Questions tagged [lattices]

Lattices in the sense of discrete subgroups of Euclidean spaces, as used in number theory, discrete geometry, Lie groups, etc. (Not to be confused with lattice theory or lattices as used in physics! For lattices (ordered sets), use the tag: [lattice-theory])

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Minimal volume of fundamental domains of lattices

Consider a full rank integer lattice in $\mathbb{R}^n$. Let $v_1$ be the shortest non-zero vector in the lattice, $v_2$ be the shortest one among those not parallel to $v_1$, $v_3$ be the shortest one ...
Yuhang Liu's user avatar
4 votes
1 answer
308 views

Computing the genus of certain ternary indefinite lattices

For $k$ a positive integer, let us consider the rank 3 ternary indefinite lattice $L=L_k$ with quadratic form $$6kx^2-2(y^2+yz+z^2).$$ Its discriminant group has length $2$. Question. Is this lattice ...
X77 Math19's user avatar
3 votes
0 answers
85 views

Volume and basis for integer lattice subject to sparse constraint

Let $k<n$ be integers. Let $A\in \mathbb{Z}^{k \times n}$ be a sparse matrix, meaning that the number of nonzero entries in every row and every column is at most $O(1)$. Further, assume that ...
Matt Hastings's user avatar
2 votes
0 answers
133 views

density of lattices

I'm looking for references pertaining to the remark at the bottom of p.18 of Conway-Sloane, "Sphere Packings, Lattices and Groups" (3rd ed), henceforth referred to as "SPLG". First,...
W Sao's user avatar
  • 509
2 votes
1 answer
264 views

Stabilizer in $G(\mathbb{Z})$ of point in fundamental domain $G(\mathbb{Z}) \backslash G(\mathbb{R}) / K$

Let $G$ be a semisimple group (the cases of primary interest to me are where $G$ is a special linear group or a special orthogonal group), let $K$ be a maximal compact subgroup of $G(\mathbb{R})$, and ...
Ashvin Swaminathan's user avatar
4 votes
0 answers
551 views

Lattices of $\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$

Edit: Thoughts updated (22/3/2021). I've come across with the following problem. Let $G=\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$ where $\varphi:\mathbb{R}^s\to \mathrm{Aut}(\mathbb{R}^k)=\mathsf{GL}(...
Alejandro Tolcachier's user avatar
1 vote
0 answers
40 views

About weight generator of quadratic lattices

I get stuck on Example 93:5 in O'Meara's book "Introduction to quadratic forms", where it is explained a method of find a weight generator of a quadratic lattice. In this question I assume ...
Yong Hu's user avatar
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104 views

Computer program which computes the automorphism group of Gram Matrix of lattice?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Fixed $n \geq 2$, given $K \in \GL(n,Z)$. One can view $K$ is a Gram matrix of Lattice. I also imposed that $K$ is symmetric i.e $K^{T}=K$. We ...
en kuo's user avatar
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123 views

Number of lattice points in a structural symmetric convex body

Let $f$ is a convex symmetric function on the interval $[-a,a]$, i.e., $f(-x)=f(x)$ for $\forall \, x\in [-a,a]$. Then we consider a $n$-dimensional convex body in Euclidean space \begin{equation} \...
RyanChan's user avatar
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2 votes
1 answer
115 views

Anchor sets for lattice polygons: Part I

Suppose $V=\{(x_1,y_1), (x_2,y_2),\dots,(x_v,y_v)\}$ is a vertex set of lattice points satisfying $$0=x_1<x_2<\dots<x_v \qquad \text{and} \qquad y_1>y_2>\cdots>y_{v-1}>y_v=0.$$ ...
T. Amdeberhan's user avatar
4 votes
1 answer
389 views

When does a subgroup of $\operatorname{GL}(n, \mathbb Q)$ have a bounded fundamental domain on $\mathbb R^n$?

$\DeclareMathOperator\GL{GL}$Let $G \subset M_{n\times n~}(\mathbb Z)$ be a finitely generated subgroup of $\GL(n,\mathbb Q)$ (i.e. $g\in G$ is an invertible matrix with entries in $\mathbb Z$). Then $...
Li Yutong's user avatar
  • 3,362
7 votes
1 answer
349 views

Vectors with minimal Hamming weight in a rational vector space?

Suppose given $n\ge 1$ and a subspace $U$ in $\mathbb{Q}^n$. It is given as $\mathbb{Q}$-span of certain known vectors. For $x \in U$, we let the Hamming weight of $x$ be the number of its nonzero ...
Matthias Künzer's user avatar
3 votes
0 answers
130 views

Understanding fusion categories from the perspective of anyons?

ncatlab defines a fusion category as "A fusion category is a rigid semisimple linear (Vect-enriched) monoidal category (“tensor category”), with only finitely many isomorphism classes of simple ...
pyroscepter's user avatar
2 votes
0 answers
90 views

Sublattices in the standard integral symplectic lattice

Let $V$ denote $\mathbb{Z}^{2g}$ with its standard integral symplectic form $\omega = \sum_{i=0}^{g-1}dx_{2i} \wedge dx_{2i+1}$ (or, the homology lattice of a genus $g$ surface with its intersection ...
Rodion N. Déev's user avatar
0 votes
1 answer
108 views

Ordering preserved by an inverse frame homomorphism

Recall that a frame homomorphism $h:L\to M$ is called ($L$ and $M$ are frames): Dense if, for any $x ∈ L$, $h(x) = 0$ implies $x = 0$. Codense if, for any $x ∈ L$, $h(x) = 1$ implies $x = 1$. ...
Biller Alberto's user avatar
2 votes
1 answer
78 views

How can I construct an element in a particular "quadrant" of a lattice (preferably short)?

Given a basis for a full-rank lattice $\mathcal{L} \subset \mathbb{R}^n$ I want to find a vector with totally positive entries, in other words an element belonging to $\mathcal{L} \cap Q$ where $Q$ is ...
wyoumans's user avatar
  • 287
2 votes
0 answers
115 views

How much Gleason type theorem do I need? Quasi states vs. states

Let $\varphi$ be a quasi state on $B(H)$. What does it mean? It means that $\varphi(cA)=c\varphi(A)$ for $c \in \mathbb{C}, A \in B(H)$, $\varphi(A) \geq 0$ for positive $A$ and $\varphi(A+B)=\varphi(...
truebaran's user avatar
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2 votes
1 answer
104 views

Bound on mutually x-ray-visible lattice points?

Say that two lattice points $a$ and $b$ of $\mathbb{Z}^d$ are $x$-visible to one another if the segment $ab$ contains at most $x$ lattice points (excluding $a$ and $b$). So $x$-visiblity is "x-...
Joseph O'Rourke's user avatar
3 votes
1 answer
282 views

There are at most four mutually visible lattice points—?

Say that two lattice points $a$ and $b$ of $\mathbb{Z}^2$ are visible to one another if the line segment $ab$ contains no other lattice points. While exploring lattice polygons all of whose vertices ...
Joseph O'Rourke's user avatar
2 votes
0 answers
128 views

A number theoretic unlikely intersection phenomenon

Let $h_1, \cdots, h_k$ be co-prime integers, $k \geq 3$. Let $n$ be a square-free positive integer, understood to be extremely large compared to the $h_i$'s, and such that $\omega(n)$, the number of ...
Stanley Yao Xiao's user avatar
0 votes
0 answers
287 views

Efficiently find at least one lattice point on hyperbola of equation $axy+bx+cy+d=0$

I need an algorithm to efficiently find at least one lattice point on a hyperbola of equation $axy+bx+cy+d=0$. Lattice point means integer coordinates and equation with integer means diophantine ...
user2548436's user avatar
9 votes
0 answers
250 views

Conway big picture for congruence subgroups of $\mathrm{SL}_3(\mathbb{Z})$

I saw in Conway’s paper "Understanding groups like $\Gamma_0(N)$" that the so-called Big Picture can give simple interpretations for important objects in number theory, such as Hecke ...
Radu T's user avatar
  • 767
2 votes
1 answer
231 views

Source on counting lattice points on a line

Looking for a book or article on the result linked below. The result tells us that the number of lattice points on a line between points $(a,b)$ and $(c,d)$ is given by $\gcd(a-c,b-d)+1$. https://math....
user6232872's user avatar
4 votes
0 answers
125 views

Deep Holes of the tensor product of two lattices

Let $L_0, L_1$ be Euclidean lattices (say full rank) of dimension $n_i$. Let $\lambda_1(L_i)$ denote the length of the shortest vector of $L_i$, and let $\rho(L_i)$ denote the covering radius of $L_i$:...
Mark Schultz-Wu's user avatar
3 votes
1 answer
240 views

Deciding isometry of unimodular lattices by Gram matrices

Say I have two unimodular lattices $A$ and $B$, represented by their Gram matrices. Question: Is there an algorithm to decide whether $A$ and $B$ are isometric, i.e. whether there exists a matrix $S \...
LeechLattice's user avatar
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1 vote
1 answer
115 views

Property of convex polygons on integer lattice structures

Another graduate student and I are working on an research project and are looking for a paper or other source that has a proof for a result about polygons on an integer lattice structure. Suppose you ...
user6232872's user avatar
2 votes
1 answer
124 views

Do discrete embeddings of surface groups not necessarily carry an embedding of SL_2?

We can get arithmetic lattices isomorphic to free groups in $\mathrm{SL}_2\mathbb{R}$, so in general we can’t expect homomorphisms of lattices into semisimple Lie groups to say much about $\mathrm{SL}...
Robin Goodfellow's user avatar
11 votes
0 answers
158 views

Characterization of certain 4-dimensional lattices

Let $\Lambda \subset {\bf Q}^4$ be a lattice, i.e., $\Lambda$ is a free abelian group and $\Lambda \otimes {\bf Q} = {\bf Q^4}$. The determinants of those dilation-rotations (i.e. linear maps of ${\bf ...
Jens Reinhold's user avatar
3 votes
1 answer
280 views

number of integer points inside a triangle and its area

Let $T$ be a triangle in $\mathbb{R}^2$ defined by $y = \alpha x$, $y = \beta$ and $x = \gamma$ where $\alpha, \beta, \gamma \in \mathbb{R}_{>0}$. I am interested in obtaining an estimate for the ...
Johnny T.'s user avatar
  • 3,547
3 votes
0 answers
92 views

Counting integral points in a rank-2 lattice

Let $\Lambda \subset \mathbb{Z}^n, n \geq 2$ be a lattice of rank 2. It is well-known (see for example Lemma 4.5 in this book) that $$\displaystyle \#\{\mathbf{x} \in \Lambda : \mathbf{x} \text{ is ...
Stanley Yao Xiao's user avatar
5 votes
0 answers
280 views

Matrix groups with two generators

Given two matrices $A,B\in{\rm{SL}}_2(\Bbb{R})$, is there any criterion guaranteeing that the subgroup they generate is discrete? What if one puts restrictions on $A,B$ e.g. they are both elliptic? ...
KhashF's user avatar
  • 2,588
7 votes
0 answers
116 views

Theta Function Associated to Kummer Lattice

This is something which I feel must be out in the literature somewhere, but I have been unable to find anything. If we let $\text{Km}(A)$ be the Kummer $K3$ surface associated to an abelian surface $A$...
Benighted's user avatar
  • 1,701
1 vote
0 answers
216 views

Does the standard arithmetic subgroup of a closed $\mathbb{Q}$-algebraic groups have non-trivial $\mathbb{Q}$-characters?

I am trying to understand the Borel-Harish Chandra theorem about arithmetic subgroups being lattices. Suppose $G$ is an algebraic group inside $GL_n(\mathbb{C})$ such that it is definable as a zero ...
Breakfastisready's user avatar
2 votes
2 answers
404 views

lattice suprema vs pointwise suprema

What is the difference between the lattice supremum and the pointwise supremum of a family of functions? I mean, given a family of real valued functions $\mathcal{F}$, is the function $\sup\mathcal{F}:...
Giuliosky's user avatar
9 votes
1 answer
276 views

Euler characteristic with compact support of spaces of Euclidean lattices

Has the Euler characteristic with compact support of $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ been computed ? References? Thanks.
sadok kallel's user avatar
6 votes
1 answer
318 views

Can all proper sublattices of $\mathbb{Z}^n$ be generated cyclically?

Let $\Lambda \subset \mathbb{Z}^n$ be a proper sublattice (so that $\Lambda \ne \mathbb{Z}^n$). We say that $\Lambda$ is cyclically generated if there exists a matrix $M \in \text{GL}_n(\mathbb{Z})$ ...
Stanley Yao Xiao's user avatar
9 votes
0 answers
447 views

A lattice with Monster group symmetries

The book Mathematical Evolutions contains the following excerpt: A last, famous, example is the following. It is known that in the space of one hundred and ninety six thousand eight hundred and ...
Adam P. Goucher's user avatar
4 votes
0 answers
77 views

Getting more out of Minkowski's convex body theorem in the case of non-convex bodies

Problem. In number theory one generally proves the finiteness of the Picard group of a number field using Minkowski's convex body theorem. The actual body $S_p$ of interest in the proof, depending on ...
MadPidgeon's user avatar
52 votes
5 answers
8k views

Why do bees create hexagonal cells ? (Mathematical reasons)

Question 0 Are there any mathematical phenomena which are related to the form of honeycomb cells? Question 1 Maybe hexagonal lattices satisfy certain optimality condition(s) which are related to it? ...
Alexander Chervov's user avatar
3 votes
2 answers
346 views

Given a lattice in $\mathbb{Z}^n$, what can be said about its 'transpose' lattice?

I apologize if this notion is well-known, but I couldn't find anything useful and I am not sure what key words to look for. Suppose we have a lattice $\Lambda \subset \mathbb{Z}^n$, given by in the ...
Stanley Yao Xiao's user avatar
1 vote
1 answer
197 views

Lattice points in hypercubes

Let $ (\Lambda_n) $ be a family of lattices, $ \Lambda_n \subset \mathbb{Z}^n $, with $ \det\Lambda_n \sim n $ as $ n \to \infty $ (meaning $ \lim_{n\to\infty} n^{-1} \det\Lambda_n = 1$). I am ...
aleph's user avatar
  • 503
5 votes
1 answer
173 views

Finding a superbase in a lattice of Voronoi first kind

An $n$-dimensional lattice in $\mathbb R^n$ is said to be of Voronoi’s first kind if it there exists $n+1$ vectors $b_1,\cdots b_{n+1}$ (called the superbase) such that $\{b_1,\ldots,b_n \}$ is a ...
user avatar
4 votes
1 answer
263 views

Number of points in a lattice and an oblong box

I have a very simple question in geometry of numbers. (It is a slight modification of Counting points on the intersection of a box and a lattice .) There's a bound I can easily prove, and it's good ...
H A Helfgott's user avatar
  • 19.3k
8 votes
1 answer
263 views

An integral Jacobson-Morozov theorem?

$\DeclareMathOperator\SL{SL}$I want to ask if there exists a version of the Jacobson–Morozov theorem for integer matrices. A first approximation would ask: given an integral unipotent matrix $m \in \...
Nate's user avatar
  • 1,992
2 votes
1 answer
110 views

Reference request: placing a set with respect to the integer grid

For $x=(x_1,...,x_n)\in \mathbb{R}^n$, let $Q_x=(x_1,x_1+1)\times ...\times (x_n,x_n+1)$ - the open cube having $x$ in its "bottom left" corner. It seems, I can prove (see a draft here) the following ...
erz's user avatar
  • 5,385
7 votes
2 answers
828 views

what is the number of paths returning to 0 on the hexagonal lattice

I am looking for an estimation of the number of paths of length $n$ going from 0 to 0 on the hexagonal (or honeycomb) lattice. I can find plenty on references on self avoiding paths, but I am looking ...
kaleidoscop's user avatar
  • 1,268
7 votes
2 answers
392 views

Homomorphisms from higher rank lattices with infinite center to $\mathbb{Z}$

Suppose that $\Gamma$ is an irreducible lattice in a semi-simple real Lie group $G$ of higher rank (with infinite center!), is every homomorphism $\Gamma \to \mathbb{Z}$ trivial? The case where $G$ ...
shurtados's user avatar
  • 1,010
0 votes
1 answer
74 views

Small linear relations in unbalanced diophantine equations from primitive Pythagorean triples

$r$ is parameter. Pick coprime $m,n\in[r,2r]$ with $mn$ even. Consider the Linear Diophantine Equation $$a^4u+b^4v+c^2z=0$$ where $a=m^2-n^2$, $b=2mn$ and $c=m^2+n^2$. Is it true that there are ...
VS.'s user avatar
  • 1,816
0 votes
0 answers
153 views

Upper bounds on the length of the shortest vector in lattices associated to polynomial congruences

We consider a lattice $\Lambda \subset \mathbb{Z}^2$, and put $\lambda_1(\Lambda), \lambda_2(\Lambda)$ for successive minima of the lattice $\Lambda$. By a well-known theorem of Minkowski, one has ...
Stanley Yao Xiao's user avatar
2 votes
0 answers
123 views

Another generalization of the Gauss circle problem

In this question I asked for a generalization of the Gauss circle problem, the type of generalization I am asking is to view the Gauss circle problem as one about counting algebraic integers of ...
Stanley Yao Xiao's user avatar

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