“Genetic Representation Theory of p-Groups”





Laurence Barker



Abstract: We are concerned with the irreducible representations of a p-group G over a field F of characteristic zero. It has long been known that, for any such representation, there is a normal subgroup K of a subgroup H of G such that the representation is induced from the section H/K and, furthermore, H/K is a basic p-group. In 2004, Bouc showed that, if F is the field of rational numbers then, by imposing a certain constraint, we can ensure that the so-called genetic section H/K is unique up to isomorphism. In this talk, we shall show that, by generalizing the constraint in a suitable way, the result can be extended to the case where F is any field of characteristic zero. The point of this genetic theory is that, firstly, the constraint ensures that the induction preserves many properties of irreducible characters; secondly the basic p-groups have been classified: they are the cyclic, dihedral, semidihedral and quaternion groups. Thus, by making observations about those groups, one is proving general results about p-groups. We shall apply the technique (but possibly next week) to obtain some new results on absolutely irreducible real representations and on minimal splitting fields.



Date:  16 September, 2005 Friday
Time: 15.40-17.30
Place: Bilkent, Mathematics Seminar Room SA-141