East Anglia

In 1993 in his paper "A new strongly minimal set" Hrushovski produced a family of counter examples to a conjecture by Zilber. Each one of these counter examples carry a pregeometry. We answer a question by Hrushovski about comparing these pregeometries and their localization to finite sets. We first analyse the pregeometries arising from different variations of the construction before the collapse. Then we compare the pregeometries of the family of new strongly minimal structures obtained after the collapse.

For the integers $a$ and $b$ let $P(a^b)$ be all partitions of the

set $N= {1,..., ab}$ into parts of size $a.$ Further, let

$\mathbb{C}P (a^b)$ be the corresponding permutation module for the

symmetric group acting on $N.$ A conjecture of Foulkes says

that $\mathbb{C}P (a^b)$ is isomorphic to a submodule of $\mathbb{C}P

(b^a)$ for all $a$ not larger than $b.$ The conjecture goes back to

the 1950's but has remained open. Nevertheless, for some values of

$b$ there has been progress. I will discuss some proofs and further

conjectures. There is a close correspondence between the

representations of the symmetric groups and those of the general

linear groups, via Schur-Weyl duality. Foulkes' conjecture therefore

has implications for $GL$-representations. There are interesting

connections to classical invariant theory which I hope to mention.