Representation groups of characters

Abstract

The subject we are concerned with has its roots in the work of Schur, who introduced the notion of a “representation group” (Darstellungs- gruppe). These groups have the property to lift (“linearize”) all projective representations of some (finite) group over an algebraically closed field [ 16). Fong [8 J noticed that Schur’s method applies in context with Clif- ford’s work, exhibiting some single invariant (stable) character or module. The familiar proof for the famous Fong-Swan theorem gives the model for this kind of representation groups (see Serre [17] or Feit [7]). Throughout we fix a finite group G with normal subgroup H and quotient group S= G/H. We also fix an absolutely irreducible (Frobenius) character x of H and a field F containing the values of x. We assume that x is G-invariant. As passage from the inertia group to the whole group usually is easy to handle, this is not too restrictive. It is a necessary con- dition for x to be extendible to G.