In the study of a non-semisimple module category, projective modules are an essential ingredient. The general theory of these modules is fairly well-organized for any finite dimensional algebra, with additional features in the case of a group algebra KG. We begin by recalling some standard facts for an arbitrary G in 9.1, then raise a number of questions in 9.2 concerning families of finite groups of Lie type.
A pivotal role is played by Steinberg modules (9.3): simple G-modules having highest weights of the form (pr – 1)ρ, where ρ is the sum of fundamental weights. Unlike other simple modules for the finite group G over a field of pr elements, L((pr –1)ρ) is its own projective cover. Tensoring arbitrary KG-modules with this one produces new projective modules, whose indecomposable summands turn out to exhaust the projective covers of all simple modules (9.4).
In the framework of Brauer characters (9.5–9.6), we see that the Steinberg character “divides” all characters of projective modules. Moreover, there is at least a rough lower bound (9.7) for the dimensions of indecomposable projectives, though this is usually far too small in practice. A more thorough study of projectives is deferred to the following chapter, where the parallel theory for Frobenius kernels comes into play.
Here we just take a detailed look at projective modules for SL(2, p) (9.8). The data can be efficiently encoded in a “Brauer tree” (9.9).