FRAGMENTED REPRESENTATIONS OF FINITE GROUPS

Abstract

Fragments over filtered rings have been introduced in [26) as partial module structures. The themy of fi•agments shows divergent characteristics depending heavily on the type ofiiltration present on the ground ring. For example, an interchange between positive and negative parts of z.- filtrations related such fragments to aspects of

projective geometry [27]', the themy may connect to valuation theory when strong l-adic filtrations are used, ... , combinatmical algebra when finite filtrations are studied.

However for some positive filtration, for example given by subrings of a ring, that look almost trivial from the point of view of filtered ring theory, the fragments are still interesting and non-trivial.

To a chain of subgroups 1 c... c Gi c ... c G of a finite group 0 we may associate a filtration FR on the group ring R=KG over a field K, by putting FjR=KGi, i =O,...,n where KG 11= KG, Fj R =0 for j < 0 and Fj R = R for jn. The fragments with respect to this

filtration reflect links between the representation themy of G and of each G i. Since there are much more fragments than modules one may hope to obtain a finer instrument for the study of represent­ ations of groups; on the other hand it is obvious that the complicated relations between representations of Gi and represent­ ations of G, for example containing_the whole Clifford theory for Gi in G, combined with particularities related to the choice of the chain between Gi and G, will make for a rather delicate themy that should be ve1y sensitive with respect to changes in all the parameters involved.