On the joins of group rings

• Published in: Journal of pure and applied algebra 2023-09, Vol.227 (9), p.107377, Article 107377
• Main Authors: Chebolu, Sunil K., Merzel, Jonathan L., Mináč, Ján, Muller, Lyle, Nguyen, Tung T., Pasini, Federico W., Tân, Nguyễn Duy
• Format: Article
• Language: English
• Subjects: Artin-Wedderburn; Augmentation map; G-circulant matrices; Group of units; Group rings; Jacobson radical
• ISSN: 0022-4049; 1873-1376
• DOI: 10.1016/j.jpaa.2023.107377

Given a collection {Gi}i=1d of finite groups and a ring R, we define a subring of the ring Mn(R) (n=∑i=1d|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.