The Green Ring and Modular Representations of Finite Groups of Lie Type’

Let G be a finite group of Lie type of characteristic p, and k a sufficiently large field of the same characteristic. The permutation module Y=ind,.(k) on the right cosets of a Sylow p-subgroup tJ has been studied by several authors; the fundamental results are due to Curtis and Richen (see [S], whose used the module Y to describe the irreducible kG-modules. Later their results were reinterpreted by Sawada [X] and Green [ 71 by consideration of the endomorphism ring E = End& Y). This was also the approach of Carter and Lusztig [4]. The indecomposable direct summands of Y were studied by Tinberg in [ 111, in which she determined their dimensions and vertices. Recently, Cabanes f 1] has computed the Green correspondents of the summands of Y by calculating the Brauer morphisms between certain endomorphism rings, thereby verifying a general conjecture of Alperin in the case of groups with split &N-pairs over fields of the defining characteristic. In this paper, starting from the basic theorem of Curtis, we shall give new calculations of the dimensions and vertices of the summands of Y (Section 4), and their Green correspondents (Section 6). In Section 8, we compute the dimensions of the modules of kG-homomorphisms among the summands of the permutation module ind,,,(k) on the right cosets of the