  For a Specht module S^\lambda for the symmetric group \Sigma_d, the
cohomology H^i(\Sigma_d, S^\lambda) is known only in degree i=0. We give a
combinatorial criterion equivalent to the nonvanishing of the degree i=1
cohomology, valid in odd characteristic. Our condition generalizes James'
solution in degree zero. We apply this combinatorial description to give some
computations of Specht module cohomology, together with an explicit description
of the corresponding modules. Finally we suggest some general conjectures that
might be particularly amenable to proof using this description.
