  During the 2004-2005 academic year the VIGRE algebra research group at the
University of Georgia computed the complexities of certain Specht modules
S^\lambda for the symmetric group, using the computer algebra program Magma.
The complexity of an indecomposable module does not exceed the p-rank of the
defect group of its block. The Georgia group conjectured that, generically, the
complexity of a Specht module attains this maximal value; that it is smaller
precisely when the Young diagram of $\lambda$ is built out of $p \times p$
blocks. We prove one direction of this conjecture by showing these Specht
modules do indeed have less than maximal complexity. It turns out that this
class of partitions, which has not previously appeared in the literature,
arises naturally as the solution to a question about the $p$-weight of
partitions and branching.
