Similar submodules and coincidence site modules

Abstract

We consider connections between similar sublattices and coincidence site lattices (CSLs), and more generally between similar submodules and coincidence site modules of general (free) $\mathbb{Z}$-modules in $\mathbb{R}^d$. In particular, we generalise results obtained by S. Glied and M. Baake [1,2] on similarity and coincidence isometries of lattices and certain lattice-like modules called $\mathcal{S}$-modules. An important result is that the factor group $\mathrm{OS}(M)/\mathrm{OC}(M)$ is Abelian for arbitrary $\mathbb{Z}$-modules $M$, where $\mathrm{OS}(M)$ and $\mathrm{OC}(M)$ are the groups of similar and coincidence isometries, respectively. In addition, we derive various relations between the indices of CSLs and their corresponding similar sublattices. [1] S. Glied, M. Baake, Similarity versus coincidence rotations of lattices, Z. Krist. 223, 770--772 (2008). DOI: 10.1524/zkri.2008.1054 [2] S. Glied, Similarity and coincidence isometries for modules, Can. Math. Bull. 55, 98--107 (2011). DOI: 10.4153/CMB-2011-076-x