Generators for Decompositions of Tensor Products of Indecomposable Modules

TL;DR

This paper identifies explicit generators for indecomposable components of tensor products of modules over cyclic groups in characteristic p, extending previous work to cases with multiplicities greater than one.

Contribution

It provides a method to explicitly find generators for all indecomposable components of tensor products, generalizing earlier results to modules with multiplicities.

Findings

Explicit generators for indecomposable components are identified.

Extension of previous results to modules with multiplicities.

Clarification that results are translations of prior work by Iiwa and Iwamatsu.

Abstract

Let $F$ be a field of non-zero characteristic $p$, let $G$ be a cyclic group of order $q =p^a$ for some positive integer $a$, and let $U$ and $W$ be indecomposable $F G$-modules. We identify a generator for each of the indecomposable components of $U \otimes W$ in terms of a particular $F$-basis of $U \otimes W$, extending previous work in which each component occurred with multiplicity one. An anonymous reviewer helped us realize that our results are really just a translation of previous results of Iiwa and Iwamatsu to our situation.