Re Q. 1 --- The recent discussion about complex & quaternion GCD illustrates that this question is non-trivial. In a more general setting, GCD must be replaced by lattice basis minimisation. Re Q. 2 --- If the lattice has full rank, I think "NIP" is equivalent to the Voronoi tiling vertex figure being a simplex: there should be a one-line proof of this, but just now I can't pin one down ... Re Q. 3 --- I don't have a copy handy to check; but Conway & Sloane must be full of this sort of stuff? WFL On 10/25/20, Dan Asimov <dasimov@earthlink.net> wrote:
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
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