-free groups, abelian groups whose countable subgroups are free, are objects of both algebraic and set-theoretic interest. Illustrating this, we note that -free groups, and in particular the question of when -free groups are free, were central to the resolution of the Whitehead problem as undecidable. In elucidating the relationship between -freeness and freeness, we prove the following result: an abelian group is -free in a countable transitive model of (and thus by absoluteness, in every transitive model of ) if and only if it is free in some generic model extension. We would like to answer the more specific question of when an -free group can be forced to be free while preserving the cardinality of the group. For groups of size , we establish a necessary and sufficient condition for when such forcings are possible. We also identify a number of existing and novel forcings which force such -free groups of size to become free with cardinal preservation. These forcings lay the groundwork for a larger project which uses forcing to explore various algebraic properties of -free groups and develops new set-theoretical tools for working with them.
I will first give a brief survey of some previous results with Louise Sutton, in which we found a large family of decomposable Specht modules for the Hecke algebra of type indexed by `bihooks'. We conjectured that outside of some degenerate cases, our family gave all decomposable Specht modules indexed by bihooks. There, our methods largely relied on some hands-on computation with Specht modules, working in the framework of cyclotomic KLR algebras.
I will then move on to discussing a recent project with Rob Muth and Louise Sutton, in which we have studied the structure of these Specht modules. By transporting the problem to one for Schur algebras via a Morita equivalence of Kleshchev and Muth, we are able to give all composition factors (including their grading shifts), and show that in most characteristics, these Specht modules are in fact semisimple. In some other small characteristics, we can explicitly determine their structures, including some in which the modules are `almost semisimple'. I will present this story, with some running examples that will help the audience keep track of what's going on.