## ABSTRACT

In the previous chapter we have seen how the global symmetry of simple lattices is described by the actions (3.7) and (3.8) of the global symmetry group GL(3, Z) on the conﬁguration spaces B and Q>3 . In this chapter we show that this global symmetry is locally compatible with the notions on crystal symmetry usually considered in crystallography and physics. Indeed, we show that, for any given conﬁguration of a simple lattice described by a basis ea ∈ B, there exists a suitable orthogonally invariant neighborhood, to be called a weak-transformation neighborhood or, in short, wt-nbhd,1 in which the global symmetry reduces to the lattice-group symmetry dictated by that crystal conﬁguration. This is because the action of the group GL(3, Z) on B, when restricted to a wt-nbhd of ea in B, reduces to the action of the (ﬁnite) lattice group L(ea). In this way the global symmetry of crystals is reconciled with their crystallographic symmetry, in the range of ‘small but ﬁnite deformations’. An analogous result also holds for multilattices (chapter 11).