pi_1\otimes\pi_2\) is primitive (resp.~irreducible and quasi-primitive). However, the results are stated and proved over arbitrary fields, where some extra assumption on the representations is needed. A representation \(\pi\) of a group \(G\) over \(F\) is said to be AI if \(\pi\) is completely reducible and all irreducible subrepresentations of all restrictions of \(\pi\) to normal subgroups of \(G\) are absolutely irreducible. The two main results cited above remain true over arbitrary fields provided one adds the hypothesis that \(\pi_1\) and \(\pi_2\) are AI; one then has the additional conclusion that \(\pi_1\otimes\pi_2\) is AI. The proof of one of the results (the one about primitive representations) requires a weak appeal to the classification of finite simple groups, but according to the author it may be possible to avoid that appeal with the right observation or some extra work.