“Biset functors and the Tornehave morphism”
ABSTRACT: The usual way of constructing a permutation module from a permutation set is linearization: one regards the permutation set as a basis for the permutation module. One can bring in some ring-theoretic structure by introducing virtual permutation modules and virtual permutation sets. One can bring in a lot more structure using the notion of a biset functor, which provides a unified way of dealing with reduction to subgroups and reduction to quotient groups. Tornehave found a way of constructing modules from precisely those virtual permutation sets that are killed by linearization. We shall introduce a refinement of his construction, expressing it as a morphism of biset functors, and we shall discuss a uniqueness property and an application of this morphism.Arithmetical investigations led me to ask: For which finite groups G is the group ring of G over a field K isomorphic to a direct product of division rings? There is a complete group-theoretical answer to this question, having been worked out in collaboration with B. Huppert. But ultimately it leads to a delicate number-theoretical problem of elementary character.
Date : September 27, 2007 (Thursday)
Time : 14.40 The second hour, , will be open to all, but more specialized in content.
Place: Mathematics Seminar Room SA141