Crystallographic Restrictions for Colour Lattices with Modular Sublattices

Abstract

The {\em d} -- dimensional $n$ -- colour lattice ${\mathbb{L}}^d$ with modular sublattices are studied, when the only one crystallographic type of sublattices does exist and the only one of the colours occupies a sublattice, which is still invariant under $k$--fold rotation $C_k$. Such kind of colouring always preserves an equal fractions of the colours composed ${\mathbb{L}}^d$. The $n$ -- colour lattice with modular sublattices allow to exist the crystallographic rotations $C_k$ for $k=p^r,r\ge 1$ and $n\le p$, where $p$ is a prime number.

18 pages, 2 figures