Recall that a lattice in is the -span of a basis of . Equivalently, letting be the matrix with rows so that , the lattice is . Given , it is easily seen that we have the equality of lattices if and only if . Thus, we can parametrize the space of all lattice by .
We say that two lattices are homothetic and write if there exists some such that . Since usually we do not consider homothetic lattices as different, we would like to have a smaller space that do not distinct between such lattices. We consider two approachs – one by taking the quotient space and the other by taking a representative from each homothetic class.
Given such that for some we get that , thus to represent a class of homothetic lattices we may consider elements in instead of . As in the previous case, two such elements define the same lattice class if and only if , thus we can paramerize the class of lattices up to homothety by .
Set of representatives – unimodular lattices:
Let be a lattice for some . The covolume of the lattice is defined to be which is the volume of a fundamental domain of in and is equal to . We call a lattice unimodular if its covolume is one, or equivalently it can be written as for some . Given a -dimensional lattice of covolume , we obtain that is a unimodular lattice homothetic to . Thus, every homothetic class contains a unimodular lattice which is easily seen to be unique. As in the previous cases, we can parametrize the space of -dimensional unimodular lattices by and by the argument above under the natural map for . We shall denote this space by .
Remark: The argument that the map above is surjective was due the fact that every positive element in has a -root. When a similar construction is done over other fields, e.g. -adic fields, this is no longer the case.
In each of the groups and we have the respective diagonal group with nonnegative entries. Namely
For we denote , and we note that the map defines an isomorphism . The restriction to
induces an isomorphism . Finally the natural map is an isomorphism. We will also use the isomorphism defined by for .
We shall be interested in the diagonal orbits in their respective spaces of lattice. Letting be the normalization to covolume 1 map and be the quotient map, it is easy to see that for any lattice we have