The Krull-Remak-Schmidt decomposition of commutative group algebras

TL;DR

This paper determines the Krull-Remak-Schmidt decomposition of group algebras over fields, including prime characteristic fields, for finite abelian groups, by analyzing circulant coordinate rings.

Contribution

It introduces a method to decompose commutative group algebras using geometric equivalence and circulant coordinate rings, extending previous results to prime characteristic fields.

Findings

Decomposition of $k[G]$ for finite abelian groups over various fields.

Introduction of circulant coordinate rings as a tool.

Extension of known decompositions to prime characteristic fields.

Abstract

We provide the Krull-Remak-Schmidt decomposition of group algebras of the form $k[G]$ where $k$ is a field, which includes fields with prime characteristic, and $G$ a finite abelian group. We achieved this by studying the geometric equivalence of $k[G]$ which we call circulant coordinate rings.