A lattice L of dimension n is a subgroup of R^n of rank n,
i.e., it has a set of generators that form a basis for R^n
as a vector space. The quotient R^n / L has finite volume
(and will always be an n-dimensional flat torus, T^n).
Examples include the lattice Z^n in all dimensions, the
triangular lattice Z[w] in R^2 = C (where w = exp(2πi/3)),
and the bcc and fcc lattices in R^3.
Also in all dimensions there are the A_n lattice, defined as
A_n = {(x_0,...,x_n) ∊ Z^(n+1) | ∑ x_j = 0}
and
D_n = {(x_1,...,x_n) ∊ Z^n | ∑ x_j ≡ 0 (mod 2)}.
I have some questions about lattices that I don't seem to
be able to find answers to in the literature:
Question 1:
-----------
Given a lattice L, what are the sublattices K ⊂ L
that are *similar* to L: rotated and uniformly scaled copies
of L ?
Question 1a:
------------
And what are their *indices* in L, i.e., the sizes |L/K|
of the quotient groups L/K ?
(In Z^2 = Z[i], these lattices are q Z[i] for any nonzero
Gaussian integer q ∊ Z[i], and their indices will be any
integer of the form a^2 + b^2. In Z[w] these lattices are
q Z[w] for any Eisenstein integer q and their indices will be
any integer of the form a^2 + ab + b^2.)
* * *
For any lattice L ⊂ R^n, the *Voronoi cell* V(p) of p ∊ L
of L is the set of points of R^n closer to p than to any
other point (or tied):
V(p) = {x ∊ R^n | ‖x - p‖ ≤ ‖x - q‖ for all q ∊ L}
V(p) is a closed and bounded n-dimensional polytope, and
all V(p) are congruent to each other by the symmetry of L.
Question 2:
-----------
For which lattices do the various Voronoi cells V(p) have the
"nice intersection property" (NIP) that for all p, q ∊ L, p ≠ q,
we have: V(p) ∩ V(q) is either a common (n-1)-dimensional face
or else it is empty?
For instance the triangular lattice Z[w] has the NIP (the V(p)
are regular hexagons). But Z^2 does not, since V(p) ∩ V(q) may
be a common vertex.
* * *
A lattice L ⊂ R^n is carried to itself by some distance-preserving
mappings R^n —> R^n that take the origin to itself; these are called
its "symmetries" and they form a group, denoted by Sym(L). Thus
Sym(L) is a subgroup of the orthogonal group O(n). (For instance,
every lattice has the symmetry x |—> -x.)
If every lattice L' with symmetries containing Sym(L) in fact has
the same symmetries:
Sym(L') ⊃ Sym(L) ⟹ Sym(L') = Sym(L),
then L is called "maximally symmetric". In 2D both Z^2 and Z[w]
are maximally symmetric; in 3D the maximally symmetric lattices
are Z^3, fcc, and bcc.
Question 3:
-----------
What is known about which lattices are maximally symmetric
in each dimension?
Presumably these also include the D_4 lattice in 4D, the E_8
lattice in 8D (defined as E_8 =
{(x_1,...,x_8) ∊ Z^8 ∪ (Z^8 + (1/2,...,1/2))) | 𝝨x_j ≡ 0 (mod 2)},
and the Leech lattice in 24D.
—Dan
