This concerns one of those "well known" facts, referred to in a recent preprint I've been looking at. In principle it's elementary, but I can't pin down an explicit textbook reference for it. Start with two finite groups
$A,B$ and their product
$G:=A \times B$, working over a splitting field
$K$ for the groups involved with prime characteristic dividing
$S_1, \dots, S_m$ and
$T_1, \dots, T_n$ be respective sets of representatives of isomorphism classes of simple modules for the group algebras
$KA, KB$. In turn let the projective covers (=injective hulls) be respectively
$P_i, Q_j$. These are the PIMs or
indecomposable projective modules for the two group algebras.
It's a standard observation (found in some books) that there is an obvious isomorphism between
$KG$ and the tensor product algebra
$KA \otimes_K KB$, while each group algebra splits into the direct sum (as a left module over itself) of the various PIMs taken with multiplicity equal to the dimension of the corresponding simple module. It's also a standard fact (found in some books) that each
$S_i \otimes T_j$ is a simple module for
$KG$. From these ingredients one can conclude that
$P_i \otimes Q_j$ is the corresponding PIM, thereby exhausting all isomorphism classes for
Is all of this written down in a self-contained way somewhere?