On the algebra of finite cyclic $p$-group representations

Let $K$ be a field with characteristic $p$ and $G$ be a finite cyclic $p$-group. It is known that the indecomposable $K$-representations of this group are Jordan cells with ones on the main diagonal. The present work is devoted to the decomposition of the tensor product of such cells. The number of Jordan cells in this expansion is found, and formulas are given for calculating the order and multiplicity of the largest of the orders of Jordan cells.