Generators for Tensor Product Components

TL;DR

This paper introduces a recursive method to explicitly find generators of indecomposable components in tensor products of modules over cyclic groups in characteristic p, enhancing understanding of module decomposition.

Contribution

It presents a novel recursive approach for identifying generators of indecomposable components in tensor products of modules over cyclic groups in characteristic p.

Findings

Recursive method successfully identifies generators

Applicable to modules over cyclic groups of prime power order

Improves explicit understanding of tensor product decompositions

Abstract

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 $U \otimes W$.