All Questions
Tagged with lattices co.combinatorics
67 questions
2
votes
1
answer
324
views
Lattice automorphisms of finite order
Are there any known examples of lattice automorphisms of finite order in indefinite lattices being classified up to conjugacy?
CommunityBot
- 1
2
votes
0
answers
89
views
Group actions on polytopes in indefinite integer lattices
Is anything at all known about polytopes in indefinite integer lattices? I'm interested in lattice automorphisms which preserve certain polytopes of "high regularity" (e.g. cones). As a first step, I'...
- 393
3
votes
1
answer
607
views
Automorphism groups of indefinite non-unimodular integer lattices
Does anyone know of any papers in which structural aspects of the orthogonal group of some indefinite non-unimodular integral lattice are calculated? The exact lattice isn't so important and they don'...
CommunityBot
- 1
1
vote
1
answer
152
views
Submodular measures on the hypercube
By the hypercube I mean the lattice formed by all n-bit strings ordered by pointwise inequality. For example, $000 \leq 110$, $010 \leq 110$, $110$ and $001$ are not comparable. Further we have the ...
- 655
5
votes
2
answers
635
views
Area of a lattice polygon in terms of its width
Let $M$ be a lattice polygon on a plane (i.e. its vertices are integer points $(i,j)\in\mathbb Z^2$).
Let us define lattice width in a direction $v=(m,n)\in\mathbb Z^2$ as $w_v(M)=\max\limits_{x,y\in ...
- 5,063
5
votes
0
answers
324
views
Lattice points inside a (n-dimensional) tetrahedron
Hi, overflowers.
I was interested in a sharp lower bound for the number of lattice points (say, integral lattice points) inside the tetrahedron defined by the coordinate hyperplanes and $x_1/a_1+...+...
- 306
4
votes
2
answers
323
views
Cancellation theorem for lattices
By a lattice, we mean a finitely generated, free $\mathbb{Z}$-module together with a symmetric bilinear form. Typical examples are the hyperbolic lattices $U$ and the root lattices $A_{n}, D_{n}, E_{n}...
CommunityBot
- 1
1
vote
3
answers
470
views
Triangulations of lattice polygons
Let L be a 2-dimensional lattice and P- a lattice polygon. Suppose, it is triangulated into lattice tiangles. What are restrictions on their areas? For instance, can a lattice triangle of even area ...
0
votes
0
answers
113
views
Question regarding contiguous forms
I read about contiguous forms in Achill Scürmann's thesis on positive quadratic forms. I am wondering about one aspect of the Voronoi algorithm presented in there, that enumerates all arithmetically ...
- 149
7
votes
2
answers
963
views
Maximal number of edges and triangular cells for n points in a triangular lattice
Consider a subset of $n$ points in an equilateral triangular lattice. Draw all the edges between nearest-neighbor points.
What is the maximum, over all such subsets, of the number of edges? This ...
- 44.4k
6
votes
1
answer
451
views
Orthogonal Complements of Root Lattices in E_8
I have a rather stupid lattice theory question. Suppose $L$ is a root lattice that can be primitively embedded in the $ E_8 $ lattice. Is the orthogonal complement of $ L$ in $E_8$ unique up to ...
- 309
10
votes
1
answer
595
views
Condition for existence of certain lattice points on polytopes
Let $a_1,\cdots, a_n$ be integers such that $a_i\geq 2$ for all $i$ and $k>0$ another integer.
I am interested in whether there exist integers $x_1,\cdots, x_n$ with $0<x_i<a_i$ satisfying:
...
CommunityBot
- 1
4
votes
0
answers
242
views
Domination in Nice Lattices
Let an integer vector be nice when it has only two nonzero components, which sum to zero. So (0, 0, 3, 0, -3) and (-1, 0, 1, 0, 0) are examples of nice vectors in $n=5$ dimensions.
Call a lattice ...
CommunityBot
- 1
12
votes
3
answers
707
views
A "round" lattice with low kissing number?
Historically, the lattices with high density were studied intensively, e.g. E_8 lattice or Leech Lattice. However, there are situations that lattices with low kissing number are required. Specifically,...
CommunityBot
- 1
0
votes
2
answers
611
views
10
votes
1
answer
803
views
Which lattices have more than one minimal periodic coloring?
The lattice $\mathbb{Z}^n$ has an essentially unique (up to permutation) minimal periodic coloring for all $n$, namely the "checkerboard" 2-coloring. Here a coloring of a lattice $L$ is a coloring of ...
CommunityBot
- 1
0
votes
1
answer
241
views
Constructing a smooth lattice from a discrete one.
I have the standard lattice L defined over partitions of $1\ldots n$ under the split-merge relation. I also have an antimonotone function from L to R that's submodular, and so gives me a metric on L ...
- 1,036