First Degree Cohomology of Specht Modules and Extensions of Symmetric Powers

Let ${\mathrm{\Sigma }}_d$ denote the symmetric group of degree $d$ and let $K$ be a field of positive characteristic $p$. For $p>2$ we give an explicit description of the first cohomology group $H^1({\mathrm{\Sigma }}_d,\mathrm{S}\mathrm{p}(\lambda ))$, of the Specht module $\mathrm{S}\mathrm{p}(\lambda )$ over $K$, labelled by a partition $\lambda$ of $d$. We also give a sufficient condition for the cohomology to be non-zero for $p=2$ and we find a lower bound for the dimension. Our method is to proceed by comparison with the cohomology for the general linear group $G(n)$ over $K$ and then to reduce to the calculation of ${\mathrm{E}\mathrm{x}\mathrm{t}}_{B(n)}^1(S^{d}E,K_{\lambda })$, where $B(n)$ is a Borel subgroup of $G(n)$, $S^{d}E$ denotes the $d$th symmetric power of the natural module $E$ for $G(n)$ and $K_{\lambda }$ denotes the one dimensional $B(n)$-module with weight $\lambda$. The main new input is the description of module extensions by: extensions sequences, coherent triples of extension sequences and coherent multi-sequences of extension sequences, and the detailed calculation of the possibilities for such sequences. These sequences arise from the action of divided powers elements in the negative part of the hyperalgebra of $G(n)$.