Generators for Decompositions of Tensor Products of Modules associated with standard Jordan partitions

TL;DR

This paper provides a method to explicitly find generators for indecomposable components in the tensor product of modules over cyclic groups in characteristic p, based on standard Jordan partitions.

Contribution

It introduces a systematic approach to determine generators for tensor product decompositions of modules associated with standard Jordan partitions.

Findings

Explicit generators for indecomposable components are constructed.

The method applies to modules over cyclic groups in finite characteristic.

The approach simplifies understanding tensor product decompositions in modular representation theory.

Abstract

If $K$ is a field of finite characteristic $p$, $G$ is a cyclic group of order $q=p^\alpha$, $U$ and $W$ are indecomposable $KG$-modules with $\dim U=m$ and $\dim W=n$, and $\lambda(m,n,p)$ is a standard Jordan partition of $ m n$, we describe how to find a generator for each of the indecomposable components of the $KG$-module $U \otimes W$.