Primitive Idempotents of Schur Rings

In this paper, we explore the nature of central idempotents of Schur rings over finite groups. We introduce the concept of a lattice Schur ring and explore properties of these kinds of Schur rings. In particular, the primitive, central idempotents of lattice Schur rings are completely determined. For a general Schur ring $S$, $S$ contains a maximal lattice Schur ring, whose central, primitive idempotents form a system of pairwise orthogonal, central idempotents in $S$. We show that if $S$ is a Schur ring with rational coefficients over a cyclic group, then these idempotents are always primitive and are spanned by the normal subgroups contained in $S$. Furthermore, a Wedderburn decomposition of Schur rings over cyclic groups is given. Some examples of Schur rings over non-cyclic groups will also be explored

On Schur Rings over Infinite Groups III

In the paper, we develop further the properties of Schur rings over infinite groups, with particular emphasis on the virtually cyclic group $\mathcal{Z}\times\mathcal{Z}_p$. We provide structure theorems for primitive sets in these Schur rings. In the case of Fermat and safe primes, a complete classification theorem is proven which states that all Schur rings over $\mathcal{Z}\times\mathcal{Z}_p$ are traditional. We also draw analogs between Schur rings over $\mathcal{Z}\times\mathcal{Z}_p$ and partitions of difference sets over $\mathcal{Z}_p$