Adam or Veit or anyone, do you know if somewhere there is a list of (some or all) maximally symmetric lattices* in R^n for various dimensions n ? For n=2 I guess these are the square and triangular lattices, and for n=3 they're the cubic, fcc, and bcc lattices. But I don't know the list in any higher dimensions, just a few of the lattices. —Dan ______________________________ * in his book "Three-Dimensional Geometry and Topology, Vol. 1", Bill Thurston considers subgroups F of GL(n,Z), and all embeddings rho: Z^n —> R^n such that F acts by isometries on the image rho(Z^n). He writes: "If some lattice with symmetry F has no other symmetries, we say that F is an *exact lattice group*. An example of an F that is not exact is the trivial group — every lattice has this symmetry, but also the symmetry -Id. If every lattice with symmetry F has no other symmetries, F is a maximal finite subgroup of GL(n,Z) and the corresponding lattices are *maximally symmetric*."
On Jan 14, 2016, at 5:01 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Note the following slightly unintuitive fact:
"There exist full-rank sublattices L of the cubic lattice Z^d such that the point-fixing automorphism group of L is greater than that of Z^d."
For instance, in Z^4 we have that:
L := {(a, b, c, d) : a + b + c + d is even}
has an origin-fixing automorphism group of order 1152 (compared with 384 for the lattice Z^4).
Similarly, in Z^8 we can take a scaled E_8 lattice:
L := {(a_0, a_1, ..., a_7) : all coordinates have the same parity and a_0 + a_1 + ... + a_7 is divisible by 4}
which has an automorphism group of 696729600, compared with 10321920 for the cubic lattice Z^8.
The reason this phenomenon is slightly unintuitive is that it doesn't occur in dimensions d <= 3 so defies our geometric intuition.