There are more related papers on http://neilsloane.com/doc/pub.html Three titles contain "Voronoi": 82. Voronoi Regions of Lattices, Second Moments of Polytopes, and Quantization, J. H. Conway and N. J. A. Sloane, IEEE Trans. Information Theory, IT-28 (1982), pp. 211-226, A revised version appears as Chapter 21 of ``Sphere Packings, Lattices and Groups'' by J. H. Conway and N. J. A. Sloane, Springer-Verlag, NY, 1988 Reprinted in ``Vector Quantization'', H. Abut, Ed., IEEE Press, NY, 1990, pp. 118-133.. 108. On the Voronoi Regions of Certain Lattices, J. H. Conway and N. J. A. Sloane, SIAM J. Algebraic Discrete Methods, 5 (1984), pp. 294-305. 169. Low-Dimensional Lattices VI: Voronoi Reduction of Three-Dimensional Lattices, J. H. Conway and N. J. A. Sloane, Proc. Royal Soc. London, Series A, 436 (1992), pp. 55-68. [Errata: 1) In the proof of Theorem 2: N(v-2w) < N(w) should read N(v-2w) < N(v); 2) Equation (1) is missing a minus sign before p_ij; 3) in equation (12), v'_13 has the wrong sign.] * Dan Asimov <dasimov@earthlink.net> [Jul 30. 2012 15:25]:
Thanks very much, Neil, for the pointer to your paper.
Now I will need to figure out which if any of these "maximally symmetric" lattices (besides A*_2 and A*_3) have Voronoi tessellations with vertex figures that are simplices.
Any further pointers to where this information might be found will be greatly appreciated.
-Dan
On 2012-07-28, at 12:39 PM, Dan Asimov wrote:
Can anyone please point me to any article or book that addresses the question of what are the maximal finite subgroups of GL(n,Z) ?
What I'm ultimately interested in is this:
Which n-dimensional lattices L (in R^n) have a maximal automorphism group Aut(L) -- maximal as a subgroup of O(n) ?
(Aut(L) here is the group of isometries of R^n that fix the origin and take L onto L.)
E.g., in 2D these are the square and triangular lattices. In 3D I think these are the cubic, bcc and fcc lattices. I don't know the answer for higher dimensions.
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