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

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How many more join-irreducibles can there be in a sub join-semilattice of a finite lattice?

Let $L$ be a finite lattice. Then $L$ is generated by its join-irreducible elements $J(L)$ or alternatively its meet-irreducible elements $M(L)$. If $S \subseteq L$ is a sub join-semilattice then $|M(...
7
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1answer
198 views

Restricting representations to lattices

Let $V$ be a finite-dimensional irreducible representation of the Lie group $\text{SL}_n(\mathbb{R})$. Must $V$ remain irreducible when you restrict the action to $\text{SL}_n(\mathbb{Z})$? More ...
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1answer
313 views

The probability a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice takes more than $N$ steps before trapping itself

What is the probability that a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice is able to take more than $N$ steps, i.e. able to take more than $N$ steps before trapping itself ...
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147 views

$T^2$-fibered K3 surface with involution

Let $S$ be a K3 surface and $f:S\rightarrow \mathbb{P}^1$ a $T^2$-fibration (not necessarily holomorphic, I have a special Langrangian fibration in mind). Assume there is a $k$-section, then a fiber ...
2
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336 views

Singular fibers of an elliptic fibered K3 surface.

Let $f:S\rightarrow \mathbb{P}^1$ be an elliptic K3 surface. Assume that $\mathrm{Pic}(S)\cong U$, where $U$ stands for the hyperbolic lattice. I think that the elliptic fibration has only singular ...
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836 views

On “The Average Height of Planted Plane Trees” by Knuth, de Bruijn and Rice (1972)

I am trying to derive the classic paper in the title only by elementary means (no generating functions, no complex analysis, no Fourier analysis) although with much less precision. In short, I "only" ...
3
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184 views

Automorphisms of Torsion Quadratic Forms

Let $L$ be an $n$-dimensional lattice with an integer valued quadratic form $q$. Fix a basis $e_i$ for $L$ and let $K_{ij} = \langle e_i, e_j \rangle$, where $\langle x,y \rangle = q(x+y) - q(x) - q(...
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2answers
277 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}...
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622 views

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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1answer
1k views

Basis of a group

Let $G$ be a finite group. I will say that a set of a subgroups $H_1,\ldots ,H_k$ defines a basis for a group $G$ if any subgroup $H$ of $G$ there exists $S\subset [k]$ such that $H=\cap_{i\in S}H_i$....
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2answers
261 views

Involution of $E_{8}$ lattice

Let $L$ be a lattice associate to the Dykin matrix of type $E_{8}$. I would like to understand involutions of $L$ and their invariant $L^{+}$ and coinvariant lattice $L^-$ (I think they are isomorphic)...
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88 views

can minimal volume rational subspaces in a lattice be arbitrarily 'close'.

Let $\Gamma$ be a cocompact lattice in $\mathbb{R}^n$, eg. $\Gamma = A \mathbb{Z}^n$ for some $A \in SL_n \mathbb{R}$. Then any $k$-dimensional subspace $P$ which is rational in $\Gamma$ has a volume: ...
4
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1answer
264 views

Reference for subsemigroups of $\mathbb{N}^n$

A well known result about the natural numbers $\mathbb{N}$ says that for any finite subset $A \subset \mathbb{N}$ there exists $R \ge 0$ such that if $n$ is in the subgroup of $\mathbb{Z}$ generated ...
2
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1answer
205 views

Expected number of identical vertex pairs with the same Euclidean distance on a randomly colored rectangular lattice

Imagine I have an $N$ by $M$ rectangular lattice where I randomly assign one of $k$ colors to every vertex in the lattice. I then write down a list of the ${N*M}\choose{2}$ possible unordered pairs ...
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102 views

Finite subgroups (lattices) in the large N limit of SU(N)

I would like to gain some information about the discrete subgroups (lattices) of SU(N) Lie groups. I have already read some answers and references concerning the N=3 and N=4 cases. I am more ...
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131 views

Comparing the volume of a rational lagrangian under a linear symplectomorphism.

Let's fix the standard symplectic structure $(\mathbb{R}^{2g}, \omega, J)$. A (marked) symplectic lattice then has the form $A\mathbb{Z}^{2g}$ for $A \in Sp_{2g}\mathbb{R}$. We say a vector subspace $...
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2answers
430 views

Lattices in $SL(n,\mathbb R)$

If $\Gamma\subseteq SL(n,\mathbb{R})$ is a lattice (i.e. discrete and finite covolume), does $\Gamma$ necessarily contain some $\mathbb{R}$-diagonalizable copy of $\mathbb{Z}^{n-1}$? I know that the ...
16
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4answers
509 views

The latice spanned by $m$ random 0-1 vectors of length $n$

Consider $m$ random 0-1 vectors of length $n$. Let $L$ be the lattice spanned by them. What is the value of $m$ (as a function of $n$) for which it is true with positive probability that $L=Z^n$? More ...
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91 views

Versions of Helly's Theorem for Unbounded Parallelpipeds

I'm studying Frobenius splitting of toric varieties based on Sam Payne's characterization of Frobenius splitting ("Frobenius splittings of toric varieties" (2008)) and I came across a sort of Helly's ...
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1answer
596 views

Is the ball reducible in some high dimension?

Let $K$ be a bounded symmetric ($-K=K$) open convex body in $\mathbb R^n$. The critical determinant $d(K)$ of $K$ is the least possible volume $|\operatorname{det}(a_1\dots a_n)|$ of the fundamental ...
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1answer
111 views

Lorentz quotient and orientation

$$ U \; = \; \left( \begin{array}{cc} 0 & 1 \\\ 1 & 0 \end{array} \right) , $$ Given a real oriented vector space $V$ with inner product, form Lorentzian $L = V \oplus U....
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171 views

Effect of Covering Radius on Shortest Vector

For "even" integral lattices in dimension at least 4, does a covering radius strictly less than $\sqrt 2$ imply that there is a vector of norm 2, also called a root? Note that this is simply false in ...
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607 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 ...
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3answers
663 views

Matrices generating non-discrete subgroups of SL(2,R)

Jorgensen's inequality $\mid \left(Tr\left(A\right)\right)^2-4\mid+\mid Tr\left[A,B\right]-2\mid\ge 1$ gives a necessary condition for two matrices A,B to generate a discrete subgroup of SL(2,R). Are ...
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132 views

Commutative quantale (Q,+) with a binary operation distributing over +

Context: In Flagg's "Quantales and continuity spaces" the notion of a continuity spaces enriched in a certain quantale is introduced. Abstracting from the set $[0,\infty]$ the essential properties ...
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1answer
152 views

dither in Leech lattice quantization!

Hi, Can you please help me how to generate a dither signal $\mathbf{U}$, where $\mathbf{U}$ is a random vector of length 24 that is uniformly distributed over the voronoi region of the Leech lattice. ...
2
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1answer
362 views

When are the join-irreducibles in a complete lattice join-dense?

A subset $A$ of a complete lattice $L$ is said to be join-dense if $L=\{\bigvee R|R\subseteq A\}$. An element $a\in L$ is said to be join-irreducible if $a\neq 0$ and if $a=x\vee y$ then $a=x$ or $a=y$...
4
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1answer
362 views

Linear symmetric spaces are spaces with ''orthogonal complements''?

The collection of marked unimodular lattices in the euclidean $n$-space corresponds to the symmetric space $S_n:=SO_n(\mathbb{R}) \backslash SL_n(\mathbb{R})$. I have only recently been made aware ...
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1answer
419 views

Octonionic reflection groups

Consider Leech lattice definition provided by Wilson (Octonions and Leech lattice, 2008). There are 819 E8 sublattices defined by $ (2\lambda, 0, 0); $ $ (\lambda \overline{s}, (\lambda \...
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2answers
307 views

Inner product spaces, Siegel's theorem and lattices: book suggestion

Background: I am a theoretical computer scientist (PhD candidate) and have done graduate level courses in Algebra. I want to understand the following theorem from the book "Symmetric Bilinear Forms" ...
2
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1answer
200 views

Spanning set for Lattice generated by an orbit of the group.

For a vector spaces it always holds that any set of vectors spanning vector space $V$ has a subset of vectors which is a basis for $V$. While for lattices it is not true. For example consider one ...
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487 views

What is the structure of a space of $\sigma$-algebras?

Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be ...
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398 views

Topology on Set of Prime Filters of a Distributive Lattice

Given a distributive lattice $A$ we can look at $Spec(A)$, whose points are prime ideals and its open sets are given similarly to the Zariski topology on Spec of a ring. That is, the basis of open ...
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2answers
239 views

Relationship of Bousfield Classes of Morava K-theories

Suppose we have $\langle K(n)\rangle$ and $\langle K(n-1) \rangle$ for some fixed prime $p$. do we know whether or not $\langle K(n) \rangle \geq \langle K(n-1) \rangle$ or $\langle K(n-1) \rangle \...
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3answers
383 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 ...
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1answer
284 views

Name of a lattice-property

Assume that we have a complete lattice $(L,\leq)$. I would like to know whether the following property has a specific name and whether lattices with this property have been studied somewhere: For ...
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0answers
218 views

Hurwitz integers and $F_4$

The Hurwitz integers are $$ \mathcal H= \{a+bi+cj+dk:a,b,c,d\in\mathbb Z\;\text{ or } \;a,b,c,d\in \tfrac12+\mathbb Z\}. $$ I want to know if there is a formula, for $m\in\mathbb Z$, for the number ...
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0answers
305 views

Constructing the Stone Space of a Distributive Lattice

Does anyone have a good reference for the method of giving a topology to a distributive lattice as outlined in M.H. Stone's "Topological representation of distributive lattices and Brouwerian logics"? ...
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1answer
808 views

Reference request: Ehrhart's conjecture on the geometry of numbers

Conjecture (Ehrhart). If a convex body $K \subset {\mathbb R}^n$ has its barycenter at the origin and contains no other point with integer coordinates, the volume of $K$ is less than or equal to $(n +...
9
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2answers
608 views

Ellipsoids and lattices: an enclosure problem.

$E \subset {\mathbb R}^2$ is an ellipse of area $1$ centered at the origin that contains no other point with integer coordinates. Is there a matrix $A \in SL(2,{\mathbb Z})$ such that the ellipse $A(E)...
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1answer
354 views

Finding lattice with short basis-vectors containing given lattice

Hello While working on understanding the space spanned by certain integer relations of real numbers I have come across the following problem. Given $v_1,\dots, v_n \in \mathbb{Z}^m$ I am would like ...
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797 views

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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1answer
128 views

Is the cardinality of occuring torsion subgroups in cofinite lattices in SL(2,R) bounded?

Let $\Gamma$ be a cofinite lattice in $PSL(2,\mathbb{R})$ with torsion subgroup $H$. Is the a uniform bound on the cardinality of $H$?
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103 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 ...
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1answer
371 views

Complete De Morgan algebra

Recall that an algebra $(A,\sim)$ is a De Morgan algebra if $A$ is a bounded distributive lattice and $\sim$ is a unary operation which satisfies: ${\sim} (x\vee y)={\sim} x\wedge {\sim} y$ and ${\...
2
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1answer
290 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 ...
5
votes
1answer
720 views

Lovász $\delta$ condition for LLL Algorithm

http://en.wikipedia.org/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm What is the importance of the $\delta$ parameter for LLL bases called Lovász condition? ...
2
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1answer
342 views

Extending a complete lattice to get a “nice” Boolean lattice

Suppose we have a complete lattice. Which additional axioms (e.g. distributivity axioms) are needed to obtain a Boolean lattice in which complement(a) = lub{b | b /\ a = bottom} = glb{b | b / a = top}?...
2
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1answer
254 views

Is there existing terminology for this technical condition on semilattices?

Given a semilattice $S$, a subset $E$, and a positive integer $n$, let $E^{[n]}$ be the set of all products of $n$-tuples in $E$. Thus $\bigcup_{n\geq 1} E^{[n]}$ is nothing but the subsemigroup of $S$...
2
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186 views

bounds for the covering radius (or diameter of the Voronoi cell) of a lattice coming from a number field

Let $K$ be a number field and $O$ an order in $K$. Denote the real embeddings of $K$ by $\sigma_1,\dots,\sigma_s$ and the non-real complex embeddings of $K$ by $\sigma_{s+1},\overline{\sigma_{s+1}},\...