The Structure of Some Permutation Modules for the Symmetric Group of Infinite Degree Original Research Article

Suppose that Ω is an infinite set andkis a natural number. Let [Ω]kdenote the set of allk-subsets of Ω and letFbe a field. In this paper we study theFSym(Ω)-submodule structure of the permutation moduleF[Ω]k. Using the representation theory of finite symmetric groups, we show that every submodule ofF[Ω]kcan be written as an intersection of kernels of certainFSym(Ω)-homomorphismsF[Ω]k → F[Ω]lfor 0 ≤ l < k, and give a simple algorithm to determine the complete submodule structure ofF[Ω]k.