Relating the cohomology of general linear groups and symmetric groups (Q2776261)

In previous work of the authors and \textit{K. Erdmann} [Extensions of modules over Schur algebras, symmetric groups and Hecke algebras, Algebr. Represent. Theory (to appear)], a theory was developed for relating the representation theories of general linear and symmetric groups in non-zero characteristic. In particular, a spectral sequence was constructed relating extensions in the two categories. In this work, homological methods and this spectral sequence are used to provide succinct new proofs of a number of fundamental results in the area.NEWLINENEWLINENEWLINEFor example, they recover G. James' result relating decomposition numbers for general linear groups and symmetric groups and a result of \textit{S. Donkin} [J. Algebra 111, 354-364 (1987; Zbl 0634.20019)] on blocks of Schur algebras. Further, explicit calculations of higher right derived functors of an adjoint to the Schur functor are used to compute the first and second cohomology groups of a symmetric group for the dual of a Specht module; computations first made by \textit{V. P. Burichenko, A. S. Kleshchev}, and \textit{S. Martin} [J. Pure Appl. Algebra 112, No. 2, 157-180 (1996; Zbl 0894.20038)]. Finally, they investigate tilting modules for Schur algebras obtaining information about composition factors and extensions. With their computations it easily follows that the \(d\)-fold tensor product of the natural representation for a general linear group does not admit any non-trivial self-extensions over the symmetric group \(\Sigma_d\). This last fact was originally proven by \textit{E. Cline, B. Parshall}, and \textit{L. Scott} [Mem. Am. Math. Soc. 591, 119p. (1996; Zbl 0888.16006)].NEWLINENEWLINEFor the entire collection see [Zbl 0977.00027].