Character distribution of simply reducible groups (Q788105)

A finite group is called simply reducible if all of its characters are real and if the tensor product of any two irreducible representations is multiplicity-free. Such groups occur in certain problems in physics. The main result is the following: If the simply reducible group \(G\) has an irreducible representation that cannot be realized over the reals then \(G\) contains a central subgroup \(W\) of order 2 such that the irreducible representations of \(G\) that can be realized over the reals are exactly those which are trivial on \(W\).