On the joins of group rings

Given a collection {Gi}di=1 of finite groups and a ring R, we define a subring of the ring Mn(R) (n = sigma di=1 |Gi|) that encompasses all the individual group rings R[Gi] along the diagonal blocks as Gi-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 JG1,...,Gd(R). In this paper, we present a systematic study of the algebraic structure of JG1,...,Gd(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 JG1,...,Gd(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. (c) 2023 Elsevier B.V. All rights reserved.