| A guiding principle in the modular representation theory of finite groups is the belief that the representation theory of a finite group G in prime characteristic p should in some sense be determined by that of its p-local subgroups (normalizers of non-identity p-subgroups). This belief takes a numerical form in Alperin's weight conjecture: the number of isomorphism classes of simple modules for G is equal to the sum of the numbers of isomorphism classes of certain kinds of simple modules for p-local subgroups.;In certain situations there is a much stronger conjecture on the table: Broue predicts a structural relationship between modules for G and modules for local sub-groups which involves derived categories. More precisely, he conjectures that if B is a p-block of G with abelian defect group D, and b is the block of the p-local subgroup ;We verify Broue's conjecture for all defect two blocks of symmetric groups. Actually for these blocks we prove a stronger statement, a refinement of Broue's conjecture due to Rickard. |
