To answer Dan's question, see here: http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/ Gabriele Nebe (following work by W. Plesken and many others) has extensively studied this kind of question Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Thu, Jan 14, 2016 at 10:45 AM, Dan Asimov <asimov@msri.org> wrote:
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.
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