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!)

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2k 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 ...
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coloring in lattice

This is a mathematical question raised from engineering and physics: Is there some established mathematical approach in filling a physical lattice with some colored basis (black and white here)? For ...
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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 ...
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Products of maximal inclusions of finite groups with a non-obvious intermediate

Let $(H_1 \subset G_1)$ and $(H_2 \subset G_2)$ be core-free maximal inclusions of finite groups. Their product, the inclusion $(H_1 \times H_2 \subset G_1 \times G_2)$, admits four obvious ...
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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. ...
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Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
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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 ...
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Are the homogeneous single chain subfactors, Dedekind?

Background: See here and there. Recall that a subfactor is Dedekind if all its intermediate subfactors are normal. A subfactor $(N \subset M)$ is Homogeneous Single Chain (HSC) if its lattice ...
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1answer
647 views

Compactness and completeness in Gödel logic

The standard proof of the completeness theorem in first-order Gödel logic is based on a first-order countable language. I want to know that is there any proof of the completeness theorem in ...
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in search of a transformation between determinants

Motivated by this MO question. Consider the two matrices $A_n$ and $B_n$ with entries $\binom{2j}i$ and $\binom{n+1}{2j-i}$, respectively; for $1\leq i, \,j\leq n$. I can show $\det A_n=\det B_n=2^{\...
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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'...
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Integrating an n-fold Cauchy product of a Fourier series

I posted this on Math Stack Exchange one month ago, but did not receive any responses. The original question (in a simplified form) can be found here. Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be ...
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Good lattice theory books?

A recent answer motivated me to post about this. I've always had a vague, unpleasant feeling that somehow lattice theory has been completely robbed of the important place it deserves in mathematics - ...
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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 ...
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A group-theoretic perspective on Frankl's union closed problem

Here is a group theoretic phrasing of a special case of the union closed conjecture: Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half ...
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Polar body of a convex body that avoids a lattice

Let $K \subset {\bf R}^d$ be a symmetric convex body (an open bounded convex neighbourhood of the origin with $K = -K$) with the property that $K + {\bf Z}^d \neq {\bf R}^d$, i.e. the projection of $K$...
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Codes, lattices, vertex operator algebras

At the end of "Notes on Chapter 1" in the Preface to the Third Edition of Sphere packings, lattices and groups, Conway and Sloane write the following: Finally, we cannot resist calling attention to ...
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Why are two “random” vectors in $\mathbb R^n$ approximately orthogonal for large $n$?

I saw that two random independent vectors are approximately orthogonal in high dimensional space. How can I prove this? And is there an intuitive explanation? Thank you.
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Maximal number of maximal subgroups

Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ ...
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Is there an Ehrhart polynomial for Gaussian integers

Let $N$ be a positive integer and let $P \subset \mathbb{C}$ be a polygon whose vertices are of the form $(a_1+b_1 i)/N$, $(a_2+b_2 i)/N$, ..., $(a_r+b_r i)/N$, with $a_j + b_j i$ being various ...
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A Priori proof that Covering Radius strictly less than $\sqrt 2$ implies class number one

It turns out that each of Pete L. Clark's "euclidean" quadratic forms, as long as it has coefficients in the rational integers $\mathbb Z$ and is positive, is in a genus containing only one ...
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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, ...
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Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)

A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$. An arithmetic subgroup is called congruence if it contains a subgroup of type $\Gamma(N)$ for ...
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On (a generalization of) the Gauss Circle Problem

Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...
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Localic locales? Towards very pointless spaces by iterated internalization.

One can think of locales as (generalizations of) topological spaces which don't necessary have (enough) points. Of course when one studies locales, one "actually" studies frames, certain sorts of ...
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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 ...
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An interesting sum over lattice points in a large disk centered at the origin

Evaluate the the limit, as $r \rightarrow \infty $, of the sum $\displaystyle \sum \limits_{(m,n) \in D_r}$ $\displaystyle (-1)^{m+n} \over \displaystyle m^2 + n^2$, where $D_r$ denotes the closed ...
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Dense sphere packings which are not lattice packings

This question is about dense sphere packings in euclidean space $\mathbb R^n$. By a sphere packing I understand any arrangement of mutually disjoint solid open spheres in $\mathbb R^n$, all of the ...
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Embedding a brouwerian lattice into a boolean lattice

I have already asked a similar question at https://math.stackexchange.com/questions/470704/can-a-brouwerian-lattice-be-extended-into-a-boolean-algebra but have received no answer. Sorry, I ask a ...
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1answer
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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 $~~ 3$. $(...
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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,...
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Chromatic number of Voronoi diagrams of lattices

Let $L$ be a Euclidean lattice. Define a graph whose vertex set is $L$ and where two points $x,y\in L$ are declared to be adjacent whenever the cells of $x$ and $y$ in the Voronoi diagram of $L$ have ...
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Even unimodular lattices with root system $32 A_1$

I'm studying Venkov's proof of the classification of even unimodular rank 24 lattices, and it prompted the following question. For an even unimodular lattice $L$, let $R(L)= \{ x \in L : (x,x) =2\}$ ...
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orbits of automorphism group for indefinite lattices

I have a question about indefinite lattices. QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice, that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form, not necessarily ...
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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 ...
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3answers
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Finite index free subgroups of $\mathrm{SL}(3,\mathbb{Z})$

Does $\mathrm{SL}(n,\mathbb{Z})$ have a free subgroup of finite index for some $n \geq 3$? I know that $\mathrm{SL}(3,\mathbb{Z})$ has many free subgroups and that in the case of $\mathrm{SL}(2,\...
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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 non-...
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Divisor bound for $r_2$ off the origin

If $r_2(n)$ denotes the number of integer solutions to $a^2+b^2=n$, we have the "divisor bound" $r_2(n) = O(n^{\epsilon})$ for any $\epsilon>0$. Another way to state this is that the number of ...
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What kind of locally symmetric space is a rational sphere

Using Dehn Surgery, we can construct compact hyperbolic $3$-manifolds with vanishing Betti numbers $b_1=b_2=0$, i.e., a rational homology $3$-sphere. My question is the following. Is there other ...
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1answer
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Siegel's Mean Value Theorem by Rogers and Macbeath

I recently became engaged in the work of Siegel, Schmidt, Rogers, Macbeath regarding random lattices and geometry of numbers, e.g. Siegel proved that $$\int_{SL(n,\mathbb{R})/SL(n,\mathbb{Z})} \sum_{ ...
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What's the ratio of inclusions of finite groups with a distributive lattice?

Definition: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the normal core of $A$ in $B$. ...
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1answer
250 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 single ...
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601 views

24 vectors in Leech lattice having scalar product $\frac{1}{4}$ pairwise

Two vectors from Leech lattice - as defined on wikipedia - have scalar product $\pm 32,\pm 16, \pm 8$ or $0$. Do there exist 24 vectors having scalar product 8 pairwise ? When we consider unit vectors ...
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Is the “Ramond sector” invariant of a 3-framed lattice always divisible by 24?

For the purposes of this question, a rank-$r$ (integral) lattice is a full-rank discrete subgroup $L \subset \mathbb R^r$ such that $\langle \ell, \ell' \rangle \in \mathbb Z$ for all $\ell \in L$. It ...
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What's in the genus of the cubic lattice?

I'll write $\mathbf{Z}^n$ for the integral quadratic form $x_1^2 + \cdots + x_n^2$. For which values of $n$ is $\mathbf{Z}^n$ unique in its genus, i.e. isolated in Kneser's graph? In particular can ...
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Are there Type III codes with small but nonzero “index”?

Recall that a Type III code of rank $r$ is a linear subspace $C \subset \mathbb F_3^r$ which is self-dual for the standard inner product. (These occur only when $r$ is divisible by $4$.) Elements of $...
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Bounded version of linear and quadratic Hasse--Minkowski theorem

The Hasse-Minkowski theorem states that if $$Q(x_1,\ldots,x_n) = \sum_{i,j=1}^n a_{ij} x_ix_j$$ is a quadratic form with $a_{ij} \in \mathbb Z$ and $\det (a_{ij}) \neq 0$, then the equation $$Q(x_1,\...
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Is every Heyting algebra the Lindenbaum algebra of an intuitionistic first order theory?

This question comes after the comments in the recent related question Sigma-complete Lindenbaum algebras?, but in its current form is sufficiently different in my opinion, and so I decided to follow ...
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diameter of Voronoi cell of the lattice ? What about R^n ? What about small n =2,3,4 ?What about random lattice ?

Consider a lattice in R^n. Consider Voronoi cell of it. What is known about diameter ? About the shape ? What are good references ? As far as I understand they are not easy to compute. May be in ...
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1answer
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Zoll Flat Finsler tori and convex bodies on a starry night

The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $ \lambda > 0$) and the "...