Resumen

Let G be a finite group and F a field of characteristic p > 0. An FG-module M is endotrivial if Hom_F(M,M) = F \oplus P, for some projective kG-module P. One main class of indecomposable endotrivial modules is given by \Omega_G := {\Omega^n(F), n \in Z}, where \Omega denotes the Heller translate. In some cases, this is the only class of indecomposable endotrivial modules, for example when G is an elementary abelian p-group. Suppose p is 2 and consider the symmetric group S_n. The set of indecomposable endotrivial FS_n-modules is just \Omega_{S_n} except for some small n. In the case when n is 4, the Specht module S^{(3,1)} is indecomposable endotrivial and S^{(3,1)} \notin \Omega_{S_4} . We examine the symmetric power Sym^nS^{(3,1)} and describe all the indecomposable nonprojective summands in Sym^nS^{(3,1)} for any n \geq 0.