Martin Herschend : On the representation ring of the polynomial algebra over a perfect field

Given two finite-dimensional modules over the polynomial algebra k[x], their tensor product over the ground field k is a k[x]-module, on which x acts diagonally. We consider the so-called Clebsch-Gordan problem of finding the decomposition of the tensor product of pairs of indecomposable k[x]-modules. Over an algebraically closed ground field of characteristic zero it was first solved by Aitken and has later been solved independently by Huppert, and also by Martisonkovsky and Vlassov. In positive characteristic it has been studied by Iima and Iwamatsu.