Permutation modules for Ramsey structures

TL;DR

This paper studies permutation modules arising from automorphism groups of $ ext{ω}$-categorical structures, providing methods to analyze their submodules, composition length, and a decision procedure in the case of Ramsey structures.

Contribution

It introduces new techniques for understanding submodule structures and properties of permutation modules associated with automorphism groups of $ ext{ω}$-categorical and Ramsey structures.

Findings

Develops methods to determine if $FW$ has the ascending chain condition on submodules.

Provides criteria for when $FW$ has finite composition length for finitely homogeneous structures.

Establishes a duality between submodules of $FW$ and definable functions when $F$ is a field.

Abstract

Suppose $R$ is a commutative ring and $G$ is a group acting on a set $W$. We consider the $RG$-module $RW$ in the case where $G$ is the automorphism group of an $\omega$-categorical structure $M$ and $W$ is, for example, $M^n$ (for $n \in \mathbb{N}$). We develop methods which may provide information about two questions in the case where $R$ is a field $F$: whether $FW$ has a.c.c. on submodules; and in the case where $M$ is finitely homogeneous, whether $FW$ is of finite composition length. In the case where $M$ is a Ramsey structure and so $G$ is extremely amenable, we give a simple `decision procedure' for membership in a submodule of $RW$ specified by a given generating set. If $F$ is a field, we show that there is a duality between submodules of $FW$ and the topological $FG$-module of definable functions from $W$ to $F$.