On the joins of group rings

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.

33 pages, Accepted for publication in the Journal of Pure and Applied Algebra