Irreducible modular representations

The idea of a group is ubiquitous in mathematics, and has a variety of applications in the sciences. As a starting point, we take a physical object, and consider all its symmetries: transformations (such as rotations and reflections) that leave that object fixed. The symmetry group of the object is the list of all these symmetries together with the operation of combining those symmetries by first applying one then the other. This combining operation satisfies certain simple rules - for example, for every symmetry there is an inverse symmetry which gets you back where you started. This leads to the idea of an abstract group, which is a set of symbols and a way of combining these symbols which satisfies the same rules. Group theory asks what different abstract groups exist and what general properties they have. The study of groups is a major branch of mathematics, and its most impressive achievement is the classification of finite groups which are simple, which means they can't be broken down into smaller groups.

Representation theory reverses the above process: given an abstract group, it asks how we can realise that group as the symmetry group of a physical object. This question is asked in an algebraic way through matrix representations: each symmetry of n-dimensional space is encoded as an n by n matrix of numbers, and combining symmetries corresponds to multiplying matrices together. As with classifying groups, we want to classify the simple (or "irreducible") representations of a given group, i.e. those which can't be broken down into smaller representations.

This project looks at what happens when we change the number system by taking a prime number p and replacing each matrix entry with its remainder modulo p. Now a representation which was irreducible can become reducible, and one of the most important tasks is to work out what its irreducible constituents are. This project looks at a special case of this question, by asking for a classification of which irreducible representations remain irreducible when reduced modulo p. This is interesting because it means that part of the task of classifying and constructing the irreducible representations modulo p (which in general is very hard) is done automatically. In this project we will look at this question for the simple groups (and some closely related groups called quasi-simple and almost-simple groups). For one important family of almost simple groups the problem is solved: these are the symmetric groups S(n), where n is a positive integer and S(n) is the group of all permutations of the numbers 1,...,n. From this starting point, we intend to solve our main problem for the double cover of S(n): this group is twice as large as S(n), consisting of permutations accompanied by a + or - sign, and has important applications in physics.

Having tackled these groups, we intend to look at a very large family, namely the groups of Lie type. These are groups which are already by definition groups of matrices, over various number systems and satisfying various algebraic conditions. The family of finite groups of Lie type is very complex and varied, so we intend to start with the easiest members of the family to begin with. These are the general linear groups, which are the groups of all n by n matrices over a given number system, with no additional condition other than having inverses. The representation theory of these groups is quite well understood, and has a remarkably close relationship to the representation theory of the symmetric groups. This means that it is an excellent initial case of our main question to solve, and acts as a test case for studying the groups of Lie type in general.

The outcomes of the project will be journal articles and conference presentations, together with a repository of information on the internet making data and the outcomes of preliminary investigations more widely available.