Representations Induced in an Invariant Subgroup

Abstract

The object of this paper is to investigate the representation 21H induced by an irreducible representation 2t1 of a group G in an invariant subgroup H of G. In ?1 it is shown that 21, is either itself irreducible or is fully reducible into conjugate irreducible representations of H. In ?2 it is shown that 2IG is imprimitive unless all the irreducible components of 21H are equivalent. In fact, if T is the representation space of Kg, and hence also of 21H , and if we lump together all equivalent subspaces of T under WH , then the resulting subspaces T1 , T2 , T. , * constitute a system of imprimitivity of WsG. If we define G' to be the subgroup of G leaving one of these invariant, say T, then the component of 21G' in T, is an irreducible representation 2<;, of G', and 21G is expressible very simply in terms of WG, . These results hold for any group G and any ground-field P. In ??3-5, however, we make the assumption that P is algebraically closed. In ?3 it is found that [' , is the direct product of two irreducible projective representations of G', one of which is actually a projective representation r of the factor-group G'/H. In ?4 some progress is made on the question of whether or not a given irreducible representation of H can be embedded in some irreducible representation of G, and in ?5 we consider all possible ways of doing this. Two irreducible representations 21G and TG of G are said to be associate if 21H and EH have an irreducible component in common; associates differ only in the projective representation r of G'/H mentioned above. In the case when the factor-group G/H is a finite cyclic group of order k, associates can be described as differing from each other only by a factor which is a one-dimensional representation of G/H, and hence just a kth root of unity. In the simplest case of all, when H is of index two in G, WG has just one associate 2* (besides itself) differing from 2tI only in that we change the sign of the matrices corresponding to elements of G not in H. The situation may then be described as follows. If 21G is not equivalent to * , then WH is irreducible. If t is equivalent to * , then 96f decomposes into two inequivalent (conjugate) irreducible components. If TG is another irreducible representation of G equivalent to neither Ads nor 2*, then TH can have no irreducible component in common with 2I(= 2H)Virtually all of this theory is known in the case of a finite group G, and the greater part of it goes back to Frobenius. For the decomposition of WH into conjugates we must refer to Frobenius' original paper.' For the results of ?2-