Representations and cohomology of general linear groups and symmetric groups

This is a report on joint work with Haralampos Geranios. The connection between representations of general linear groups and symmetric groups goes back to Schur's Thesis in 1901, where it was used to  describe the representations of general linear groups over fields of characteristic $0$. Much  can still be gained from this connection in positive characteristic, especially as there are now many methods available on the general linear side not available to Schur, such as high weight theory, cohomology theory and the algebraic geometry of quotient spaces. In our current work we exploit this connection to give a complete description of the first cohomology of all Specht modules for symmetric groups  over fields of  characteristic different from $2$.  The solution involves a very detailed analysis of "extension multi-sequences" arising via the action of divided powers operators coming from the general linear group side of things.