Solution to the Clebsch-Gordan problem for string algebras

TL;DR

This paper investigates the tensor product structure in string algebra modules, solving the Clebsch-Gordan problem, and describing the representation ring and tensor ideals, advancing understanding of their module categories.

Contribution

It provides a solution to the Clebsch-Gordan problem for string algebras and describes their representation ring and tensor ideals.

Findings

Solved the Clebsch-Gordan problem for string algebras

Described the representation ring of string algebra modules

Analyzed tensor ideals in the module category

Abstract

The category of modules over a string algebra is equipped with a tensor product defined point-wise and arrow-wise in terms of the underlying quiver. In the present article we investigate how this tensor product interacts with the classification of indecomposables. We apply the results obtained to solve the Clebsch-Gordan problem for string algebras. Moreover, we describe the corresponding representation ring and tensor ideals in the module category.