Lattices as they are used in number theory. (Not to be confused with lattice theory or lattices as used in physics!)

learn more… | top users | synonyms

5
votes
1answer
117 views

Is the braid group with $n$ strings $\mathcal{B}_n$ a lattice in a connected semi-simple Lie group?

Is the braid group with $n$ strings $\mathcal{B}_n$ known to be a lattice in a connected semi-simple Lie group ? (for $n$, say, bigger than $3$) Or is it known that it cannot be such a lattice ?
4
votes
1answer
394 views

A dual version of a theorem of Øystein Ore in group theory

Let $(H \subset G)$ be an inclusion of finite groups. This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved: Theorem: $\mathcal{L}(H\subset G)$ ...
0
votes
0answers
46 views

Hexagonal lattice in a disk when the distance between points is $R_l$ [on hold]

Consider a hexagonal tiling of a 2D plane where hexagons are of identical size and of radius $R_l$. I assume we can say that the vertices together with the center of each hexagon form an integer ...
5
votes
2answers
186 views

Ordered lattice point enumeration

I initially asked this question over at StackOverflow as it has algorithmic flavor to it, but I haven't been getting much traction so I thought I would probe the mathematics community. Setup: Let ...
4
votes
1answer
115 views

Is the product of two supermodular functions supermodular?

The definition of Supermodularity is that for every $x′>x$ and $y′>y$, we have \begin{equation*} f(x′,y′)+f(x,y)>f(x′,y)+f(x,y′). \end{equation*} Suppose $f$ and $g$ are supermodular, ...
0
votes
0answers
58 views

mean value estimate $ \frac{1}{T} \int_0^T | \zeta( \tfrac{1}{2} + it)|^2 \, dt = \log \frac{T}{2\pi } + (2\gamma - 1) + O(T^{\delta})$

I was browsing through the recent paper of Bourgain and Watt on mean value estimates of the Riemann zeta function on the critical line. $$ \frac{1}{T} \int_0^T | \zeta( \tfrac{1}{2} + it)|^2 \, dt = ...
5
votes
1answer
126 views

Number of matrices with given Smith normal form

Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...
9
votes
2answers
908 views

The Gauss circle problem on a hexagonal lattice

Take an infinite hexagonal lattice (or equivalently, an equilateral triangular lattice), with unit spacing between the closest lattice point pairs, and draw a disc of radius $r$ centered on a lattice ...
10
votes
1answer
511 views

Inequality regarding sum of gaussian on lattices

When S is a subset of an inner product space, let d(S) denote ${\sum\limits_{s \in S} e^{- \langle s,s \rangle}}$ Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, ...
5
votes
1answer
130 views

Bounding the number of lattice points inside an $n$-dimensional ellipsoid

I am wondering if it is possible to produce an upper and/or lower bound on the number of integer lattice points that lie inside an $n$-dimensional ellipse. That is, given an $n$-dimensional ellipsoid ...
4
votes
2answers
174 views

Counterexample on completely distributive lattices

I would like to see an example of a complete lattice $C$ which is both a frame and a dual-frame, i.e. finite meets distribute over arbitrary joins and finite joins distribute over arbitrary meets ...
54
votes
5answers
2k views

How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?

This question is about the space of all topologies on a fixed set X. We may order the topologies by refinement, so that τ ≤ σ just in case every τ open set is open in σ. ...
-2
votes
2answers
70 views

does a lattice have a minimal item [closed]

A lattice (L, ≤) is a partially ordered set, where each two elements of the set have a least upper bound and a greatest lower bound. Let's consider the situation where L is finite. I think that ...
2
votes
1answer
40 views

covering radius of a lattice from cyclotomic extension

Given a finite field extension $L$ of $\mathbb{Q}$ of dimension $n$, there is a natural way to embed it into $\mathbb{R}^n$ such that the image of its ring of integers $\mathcal{O}_L$ is a lattice. If ...
5
votes
0answers
174 views

When do two lattices have the same stabilizer in the diagonal torus?

This is moved from MSE, where I asked and didn't receive an answer (see http://math.stackexchange.com/questions/1145151/lattices-in-mathbbq-pn-with-the-same-stabilizer) Let $T$ be the diagonal torus ...
1
vote
1answer
224 views

Existence of homogeneous single chain compositions of a given maximal subfactor?

All the subfactors here are irreducible inclusion of hyperfinite II$_1$ factors. A subfactor $(N \subset M)$ is Homogeneous Single Chain ($HSC$) if its lattice of intermediate subfactors is a ...
10
votes
2answers
377 views

Inequalities for averaging over partially ordered sets

Let's start from a classical inequality: If $0\le a_1\le\cdots\le a_k$ and $0\le b_1\le\cdots\le b_k$ then $(a_1+\cdots+a_k)(b_1+\cdots+b_k)\le k(a_1b_1+\cdots+a_k b_k)$. It can be written also in ...
10
votes
3answers
395 views

Is a distributive lattice planar iff it admits no B3 sublattice?

A finite lattice is planar if it admits a Hasse diagram which is a planar graph (we want the Hasse diagram to be represented in the plane so that the $y$-coordinate of each element respects the ...
1
vote
1answer
120 views

Uniqueness of minimal completions of a partially ordered set

The partially ordered set $(Y,\le)$ is called a superspace of the partially ordered set $(X,\le')$ iff $X\subseteq Y$. $\le'=(\le\cap X^2)$. and a completion of $(X,\le')$ if in addition $~~ 3$. ...
8
votes
2answers
903 views

Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive. Proof: see below. Let $(H \subset G)$ be an inclusion of finite groups and ...
4
votes
1answer
122 views

Can any finite distributive weighted lattice be realized by inclusion of groups?

By theorem 2.1 here, any finite distributive lattice $\mathcal{L}$ can be realized as an intermediate subgroups lattice. A weighted lattice $(\mathcal{L},\tau)$ is a lattice $\mathcal{L}$ with a ...
2
votes
1answer
236 views

Implementation of the bounded-distance decoder of Leech-lattice?

Hi all, I am wondering, anybody can help me how can I find an implemented version of Leech-Lattice quantizer/decoder, i.e., "Matlab", "C++" or "Python" code, using the approach proposed by Ofer ...
32
votes
16answers
2k views

Counterexamples in universal algebra

Universal algebra - roughly - is the study, construed broadly, of classes of algebraic structures (in a given language) defined by equations. Of course, it is really much more than that, but that's ...
0
votes
0answers
33 views

kernel lattice example

Could anyone give an example of the following? Suppose that $A \in \mathbb{Z}^{m\times n}$, where $m \leq n^{1-\epsilon}$ for some $\epsilon > 0$. The entries of $A$ have size $poly(n)$, meaning ...
3
votes
1answer
173 views

n-dimensional Delaunay Triangulation of Lattices

I have several questions concerning the Delaunay triangulation of a high dimensional lattice. Given an $n$-dimensional lattice $L$ and its Delaunay triangulation (partition of $R^n$ into simplices ...
2
votes
1answer
103 views

lattice orthogonal complement

Let $A\in \mathbb{Z}^{m\times n}$ ($m<n$) be a matrix with orthogonal rows. Further assume that the gcd of the coefficients in each row of $A$ is $1$. Consider $\ker A\cap \mathbb{Z}^n = ...
1
vote
0answers
77 views

Are lattices in the special real linear group subgroup seperable?

Let $G \leq SL_2(\mathbb{R})$ be a lattice, let $H \leq G$ be a finitely generated subgroup of infinite index, and let $n \in \mathbb{N}$. Must there be some $H \leq U \leq G$ such that $n \leq [G : ...
-1
votes
1answer
159 views

How to recognize if a lattice is distributive? [closed]

I know that a Boolean lattice must be distributive. But what with these lattices? Are these distributive? $\hskip0.7in$ How to recognize which lattices are distributive or not only by looking on ...
11
votes
6answers
1k views

Giving $Top(X,Y)$ an appropriate topology

I am not sure if its OK to ask this question here. Let $Top$ be the category of topological spaces. Let $X,Y$ be objects in $Top$. Let $F:\mathbb{I}\rightarrow Top(X,Y)$ be a function (I will ...
6
votes
1answer
287 views

Are the distributive permutation groups linearly primitive?

An action of a group $G$ on a set $X \neq \emptyset$ is called transitive if $\forall x,y \in X$, $\exists g \in G$ such that $g.x = y$. It is called primitive if it is transitive and preserves no ...
1
vote
1answer
123 views

basis of the lattice generated by the integer points inside a subspace of R^L

Consider $K$ linearly independent vectors $\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_K \in \mathbb{Z}^L$, where $1 \leq K<L $. Hence, the span of $\lbrace\mathbf{a}_1, \mathbf{a}_2, ..., ...
0
votes
0answers
152 views

Number of lattice points in a given triangle

Given a triangle with real coordinates, does anybody know how to find the number of lattice points contained within it? What if the points are only rational? I know Pick's formula can be used for the ...
2
votes
1answer
500 views

Abelian subfactors, a relevant concept?

Through the questions below, this post asks whether the concept of abelian subfactor is relevant. Remark : here abelian qualifies an inclusion of II$_1$ factors $(N \subset M)$, $N$ is not an abelian ...
2
votes
2answers
217 views

Is certain topology-related set a distributive lattice?

In my research the following problem appeared (and if it is true, this solves positively several my conjectures): Let $U$ be a fixed set (usually $U$ is infinite). Let $n$ be a fixed index set ($n$ ...
0
votes
1answer
57 views

Lower bounds on the rank of a unimodular lattice, given the binlinear pairing of a subset of basis vectors

I have some questions on lower bounds on the rank of unimodular lattices given the bilinear pairing of a subset of its basis is known. Let $\Lambda$ be an odd, unimodular matrix of signature ...
9
votes
4answers
578 views

Applications of n-dimensional crystallographic groups

I would like to know what are the applications of the theory of $n$-dimensional crystallographic groups (aka space groups) 1) in mathematics 2) outside of mathematics, besides the applications to ...
2
votes
0answers
108 views

$\mathbb{Q}$-forms of $\mathrm{SL}_2\times \mathrm{SL}_2$

I am learning something about lattices in algebraic groups. Consider the algebraic group $\mathrm{SL}_2\times \mathrm{SL}_2$. What are the $\mathbb{Q}$-forms of such groups?
3
votes
2answers
141 views

Number of *distinct* dot products of an integer vector by elements of a hyper-rectangle

Imagine a vector $\boldsymbol{v}$ composed of integers, and the set $S$ of all integer vectors within a hyper-rectange, with one corner at the origin and other at $\boldsymbol{m}$. In other words: $S ...
18
votes
6answers
693 views

Why is the right permutohedron order (aka weak order) on $S_n$ a lattice?

This is one of those things I never expected to be hard until I tried to prove it. Why is the right permutohedron order (a.k.a. weak Bruhat order, a.k.a. weak order -- not to be confused with the ...
4
votes
3answers
475 views

Product-Decomposition of distributive lattices

EDIT I now (strongly) believe that the following claim answers my question (see the text below). However, if it does, then I am sure that it is known. It is not difficult to prove and the question ...
5
votes
1answer
238 views

Given a positive-definite integral unimodular Gram matrix, how to find a basis of the associated lattice (over $\mathbf Q$)?

Let $G$ be a $n\times n-$symmetric matrix with integral coefficients and determinant $1$ (i.e. unimodular) such that the associated quadratic form is positive-definite. I am interested in having an ...
13
votes
3answers
621 views

Conjecture regarding closest point inside a discrete ball to a line

I'm a PhD student in image processing, where I've stumbled into a problem that seems to be essentially number theory. I've hunted around online and while I've found many results on similar problems, ...
6
votes
1answer
125 views

Translative packing constant strictly larger than lattice packing constant

Simply put, my question is this: what is the smallest dimension, if any, where we can know for sure that a convex body exists whose translative packing constant is strictly larger than its lattice ...
3
votes
2answers
192 views

Cohomology of SL(2,R) with coefficients given by linear action

Let $SL(2,{\mathbb R})$ act on ${\mathbb R}^2$ by matrix multiplication. What is known about group cohomology $H^*(SL(2,{\mathbb R}),{\mathbb R}^2)$? And about $$H^*(\Gamma,{\mathbb R}^2)$$ for a ...
5
votes
2answers
199 views

Decomposing polyhedral cones into “direct sums” and a polynomial

This question consists of two parts. I'm not breaking it up into two separate ones because posing the second question would essentially require me two rewrite the first one. Also, to some extent, the ...
2
votes
0answers
77 views

Generators for lattice polytopes

Let $V$ be a finite set of integral vectors in $\mathbb{R}^n$. Consider the real (positive) convex cone $\mathbb{R}_+V$ spanned by $V$. We can assume that all vectors in $V$ are extremal in ...
0
votes
2answers
76 views

Information needed to distinguish combinatorially isomorphic polytopes (up to affine equivalence)

I originally posted this question on Stack Exchange, thinking it perhaps does not qualify as "research-level" but it received no answers... hopefully someone here can help. The title pretty much ...
5
votes
0answers
102 views

Bound on the number of lattice points in d-dimensional ball

The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit ...
12
votes
0answers
215 views

Who first showed that $SL(n,O_K)$ is a lattice for a number ring $O_K$?

Let $O_K$ be the ring of integers in an algebraic number field $K$. Assume that $K$ has $r$ real embeddings and $s$ pairs of complex conjugate complex embeddings. There is then an injective ...