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arXiv · 2208.07413

On the joins of group rings

Sunil K. Chebolu·Jonathan L. Merzel·Ján Mináč·Lyle Muller·Tung T. Nguyen·Federico W. Pasini·Nguyen Duy Tân
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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.

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Sunil K. Chebolu, Jonathan L. Merzel, Ján Mináč, Lyle Muller, Tung T. Nguyen, Federico W. Pasini, Nguyen Duy Tân. 2022-08-15. On the joins of group rings. https://arxiv.org/abs/2208.07413

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