Group representation theory often relates quite different areas of mathematics and we shall give yet another example of this phenomenon. A construction from finite geometries will lead us to a new concept in representation theory which we shall then apply to the representation theory of Lie type groups. This, in turn, will involve ideas from the homological approach to modular representations. We shall, therefore, cover a spectrum of ideas.

One construction of finite projection planes involves the use of spreads. Suppose that V is a 2n-dimensional vector space over a finite field k of characteristic p. A spread S is a collection of n-dimensional subspaces whose (set-theoretic) union is all of V but where the intersection of any two members of the collection is zero. A group of linear transformations of V preserves the spread S if its elements permute the members of S.

Proposition 1.If E is an elementary abelian 2-group of linear transformations of V which preserve S and p = 2 then there is a subgroup F of E with the following two properties:

1. i) V is free as a kF-module;

2. ii) The space of fixed-points VF, of V under F, equals VE.

This is the key idea and we shall now formulate it in more generality. If E is an elementary abelian p-group and k is any field of characteristic p then the kE-module M is said to be subfree if there is a subgroup F of E with the two properties of the proposition.