Generators for Tensor Product Components

Let $p$ be a prime number, $F$ a field of characteristic $p$, and $G$ a cyclic group of order $q =p^a$ for some positive integer $a$. Under these circumstances every indecomposable $F G$-module is cyclic. For indecomposable $F G$-modules $U$ and $W$, we present a new recursive method for identifying a generator for each of the indecomposable components of $U \otimes W$ in terms of a particular $F$-basis of $U \otimes W$.