On the cohomology of Specht modules. (Q855989)

Let \(k\) be an algebraically closed field of odd characteristic, \(\Sigma_d\) be the symmetric group on \(d\) objects, and \(\text{GL}_n(k)\) be the general linear group with \(n\geq d\). The representation theories of these two groups are related by the Schur functor. This highly used relationship is used here to make explicit cohomological connections and computations involving Specht modules for the symmetric group. Under the Schur functor, Specht modules correspond to induced (or dual Weyl) modules for \(\text{GL}_n(k)\). The Schur functor admits a right adjoint. Using this adjoint functor, \textit{S. R. Doty, K. Erdmann}, and the second author [Algebr. Represent. Theory 7, No. 1, 67-99 (2004; Zbl 1084.20004)] constructed a spectral sequence relating extensions over the Schur algebra (associated to \(n\) and \(d\)) to extensions over \(k\Sigma_d\). When higher right derived functors of the adjoint functor vanish, this spectral sequence can be used to identify symmetric group extensions with extensions over the Schur algebra. The latter extensions can then be identified with extensions over \(\text{GL}_n(k)\). When this general relationship is applied with a Specht module, with its correspondence to an induced module, one obtains a relationship with extensions over the Borel subgroup \(B\) of upper triangular matrices in \(\text{GL}_n(k)\). The authors then use known vanishing results on the aforementioned higher right derived functors to give a number of explicit extension relationships. In particular, the cohomology of a Specht module over \(k\Sigma_d\) (in small degrees) can be identified with the cohomology of a one-dimensional \(B\)-module (for \(n=d\)) in degree shifted by \(d(d-1)/2\). For a fixed \(d\), since one has identifications of certain \(B\)-extensions for arbitrary \(n\geq d\) with extensions over \(k\Sigma_d\), the authors first observe that one can necessarily obtain identifications of various \(B\)-extensions. Then more explicit computations of \(B\)-cohomology groups are made using known computations of Specht module cohomology. Some new computations of Specht module cohomology are also made. Lastly, through the use of a spectral sequence, the authors demonstrate how knowledge of the cohomology of the first Frobenius kernel of \(B\) (for one-dimensional modules) could potentially be used to make computations of the first (and possibly higher) degree cohomology of Specht modules.