In our database we give lists in lower dimensions that include all those irreducible lattices whose groups are locally maximal. You can go through them and take your pick. There are also various sequences in the OEIS that give the max order 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 12:42 PM, Dan Asimov <asimov@msri.org> wrote:
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*."
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