The following is a description of the project that will be directed by Dr. Thomas Keller. A student need not understand the details of the description in order to work on this project. It is expected that some background material will need to be covered. A student who wishes to work on this project should have completed at least one semester of Modern Algebra and of Linear Algebra. Facility with computers might also prove beneficial.

Representations and Characters of Groups
The goal of the planned project, which will be supervised by Thomas Keller, is to obtain new results on an old problem in the representation theory of finite solvable groups, namely the question of how the derived length is bounded in terms of the number of irreducible complex character degrees. As has become clear by Keller's recent work, the core of the problem is the case of p-groups.

Let G be a finite p-group and let dl(G) be its derived length, i.e., the length of the shortest possible normal series of G such that all factors of consecutive members of the series are abelian. dl(G) is a measure for the "height" of the group; the larger it is, the more complex is the group. Therefore it is an important problem to study by which parameters the derived length can be controlled. One of the parameters stems from the representation theory of finite groups and is the number of the irreducible character degrees of G. These character degrees are the dimensions of the irreducible representations of G over the complex numbers. Denote by cd(G) the set of all the irreducible character degrees of G. By an old result of Taketa it is known that dl(G) is bounded above by the cardinality of cd(G), which is the best general bound to date. However, one conjectures that the true bound in fact is logarithmic in the cardinality of cd(G).

While this has been proven for some families of groups (e.g. Sylow subgroups of some classical groups), there is not a more general class of groups for which such a bound is known, and a general proof of the conjecture seems to be far out of the reach of today's techniques.

The proposed project aims at investigating a special class C of groups for which there is some hope of establishing the desired bound. Due to the enormous complexity of p-groups in general, such a class C has to be chosen very carefully to be tractable at all. For this project Keller suggests looking at the class C of normally monomial p-groups of maximal class. "Normally monomial" is a representation theoretic restriction while "maximal class" is a group theoretic condition.

While the difficulty of the problem even for groups in C is still considerable, by the choice of C it becomes accessible to undergraduate students mainly for the following two reasons:

  1. For groups in C, the representation theoretic quantity |cd(G)| can be described entirely in group theoretic terms; so the students will not have to deal with representation theory at all.
  2. The groups in C essentially can be treated by looking at certain Lie algebras over prime fields (via the Lazard--correspondence). So no deep group theory has to be known; only basic notions of Lie algebras need to be learned before the research can begin.
Therefore to participate successfully in the project, no more background than an undergraduate group theory course and a linear algebra course will be needed. Within the first few days of the project, Keller plans to give the students a basic introduction to the problem. After that they will already be in a position to do independent research. One of their endeavors will be the search for new and illuminating examples of Lie algebras relevant to the problem. To construct nontrivial examples can already be quite challenging, but in a former student project that Keller conducted a student wrote a computer program for constructing such examples. This program will be available and will provide a useful tool in the search for examples. The students, however, may also develop additional programs to construct Lie algebras -- in any case, computers will prove very helpful in this part of the project. New and interesting examples would definitely constitute publishable work.

Moreover such examples should give clues on how to deal with the general question of what the right bound for dl(G) in terms of |cd(G)| for groups in C is and how to prove it. Thus the analysis of the examples with respect to patterns and structural behavior would be the third part of the project. The hope is to obtain best possible bounds for small values of dl(G) and to figure out and prove a general better than linear asymptotic bound (possibly logarithmic).