Permutation modules for omega-categorical structures

Suppose F is a field and G is a group acting on a set W. We consider the FG-module FW in the case where G is the automorphism group of an omega-categorical structure M, and W is, for example, M^n. Two main questions here are: whether FW has the ascending chain condition on submodules; and in the case where M is finitely homogeneous, whether FW is of finite composition length. 

 

I will say something about these questions from a model-theoretic viewpoint. The main theme will be to look at what can be proved under the additional assumption that M is a Ramsey structure. In particular, in this case, we give a simple ‘decision procedure’ for membership in a submodule of FW specified by a given generating set. We also show that there is a duality between submodules of FW and the FG-module of definable functions W \to F (suitably interpreted).