On Some Lattices Connected with a Finite Group

Let $\mathbb C[G]$ be the group ring of a finite group $G$, $\pi _r$ be a minimal central idempotent of this group ring, and $W_r=\mathbb C[G]\pi _r$ be the corresponding minimal central two-sided ideal. The ring $\mathbb C[G]$ contains the group ring $\mathbb Z[G]$, whereby the ideal $W_r$ contains a subring $A_r=\mathbb Z[G]\pi _r$. This article concerns the geometrical properties of location of the subring $A_r$ in the ideal $W_r$. The following facts are proved: (1) generally, the subgroup $A_r$ is not discrete in $W_r$; (2) if the associated irreducible character $\chi _r$ has integer values, then $A_r$ is a lattice in $W_r$; (3) if the irreducible character $\chi _r$ is real, the converse is true as well; (4) for a symmetrization $W_r^{\bullet }$ with respect to an action of a certain Galois group, the subgroup $\mathbb Z[G]\pi _r^{\bullet }$ is a lattice in $W_r^{\bullet}$.