# Talk:Lattice (group)

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Field: Algebra

## Self-dual

hey, can someone explain what it means for a lattice to be self-dual? -Lethe

I think the duality theory is what the crystallographers call the reciprocal lattice. Anyway, one can look at it via Pontryagin duality (ie Fourier theory) so that Λ in Rn has a dual lattice Λ* in the dual Rn, which is its annihilator in the natural pairing. The other way is simply to take linear functionals taking integer values on Λ - not much difference except for some factors of 2π.

Charles Matthews 08:25, 17 Jul 2004 (UTC)

Given a lattice L in Rn, mathematicians usually define its dual lattice L*  := { x ∈ Rn | x⋅v ∈ ℤ for all v ∈ L }. Or in English: The dual lattice L* of L is the set of vectors in Rn whose inner product with each element of L is an integer.
An "integral lattice" L is one for which L* contains L as a sublattice. This is equivalent to saying that for any v ∈ L we have v⋅w ∈ ℤ for all w ∈ L.
Then L is self-dual (L = L*) if and only if both L and L* are integral.
A lattice L is unimodular if for some set B of generators of L, given by B = {e1, . . ., en}, its Gram matrix M(B) (defined by M(B)(i,j) := ei⋅ej) satisfies det(M(B)) = 1. An integral lattice L is self-dual if and only if it is unimodular.
For technical reasons, unimodular lattices having some vectors of norm 1 (which include the cubic lattices) are less interesting. The first nontrivial unimodular lattice with no vector of norm 1 is the 8-dimensional E8 lattice.Daqu (talk) 06:45, 25 July 2012 (UTC)

## Source of the moved material

The material that I moved here from the lattice disambiguation page appears to have been written by User:Patrick. Michael Hardy 20:03, 4 October 2005 (UTC)

Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as lattice groups, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006

Please post at the bottom of talk pages. Anyway, do you know what a group (mathematics) is? If not, you may want to start by getting a book on introductory group theory. I don't know of any I can recommend, unfortunately. linas 16:22, 27 January 2006 (UTC)

I could use a good book on group theory. Problem is, I live in the Dominican Republic and it would cost me almost \$100 just to ship the book here! And, after all, the idea of Wikipedia is to BE the book! If the book isn't complete without the examples, do you think you guys could work on some examples for those of us trying to study who have nothing more than the Wikipedia? TIA. beno 31 Jan 2006

First of all, it's not exactly true that the point of wikipedia is to "be the book". We're aiming to write an encyclopedia, not a textbook. So wikipedia isn't necessarily optimal for learning stuff from scratch. In principle, that's what wikibooks is for. In practice, wikibooks is very much less developed than wikipedia. And a lot of wikipedia is quite approachable even if you start from scratch, so your goal isn't entirely unreasonable. That said, some articles are better than others. If some article seems not so readable to you, there are people who'd like to help. If this article isn't "sinking in" for you, maybe we can improve it, so that it will! In order for that to happen, it would be better if you could explain exactly what parts you're not getting. For those of us who know the material already, it can be hard to know exactly where we lose the beginner. Of course, any article is hard if you don't know the prerequisites, so as linas suggests, you should make sure you know what a group is before you tackle lattice groups. So with these points in mind, are there any specific points that I can help you understand? -lethe talk + 15:13, 31 January 2006 (UTC)
A group is a thing in higher mathematics, which roughly corresponds to 'walking' the symmetry. Basically, you reduce a symmetry to (say), three mirrors. In a regular figure, the mirrors go from the centre of the edge to the centre of the face, and one each along the edge and from the face to the vertex. You can call these V, E, and F, according to which one you step away from.
Stepping over and back across a mirror, like VV or EE, is going to bring you back to where you started from. To go around a vertex or some other corner, you need to step across alternating edges like VEVEVE.. to walk around a face. Since there are 2p, 4 and 2q of these at a corner, you can walk, eg VEV = EVE, VF = FV, and FVFVF = VFVFV (for an icosahedron).
One can make a 'graph' or picture of this, by putting a point (or 'node') for each mirror, eg V, E, and H. These are connected by 'branches', for each connection greater than AB = BA, ie an unmarked branch is ABA = BAB, a marked branch has as many alternating letters as the number is, so a branch marked 5, would equate ABABA = BABAB, or ABABABABAB = 1.
The total combination of different places one can reach with the allowed mirrors is the 'order' of the group. If a group is made of unconnected bits, (ie two parts, where aA = Aa for all mirrors), the order is the simple product. This is the reflection of the prism product, for example.
Any subset of mirrors, and their interconnecting branches, also makes a group. It corresponds to walking, for example all the possible steps in V and E, without using F. The F mirrors then become 'walls', and the VE becomes a room. The number of rooms then is the order of group (VEF), divided by the order of the subgroup (VE). These numbers are easy to come by.
Since our group is a kaleidoscope, we can put a motif, or dercoration inside it. One marks one or more nodes, to represent that the vertex is connected to that node: that is, a non-zero edge exists there. The edge is a mirror-edge, the ends are images of each other. If there is only one edge, falling on eg B, then a polygon will form only where node B is connected to another node where the walk ABABAB produces three or more B's. (edges are only laid when B steps are made).
Likewise, if several nodes are marked, then edges are placed down for each of these steps. Marking edges B and C, for example, will still produce a pentagon on ABABABABAB = 1 (since edges are laid at BBBBB), but also hexagons BCBCBC = 1 (where edges are laid on each step, but of alternating kinds). The resulting polyhedron here has pentagonal and hexagonal faces.
It's a supprisingly accurate and extendable construction for nearly every known uniform polyhedron (just three misses).
--Wendy.krieger (talk) 07:42, 6 July 2012 (UTC)
lethe responded to beno's expressed wish for Wikipedia to "be the book", above, saying: "So wikipedia isn't necessarily optimal for learning stuff from scratch."
Of course I agree with lethe that Wikipedia is an encyclopedia and not a textbook. But I strongly believe that this encyclopedia should try much harder to explain things from scratch as much as possible (making use of links to related articles, of course, when relevant). There are far too many articles where not only are they written solely for the expert, with no accommodation for anyone else, but the people who wrote it will vigorously defend that practice. In my opinion such people are unclear on the concept of an encyclopedia. (One of the worst such articles I've ever seen -- slightly better now than it was then -- is the one on model theory.)
This article could certainly do better, by giving a few simple concrete examples in the plane before launching into purely rigorous math. (E.g., the figure showing an unnamed (but triangular) lattice gives no origin and no point coordinates at all. This is likely to give the mistaken impression that a lattice is nothing but a wallpaper pattern.)
Not to go off on a rant here, but the first sentence of the article reads:
"In mathematics, especially in geometry and group theory, a lattice in Rnis a discrete subgroup Rn of which spans the real vector space Rn." But where will one find what it means for a discrete subgroup of a group to "span" the whole group? It would be far, far clearer if it just said the discrete subgroup is isomorphic to Zn. And to give an example like the subgroup of Rn generated by (1,0) and (√2,0) to show why the "discrete" condition is necessary to obtain the intended concept. Daqu (talk) 14:00, 24 July 2012 (UTC)

## In Lie groups

I have written an article on lattices in Lie groups (and generalizations), which largely eliminates the need for a section on this subject here (modulo redirects). If there are no objections, I will remove this section entirely, it does not fit with the rest of the article. Arcfrk (talk) 09:49, 16 February 2008 (UTC)

## moved comment from article space

We need a disambiguation statement as 'lattice' is used in groups and elsewhere in mathematics in two senses, one (a) as an ordered set, and (b) as a regular spacing (namely, a trellis). We may talk of the lattice (a) of all sublattice (b) stabilizers in the case of the symmetries of the Leech lattice (b).

66.130.86.141 (talk) 04:30, 16 April 2009 (UTC)John McKay

There is one already, see the first line of the article. McKay (talk) 05:12, 16 April 2009 (UTC)

## Erroneous reference to "the" fundamental domain of a lattice

One section of the article reads in part:

"If one thinks of a lattice as dividing the whole of into equal polyhedra (copies of an n-dimensional parallelepiped, known as the fundamental region of the lattice), then d(Λ) is equal to the n-dimensional volume of this polyhedron. This is why d(Λ) is sometimes called the covolume of the lattice."

There is no assignment of a specific n-dimensional parallelepiped that serves as a natural fundamental domain of a lattice, that is a smooth function on the space of lattices and is invariant under rotations. (This is easily seen from the existence of the Eisenstein lattice in 2D, generated by the complex numbers {1, exp(2πi/3)}, because of its 6-fold rotational symmetry.) For this reason it is wrong to refer to "the fundamental domain" of a lattice. (Note that any lattice also has infinitely many non-congruent choices of its set of generating vectors, and hence infinitely many non-congruent parallelepipeds that can each serve as a fundamental domain.)

It should also be noted that there are many fundamental domains for lattices that are *not* parallelepipeds. So the parenthetical phrase quoted above should not be parenthetical, either: If one wants a parallelepiped as fundamental domain, it is necessary to specify that non-parenthetically. (E.g., using the Voronoi region of the origin of the Eisenstein lattice as fundamental domain, it is a regular hexagon.)Daqu (talk) 06:35, 10 July 2012 (UTC)

## Points or vectors

Most of the article suggests the second, but the drawing for instance at the beginning displays points. Of course, the relation is simple, but it is still confusing (and linked to the preceding issue). All the best, --Cgolds (talk) 15:14, 29 March 2013 (UTC)

## Intentionally identical examples for 2D lattices?

Are p6m and p3m1 supposed to have identical layouts? The dots are in the exact same locations. 75.70.89.124 (talk) 06:03, 31 July 2013 (UTC)