Minimal Resolutions

Abstract

The theory of blocks with cyclic defect groups is one of the high points of group theory; its extension to the non-cyclic case is the main problem of representation theory. Recently, the author and G. Janusz observed the consequence of the theory: if G is a group with cyclic Sylow p-subgroup and F is a splitting field of characteristic p for G, while . . . → P n → P n–1 → . . . . → P 0 → F → 0 → . . . is a minimal projective resolution for F over F[G], then each projective module P n is indecomposable. The periodicity and the precise period are immediate. A generalization of this result is presented in this chapter for the case of dihedral Sylow 2-subgroups, in which the one-dimensional array of indecomposable projective modules is replaced by a two dimensional array.