Coincidence indices of sublattices and coincidences of colorings

Manuel Joseph C. Loquias, Peter Zeiner

Published 2015-05-29Version 1

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 $\Gamma_1$ and to a sublattice $\Gamma_2$ may differ. Here, we examine the coloring of $\Gamma_1$ induced by $\Gamma_2$ to identify how the coincidence indices with respect to $\Gamma_1$ and to $\Gamma_2$ 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.