  Cohomology of Specht modules for the symmetric group can be equated in low
degrees with corresponding cohomology for the Borel subgroup B of the general
linear group GL_d(k), but this has never been exploited to prove new symmetric
group results. Using work of Doty on the submodule structure of symmetric
powers of the natural GL_d(k) module together with work of Andersen on
cohomology for B and its Frobenius kernels, we prove new results about
H^i(\Sigma_d, S^\lambda). We recover work of James in the case i=0. Then we
prove two stability theorems, one of which is a "generic cohomology" result for
Specht modules equating cohomology of S^{p\lambda} with S^{p^2\lambda}. This is
the first theorem we know relating Specht modules S^\lambda and S^{p\lambda}.
The second result equates cohomology of S^\lambda with S^{\lambda + p^a\mu} for
large a.
