  We give an algorithm for working out the indecomposable direct summands in a
Krull--Schmidt decomposition of a tensor product of two simple modules for
G=SL_3 in characteristics 2 and 3. It is shown that there is a finite family of
modules such that every such indecomposable summand is expressible as a twisted
tensor product of members of that family.
  Along the way we obtain the submodule structure of various Weyl and tilting
modules. Some of the tilting modules that turn up in characteristic 3 are not
rigid; these seem to provide the first example of non-rigid tilting modules for
algebraic groups. These non-rigid tilting modules lead to examples of non-rigid
projective indecomposable modules for Schur algebras, as shown in the Appendix.
  Higher characteristics (for SL_3) will be considered in a later paper.
