Coincidence indices of sublattices and coincidences of colorings

Abstract

Even though a lattice and its sublattices have the same group of coincidence isometries, the coincidence index of a coincidence isometry with respect to a lattice $\Lambda_1$ and to a sublattice $\Lambda_2$ may differ. Here, we examine the coloring of $\Lambda_1$ induced by $\Lambda_2$ to identify how the coincidence indices with respect to $\Lambda_1$ and to $\Lambda_2$ are related. This leads to a generalization of the notion of color symmetries of lattices to what we call color coincidences of lattices. Examples involving the cubic and hypercubic lattices are given to illustrate these ideas.