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

Supercharacter Theories and Semidirect Products

We describe the supercharacter theories of the semidirect product of H and K, $H\rtimes K$ in terms of the supercharacter theories of the direct product of H and K in the case when both H and K are Abelian groups. To do this we introduce the concept of a homomorphism of supercharacter theories. This provides a classification of the supercharacter theories of the dihedral groups of order 2m when m is odd using the known classification of the supercharacter theories of cyclic groups.Comment: 9 page