Coincidence indices of sublattices and coincidences of colorings

Abstract

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 Λ1 and to a sublattice Λ2 may differ. Here, we examine the coloring of Λ1 induced by Λ2 to identify how the coincidence indices with respect to Λ1 and to Λ2 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.