On tensor induction for representations of finite groups

Abstract

It is well known that if D is an irreducible complex representation of a finite group G, then every direct summand of the restriction of D to a subgroup H must have degree at least as large as the degree of D divided by the index | G : H | ; moreover, D is induced from H if and only if the restriction does have a direct summand whose dimension is equal to this quotient. This paper explores the possibility of an analogous result for tensor induction, under the additional assumption that D is faithful, quasi-primitive and not a tensor product (of projective representations of degree greater than 1), and that the Fitting subgroup F ( G ) is not in the centre Z ( G ) . The main question is this: if the restriction has a (projective) tensor factor whose degree is the | G : H | th root of the degree of D, does it follow that D is tensor induced from H? Among other results, examples are given to show that the answer can be negative when the index is 2. An affirmative answer is proved for normal subgroups of odd index, and also for arbitrary subgroups of odd prime index. As might be expected, the key lies in the study of F ( G ) / Z ( G ) as a symplectic module over a finite prime field; in particular, in exploring the connection between (ordinary) induction and form-induction of such modules.