On the joins of group rings

TL;DR

This paper introduces and systematically studies the algebraic structure of the join of group rings, a new ring construction inspired by graph theory, analyzing its properties, modules, and diagonalization techniques.

Contribution

It defines the join of group rings, characterizes its algebraic properties, and extends Fourier analysis to this new structure, generalizing classical results.

Findings

The join of group rings forms a well-defined ring with specific algebraic properties.

A formula for the number of irreducible modules over the join ring when over an algebraically closed field.

A blockwise Fourier transform generalizes circulant diagonalization to join of circulant matrices.

Abstract

Given a collection $\{ G_i\}_{i=1}^d$ of finite groups and a ring $R$, we define a subring of the ring $M_n(R)$ ($n = \sum_{i=1}^d|G_i|)$ that encompasses all the individual group rings $R[G_i]$ along the diagonal blocks as $G_i$-circulant matrices. The precise definition of this ring was inspired by a construction in graph theory known as the joined union of graphs. We call this ring the join of group rings and denote it by $\mathcal{J}_{G_1,\dots, G_d}(R)$. In this paper, we present a systematic study of the algebraic structure of $\mathcal{J}_{G_1,\dots, G_d}(R)$. We show that it has a ring structure and characterize its center, group of units, and Jacobson radical. When $R=k$ is an algebraically closed field, we derive a formula for the number of irreducible modules over $\mathcal{J}_{G_1,\dots, G_d}(k)$. We also show how a blockwise extension of the Fourier transform provides both a generalization of the Circulant Diagonalization Theorem to joins of circulant matrices and an explicit isomorphism between the join algebra and its Wedderburn components.