My area of study within mathematics is the representation theory of finite groups. The study of groups is really the abstract study of symmetry, and representations of groups are realizations of them as groups of matrices. This is an important discipline because of its strong connections with other branches of mathematics, as well as with fields outside mathematics. It has been a very active area of research in the past 15 years due to its far-reaching implications. I work in particular with the symmetric groups. These groups appear often, even in non-mathematical fields such as chemistry and quantum physics, not to mention such important mathematical areas as combinatorics and topology.
The main goal of my research is to provide a strengthening of a conjecture by Professor J. L. Alperin (University of Chicago) for symmetric groups. Roughly speaking, Alperin's Conjecture [A] asserts the existence of a correspondence between the irreducible representations of a group (which in some sense are the building blocks of all representations) and a certain set of objects which Alperin calls ``weights''. This conjecture has been one of the main focuses of attention of representation theory in recent years, and has been shown to be valid in many cases (including the symmetric groups) but remains unproven in general. In the case of symmetric groups, no explicit construction of a correspondence between irreducible representations and weights has ever been given, although such explicit constructions are known for some other important classes of groups. I attempt to find a procedure to get an explicit correspondence in a general setting, which will in particular encompass Alperin's Conjecture for the symmetric groups.
If successful, I shall provide much more than just an alternative proof of Alperin's Conjecture. I do not seek simply to embellish the present literature by adding a new proof of a known result for the sake of variety. This approach will provide a new perspective, with the potential to reveal structural connections that had gone unnoticed before, thereby unifying the existing theory and expanding it in new directions. A result like the one I aim to prove can offer a fresh vantage-point that will consolidate what is known while serving as a firm basis for further, deeper development of the discipline. A canonical correspondence (rather than just numerical equality) will provide a clearer picture of the structure of these representations.
My work can be divided in two parts: theoretical and practical. On the practical side, I have created a library of irreducible representations for symmetric groups using the computer algebra package SYMMETRICA [KK], and I have been writing various routines within the computer algebra package GAP ( ``Groups, Algorithms and Programming'', [S]) which have enabled me to produce substantial computational data to deal with the irreducible representations of the symmetric groups. This software is continually being improved and updated, the end result being software that is robust and free of bugs, and which includes several functions of considerable interest to people who work on representation theory (for example, functions that compute relative traces and Brauer quotients, or that test whether a module is projective or not).
On the theoretical side I have proved some of the conjectures that I drew from my experimental data, and I shall continue to work on other conjectures. The area that I have been studying has many connections to other fields, and as a consequence I am able to consider problems from various fields such as combinatorics, modular representation theory and finite groups. For instance, the tables of dimensions of Brauer quotients that I obtained experimentally can always be arranged in a triangular shape, and this in turn suggested a possible way to arrange 2-regular partitions in an infinite matrix indexed by 2-cores and partitions with empty core (a purely combinatorial conjecture that originated from facts in modular representation theory).
It is worth noticing that many of the subroutines that I have written, as well as the results I have proved, are important in their own right, irrespective of the final outcome (i.e., even if Alperin's conjecture cannot be proved from them). This is always a desirable feature, because in mathematics - unlike most experimental sciences - the usefulness of a tool or the impact of an experiment need not transcend that of the driving force behind their creation.
[A] J. L. Alperin, Weights for Finite Groups, ``The Arcata Conference on Representations of Finite Groups'', Proc. Sympos. Pure Math. 47 (1987), 369-379.
[KK] A. Kerber and A. Kohnert, SYMMETRICA 1.0, Lehrstuhl II für Mathematik, Department of Mathematics, University of Bayreuth, 1993.
[S] Martin Schönert et al, GAP - Groups, Algorithms and Programming, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hocschule, Aachen, Germany, third edition, 1993.