Notes on trivial source modules (Q1912845)

Let \(G\) be a finite group and \(k\) an algebraically closed field of characteristic \(p\). An indecomposable \(kG\)-module with a vertex \(Q\) is said to be a weight module if its Green correspondent with respect to \((G, Q, N_G(Q))\) is simple. Let \(B\) be a block of \(kG\). Alperin conjectured that the number of the weight modules belonging to \(B\) equals that of the simple modules in \(B\). If this is the case and a defect group of \(B\) a TI set, then it can be shown under some additional assumption that the socles of weight modules are simple, which in turn determine the isomorphism classes of the weight modules; this holds if \(G\) is a simple group with a cyclic Sylow \(p\)-subgroup. This rather surprising property has been known to hold for finite groups of Lie type of characteristic \(p\). However little is known about general properties of weight modules. In the final section we shall study solvable groups that have only simple weight modules.