We consider the symmetric group S_n of degree n and an algebraically
closed field F of prime characteristic p.
As is well-known, many representation theoretical objects of S_n
possess concrete combinatorial descriptions such as the simple
FS_n-modules through their parametrization by the p-regular partitions of n,
or the blocks of FS_n through their characterization in terms of p-cores
and p-weights. In contrast, though closely related to blocks and their
defect groups, the vertices of the simple FS_n-modules are rather poorly
understood. Currently one is far from knowing what these vertices look
like in general and whether they could be characterized combinatorially
as well.
In this talk I will refer to some theoretical and computational
approaches towards the determination of vertices of simple FS_n-modules.
Moreover, I will present some results concerning the vertices of
certain classes of simple FS_n-modules such as the ones labelled by
hook partitions or two part partitions, and will state a series of
general open questions and conjectures.