**ALGEBRAIC GEOMETRY
SEMINAR**

#

**“Genetic
Representation Theory of p-Groups”**

by

**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