Geodesic
On Some Lattices Connected with a~Finite Group
Informatics and Automation, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 179-194.

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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 $W_r^{\bullet}$. 
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     author = {A. V. Zarelua},
     title = {On {Some} {Lattices} {Connected} with {a~Finite} {Group}},
     journal = {Informatics and Automation},
     pages = {179--194},
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     volume = {239},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2002_239_a11/}
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A. V. Zarelua. On Some Lattices Connected with a~Finite Group. Informatics and Automation, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 179-194. http://geodesic.mathdoc.fr/item/TRSPY_2002_239_a11/

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