The lattices tag has no wiki summary.

**5**

votes

**1**answer

142 views

### Lattice in motion group

Let $\Gamma$ be a discrete cocompact subgroup of the euclidean motion group
$$
G={\mathbb R}^d\rtimes O(d).
$$
Let $\phi:G\to O(d)$ the projection homomorphism.
Is it true that $\phi(\Gamma)$ is ...

**10**

votes

**2**answers

540 views

### discrete subgroups of Lie groups and actions on homogeneous spaces

Let $\Gamma$ be a discrete subgroup of a connected finite dimensional Lie group $G$. Let $K$ be a maximal compact subgroup of $G$ and denote $X=G/K$. It is well-known that $\Gamma$ acts properly on ...

**0**

votes

**1**answer

98 views

### orthogonality in a lattice

Let $\Lambda$ be a lattice with a quadratic form $q$ of signature (3,19).
Let $\Lambda_{\mathbb{R}}:=\Lambda\otimes \mathbb{R}$ and $W\subset \Lambda_{\mathbb{R}}$ a positive subspace of dimention 3.
...

**1**

vote

**1**answer

113 views

### orthogonal base in unimodular lattice

Let $\Lambda$ be an unimodular lattice with a quadratic form $(-,-)$ of signature $(m,n)$ , $m,n>0$.
I know that, fixed a base $e_1,\cdots,e_{m+n}$ for $\Lambda$, the matrix which has entries ...

**4**

votes

**2**answers

503 views

### Property of lattices in Lie groups

Let $\Gamma$ be a lattice in a (real or p-adic) Lie group.
Is it true that for a given natural number $n$ there exists a finite index subgroup $\Sigma\subset\Gamma$ such that each $\sigma\in\Sigma$ is ...

**2**

votes

**1**answer

115 views

### Distributive lattice embedding into a finite lattice.

Suppose one has an inclusion $\iota : D \hookrightarrow S$ where $D$ is a finite distributive lattice and $S$ is a finite join-semilattice.
If $\iota$ preserves all meets and joins one can show that ...

**2**

votes

**2**answers

296 views

### Banach lattice subspace of $C([0,1])$ not a sublattice

This is probably easy, but I did not see it in standard texts. Describe a closed subspace $V$ of $C([0,1])$ such that $V$ is a Banach lattice (in the pointwise ordering), but $V$ is not a sublattice ...

**1**

vote

**1**answer

99 views

### distributive sublattices of atomistic ortholattices

Let $L$ be an atomistic ortholattice (i.e. every element can be written as a join of atoms) with top and bottom elements 0 and 1, and let $M$ be a distributive atomic sub-ortholattice of $L$.
Is $M$ ...

**6**

votes

**1**answer

389 views

### genus and spinor genus over a number field

Let $F$ be a number field with ring of integers $\mathfrak{o}$. Let $(V,Q)$ be a quadratic space of dimension $n$ over $F$, and let $L$ be a free lattice in $V$ (i.e. $L\cong\mathfrak{o}^n$). If the ...

**1**

vote

**0**answers

67 views

### Reference request for gluing construction of lattices

I would like to study gluing method of lattices (such as constructing Niemeier lattices from certain root lattices etc) and am looking for good references. I am aware of the book "Sphere Packings, ...

**9**

votes

**1**answer

302 views

### Lattice in a certain Lie group

Let $G_n$ be the Lie group consisting of $n \times n$ upper triangular matrices of determinant $1$ with real entries. In other words,
$$G_n = \{\text{$\left(\begin{matrix} a_{11} & a_{12} & ...

**1**

vote

**0**answers

173 views

### existence of order preserving map [closed]

suppose A is a linear order set with a copy of rationals in it that is $A=B\cup\{\bar{r}:r\in\mathbb{Q}\cap[0,1]\}$. is there an orde preserving map that preservs sup and inf betwwen A and $[0,1]$ in ...

**1**

vote

**1**answer

102 views

### Help with (Coxeter?) lattice identification.

I'm trying to find information about a specific lattice, which is proving difficult since I am not sure what its standard name is.
Consider the regular $n$-Simplex embedded in $\mathbb{R}^n$ with one ...

**1**

vote

**0**answers

82 views

### Information about mutant Leech lattice related to smallest perfect squared square

What happens if we follow the construction of the Leech lattice but replace the relation
$\displaystyle \sum_{n=1}^{24} n^2 = 70^2$
with the smallest perfect squared square? Explicitly, if we set ...

**2**

votes

**1**answer

278 views

### Probability that a “closable” self-avoiding random walk forms a polygon

Consider a self-avoiding random walk on an infinite graph (for concreteness, the grid of 2-dimensional lattice points $\mathbb{Z}^2$), in which on each step, the next position is chosen uniformly at ...

**5**

votes

**1**answer

212 views

### Complete anti-chain lattices and the axiom of choice

Hello, everyone. I'm trying to find out about lattices of anti-chains, and was wondering whether you could help me with getting to grips with a Comp. Sci. paper I'm struggling with.
I've been reading ...

**2**

votes

**4**answers

362 views

### Minimal (semi)lattice containing a given poset

For a given poset, (I think that) it is easy to construct the minimal join-semilattice containing that poset. I wonder whether the minimal lattice containing that poset is also easy to construct. I ...

**2**

votes

**2**answers

246 views

### Lattice reduction on an orthonormal lattice?

Suppose you are given an inner product on a vector space and given a set of linearly independent vectors, and that you have been promised that the lattice they span has an orthonormal basis. Can you ...

**6**

votes

**2**answers

320 views

### Measuring how far from being cocompact a lattice is

Let $G$ be a locally compact group and $\Gamma$ a lattice (=discrete
subgroup of $G$ such that $G/\Gamma$ carries a probability measure $\mu$
that is invariant under the action of $G$ by ...

**11**

votes

**1**answer

274 views

### Defining measures over frames in place of $\sigma$-algebras

Normally, measures and probability spaces are defined over $\sigma$-algebras. I was wondering what would happen if one tries to define it over frames in place of $\sigma$-algebras? Specifically, ...

**3**

votes

**1**answer

427 views

### Maximizing the number of lattice points in a circle of radius $r$ placed on a lattice

I have a circle of radius $r$, and I wish to place this circle of a $Z^2$ integer lattice or an $A_2$ hexagonal lattice s.t. I maximize the number of lattice points within or along the contour of the ...

**2**

votes

**2**answers

569 views

### An exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in a $A_2$ hexagonal lattice

In a previous question: (The Gauss circle problem on a hexagonal lattice) I asked for an analytic approximation for the number of lattice points in or along the contour of a circle centered on a ...

**1**

vote

**0**answers

56 views

### variant on ring objects in the category of complete lattices

Let $L$ be a complete lattice and denote its top and bottom elements by $0$ and $\infty$ respectively. Consider two binary operations $+$ and $\times$ defined on $L$ such that $(L^{op},+,0)$ is a ...

**4**

votes

**1**answer

217 views

### Most orthogonal lattice basis

Let $n \in \mathbf{N}$ be a natural number and $v_1,\cdots,v_n$ a set of basis vectors in $\mathbb{R}^n$. How does one find the matrix $g \in \mathbf{GL}_n(\mathbb{Z})$ orthogonalizing these best ...

**2**

votes

**1**answer

101 views

### Is the complete lattice of inflators on a frame a frame?

Yup, it may sound like an inocent question to many of you; but a very good friend of mine is completely baffled in his research about lattices of inflators on a frame. He asked me very kindly to post ...

**3**

votes

**0**answers

125 views

### references for properties/examples of breadth in (semi)lattices

This is in some sense following up on my earlier question and the answer given by NN.
I am currently revising the paper which used the condition mentioned in my question. It was pointed out in NN's ...

**4**

votes

**1**answer

278 views

### Invariant lattice of algebraic surface.

Given an algebraic surface $S$ with action of a finite group $G$. Is it true that the invariant lattice $H^2(X,\mathbb{Z})^G$ is generated by elements pulled back from the $H^2(X/G,\mathbb{Z})$ (or ...

**1**

vote

**1**answer

138 views

### Covering a $d$-dimensional integer lattice by repeating a minimal set of deterministic moves

Imagine I place a turtle on some desired vertex, $v_i$, of a bounded $d$-dimensional integer lattice, $Z^d$, with dimensions $(l_1, ..., l_d)$. The turtle is able to travel from vertex to vertex ...

**1**

vote

**0**answers

115 views

### 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 ...

**7**

votes

**1**answer

191 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 ...

**1**

vote

**1**answer

259 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 ...

**0**

votes

**0**answers

143 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**

votes

**0**answers

253 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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votes

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657 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**

votes

**0**answers

155 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) - ...

**3**

votes

**2**answers

265 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}, ...

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votes

**2**answers

541 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 $G$ ...

**6**

votes

**1**answer

981 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 ...

**1**

vote

**0**answers

157 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 ...

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vote

**0**answers

75 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**

votes

**1**answer

238 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**

votes

**1**answer

183 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 ...

**2**

votes

**0**answers

94 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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vote

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125 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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votes

**2**answers

367 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 ...

**15**

votes

**4**answers

472 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 ...

**3**

votes

**0**answers

85 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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votes

**1**answer

549 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 ...

**1**

vote

**1**answer

95 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 ...

**5**

votes

**0**answers

159 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 ...