I should have mentioned that in that terrific database, Nebe lists "Lattices from the maximal finite subgroups of GL(n,Q)", which I suspect are the same ones I'm seeking. But, these are listed under that phrase only for various n >= 12. Ideally I'd like to know about dimensions 4 through 11 as well. —Dan
On Jan 14, 2016, at 9:35 AM, Dan Asimov <asimov@msri.org> wrote:
Thanks, Neil! That is quite a database, which I was not aware of.
But, do you know if it uses a particular term for "maximally symmetric" as defined below?
—Dan
On Jan 14, 2016, at 9:32 AM, Neil Sloane <njasloane@gmail.com> wrote:
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.
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*."