Endomorphism rings of permutation modules

Let k be an algebraically closed field of characteristic p and G a finite group. Then the permutation module of G on the cosets of a Sylow p-subgroup P is via Fitting correspondence strongly related to its endomorphism ring . On the other hand, each Green correspondent in G of a weight module of G occurs as a direct summand of . This fact suggests to analyze both structures, the permutation module and the associated endomorphism ring towards hints at a proof for Alperin's weight conjecture. We present a selection of such investigations for different groups and characteristics. In particular we focus on the socle and head constituents of the indecomposable direct summands of and of the PIMs of EE.