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])

Filter by
Sorted by
Tagged with
1 vote
0 answers
58 views

Convex polyhedron in lattice points

Given $P$ a convex polyhedron with vertices in lattice points with $n$ faces. a) what is the minimum volume of $P$. b) what is the minimum area surface of $P$. Paper: The minimum area of convex ...
Yessir03's user avatar
  • 571
1 vote
0 answers
135 views

Obstruction in construction of some lattices, related with $K3$ surfaces

I am considering a certain $K3$ surface that is lattice-polarized in two ways. This leads to the following simple problem in lattice theory: (Let me borrow notations for lattice from Ch.14 of this ...
Basics's user avatar
  • 1,821
3 votes
0 answers
100 views

Minimal Norm Vectors in certain Cyclotomic Ideal Lattices

Let $q$ be an odd prime and let $\zeta$ be a primitive $q^{th}$ root of unity. Let $I$ be the ideal in $\mathbb{Z}[\zeta]$ generated by $1-\zeta$. It is known that we can give $I$ the structure of an ...
Tommy Occhipinti's user avatar
5 votes
3 answers
547 views

Finite index subgroup of $\mathbb{Z}^2$ that is invariant under a non-singular matrix

Let $M $ be a matrix in $ \operatorname{GL}(2, \mathbb{Z})$ that has at least one eigenvalue of absolute value strictly bigger than $1$. What are the finite index subgroups $H$ of $\mathbb{Z}^2$ such ...
ghc1997's user avatar
  • 763
0 votes
0 answers
103 views

Optimal covering trails in 3 and 4 dimensions

A couple of years ago, I constructively solved (inside the $AABB$ $[0,3]$ X $[0,3]$ X ... X $[0,3]$) the $k$-dimensional generalization of the infamous Nine-Dot Problem by S. Loyd (see Cyclopedia of ...
Marco Ripà's user avatar
  • 1,102
8 votes
2 answers
727 views

The smallest volume possible for a lattice with integer distances?

Let $\Lambda \subset\mathbb{R}^n$ be a lattice satisfying $\|x-y\|_2^2 \in \mathbb{Z}$ for all $x,y\in\Lambda$. How small can $\text{vol}(\Lambda)=\det(\Lambda)$ be? For example, in dimension $2$, the ...
Eric Naslund's user avatar
  • 11.2k
3 votes
0 answers
64 views

Stability of successive minima with respect to the metric on the space of lattices

Let $\mathcal L$ be the space of unimodular (covolume one) lattices in $\mathbb R^d$. The $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the ...
No One's user avatar
  • 1,545
14 votes
1 answer
408 views

Geometric explanation for coincidence in lengths of 16-dimensional even unimodular lattices?

Question Up to equivalence, there are two positive-definite even unimodular lattices in $16$ dimensions: $D_{8}^{+}\oplus D_{8}^{+}$ and $D_{16}^{+}$. As observed by Witt in 1941, the theory of ...
Ben Mares's user avatar
  • 401
2 votes
2 answers
129 views

The range of each of successive minima for all unimodular lattices

Let $\mathcal L$ be the space of unimodular (covolume one) lattices in $\mathbb R^d$. The $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the ...
No One's user avatar
  • 1,545
1 vote
1 answer
86 views

Affine semigroup generating a lattice

This is a cross-post from MSE. Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$ of a Euclidean space). Assume that ...
Grisha Taroyan's user avatar
2 votes
0 answers
86 views

Is there an exact solution for the number of points within a circle of radius r for an honeycomb lattice?

I want to ask if exists an exact solution for the number of points within a circle of radius r for an honeycomb lattice. I know that it is exist for an square lattice https://mathworld.wolfram.com/...
Mihaela's user avatar
  • 31
0 votes
1 answer
530 views

The exact number of points within a circle of radius r centered on a lattice point in a hexagonal lattice? Review expression Gauss circle problem

In the case of a square lattice, the exact number of points within a circle of radius r centered in the center is (see: http://mathworld.wolfram.com/GausssCircleProblem.html: $$N(r)=1+4Floor(r)+4 \...
Mihaela's user avatar
  • 31
2 votes
0 answers
55 views

Does a transitive action of $O(L^{\#}/L)$ imply a transitive action of $O(L)$ on $L^{\#}/L$?

Given an even lattice $L$ the orthogonal group $O(L)$ acts on the discriminant group $L^{\#}/L$ as is known. Hence, for every $\gamma \in O(L)$ there is a $\overline{\gamma}\in O(L^{\#}/L)$ that acts ...
F_S's user avatar
  • 21
3 votes
1 answer
119 views

Lattice-like structure with maximum spacing between vertices

I'll first describe my problem in layman's terms. I have a map with $m$ countries and I want to color each country with a different color (this has nothing to do with the 4-color theorem). How do I ...
Vincent Granville's user avatar
9 votes
1 answer
374 views

Compact flat orientable 3 manifolds and mapping tori

There are 10 compact flat 3 manifolds up to diffeomorphism, 6 orientable and 4 non orientable. I am looking to better understand how to construct the orientable ones. The six orientable ones are ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
133 views

Relationships among lattices U14, C2xG23, A15+ and their Delaunay polytopes

Do you have any references explaining the relationships among the lattices U14, C2xG23 aka Q14, and A15+? Do you have any references explaining the relationships among these lattices and the 7D ...
Dan Haxton's user avatar
0 votes
1 answer
341 views

Standard Gram matrices for lattices

I would like to define standard Gram matrices, and use them to help me understand the symmetries of lattices. I define "standard Gram matrix" as the Gram matrix g that minimizes the ...
Dan Haxton's user avatar
1 vote
1 answer
139 views

On parametrization of a type of unimodular $2\times2$ integral matrix

A matrix $\begin{bmatrix}w&x\\y&z\end{bmatrix}\in\mathbb Z^{2\times 2}$ is unimodular if $$|wz-xy|=1$$ holds. Is there a parametrization of such matrices with $|w||z|-xy=1$ $$w,z<0<\max(...
Turbo's user avatar
  • 13.6k
0 votes
0 answers
74 views

Is $(\omega+1)^\omega/{\cal U}$ "unique"?

If ${\cal U}_i$ free ultrafilters on $\omega$ for $i = 1,2$ , are the ultrapowers $(\omega+1)^\omega/{\cal U}_i$ necessarily isomorphic as lattices for $i = 1,2$?
Dominic van der Zypen's user avatar
3 votes
2 answers
188 views

Limit of the Schröder numbers ratio

I have been playing around with interesting integer sequences and came across Schröder number which defines the number of lattice paths of n x n grid. The recurrence formula to calculate these numbers ...
Justin van Zyl's user avatar
7 votes
1 answer
323 views

The Tarski-Lindenbaum theorem of the middle value

In Ettore Casari, La Matematica della Verita' (2006), the following theorem (attributed to Tarski and Lindenbaum 1926), is stated, which Casari calls the theorem of the middle value of Tarski-...
user65526's user avatar
  • 629
1 vote
0 answers
76 views

Intersecting lattices with surfaces in R^d

Let us fix some bounded surface $S\subset \mathbb{R}^d$. Let $x_1,\ldots, x_m$ be some non-zero vectors in $\mathbb{R}^d$. I am interested is the maximum number of points that the lattice $L_m=\{\sum ...
TOM's user avatar
  • 2,218
1 vote
0 answers
74 views

Conjectures about the automorphism group of integer lattice by enlarging the matrix

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Notation: $\GL(n, \mathbb{Z})$ to be the set of $n \times n$ invertible matrix, and ...
en kuo's user avatar
  • 145
3 votes
0 answers
50 views

Asymptotical equidistribution of index $p$ subgroups of $\mathbb Z^2$ on the unit tangent bundle of the modular curve

Given a prime $p$ we get $p+1$ sublattices of index $p$ in $\mathbb Z^2$ (identified with $\mathbb Z[i]\subset \mathbb C$) which correspond to some points on the moduli space of such lattices up to ...
Roland Bacher's user avatar
1 vote
0 answers
73 views

Canonical representation of the a probability distribution for Hammersley Clifford Theorem

I'm reading the following paper http://www2.stat.duke.edu/~scs/Courses/Stat376/Papers/GibbsFieldEst/BesagJRSSB1974.pdf On page 7 they give the result that $$Q(\textbf{x}) = \sum_{1 \leq i \leq n} ...
Pavan Sangha's user avatar
7 votes
2 answers
489 views

Is the group of integer points of a simple real linear algebraic group a maximal closed subgroup?

Let $ G $ be a simple linear algebraic group. Let $ G_\mathbb{R} $ be the real points of $ G $. Let $ G_\mathbb{Z} $ be the integer points of $ G $. Is $ G_\mathbb{Z} $ a maximal closed subgroup? In ...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
375 views

Tri-coloring of E8 lattice? Why is the Gram matrix of E8 not unique?

This question is about euclidean lattices, regular arrays of points in $\mathbb R^N$. Why are there 3 Gram matrices for the E8 lattice? They are not related by a similarity transformation; they have ...
Dan Haxton's user avatar
4 votes
0 answers
146 views

Drinfeld center of a tensor category

Firstly, apologies for the imprecise language, I'm a physicist trying to understand anyonic excitations from the lens of category theory. If I have a category (say $\operatorname{Rep}(\mathbb{Z}_2)$) ...
pyroscepter's user avatar
1 vote
0 answers
48 views

Short lattice vectors in the complement of a hyperplane

Suppose that $\Lambda \subseteq \mathbb{R}^n$ is a lattice and $H \subseteq \mathbb{R}^n$ is a hyperplane such that $H \cap \Lambda$ has rank $n - 1$. I would like to know an upper bound on the ...
Russ Weterson's user avatar
5 votes
1 answer
186 views

Number of distinct normalized vectors from the center of a hexagon in a hexagonal grid

Consider an infinite hexagonal grid composed of regular hexagons. Choose any hex to be the origin hex. Let n be a natural number. Find an expression, in terms of n, for the number of distinct ...
Gabriel Schweitzer's user avatar
0 votes
1 answer
173 views

Fourier transform on lattice strip

I am working with a triangular lattice $L=\{n_1 a_2 + n_2 a_2 : n\in\mathbb{Z}^2 \}$ and $a_1 = \pmatrix{1 \\ 0}$ and $a_2 = \frac{1}{2} \pmatrix{-1 \\ \sqrt{3}}$, and I want to compute the Pontryagin ...
spaceman's user avatar
  • 575
8 votes
0 answers
251 views

Positive integer solutions of $ab+ac+ad+bc+bd+cd=n$

Consider a quadratic form $Q(a,b,c,d)=ab+ac+ad+bc+bd+cd$ on $\mathbb Z^4$. For some reason I am interested in the number of solutions $(a,b,c,d)\in\mathbb Z_{> 0}^4$ of $Q(a,b,c,d)=n$ as a function ...
Nikita Kalinin's user avatar
4 votes
2 answers
387 views

Lattices containing $A_n$ and $D_n$

How many lattices are there which contain both the $A_n$ and $D_n$ lattices of the same dimension as sublattices? So far, I’ve found examples in 1D, 3D, 8D, and 24D.
Daniel Sebald's user avatar
4 votes
1 answer
190 views

Roots of reverse Bessel polynomials define asymptotically a lattice

(Title changed after comment from Nemo). Set $$P_n=\sum_{k=0}^n\frac{(n+k)!x^{n-k}}{k!(n-k)!}\ .$$ (The polynomial $\theta_n(x)=2^{-n}P_n(2x)$ is the so-called reverse Bessel polynomial, see comment ...
Roland Bacher's user avatar
2 votes
1 answer
74 views

Reference request: Given a non-degenerate integral quadratic lattice $L,q$ over a PID, the quotient $L^*/L$ is given by SNF of $q$

Let $R$ be a PID with field of fraction $K$. Let $L$ be a lattice with non-degenerate quadratic form $q:L\times L \to R$. Let $$ L^* = \{x \in L\otimes K \text{ s.t. } q(x,l) \in R \text{ for all } l \...
user148575's user avatar
0 votes
1 answer
381 views

Finding a subset of vectors whose sum is close to a given vector

Given a set of vectors $x_1,...,x_n\in\mathbb{R}^d$ and a vector $y$, find a subset $I\subset\{1,2,...,n\}$ such that $\|\sum_{i\in I} x_i-y\|$ is as small as possible. Here $\|.\|$ can be any norm, ...
legon's user avatar
  • 31
9 votes
1 answer
200 views

Which unimodular lattices $L\subset \mathbb R^2$ minimize $f_t(L):=\sum_{ v\in L} e^{-t \|v\|_2}$? (for parameters $t>0$)

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Consider the lattices in $\SL(2,\mathbb R)(\mathbb Z^2)$ up to rotation. The space of such lattices can be identified with the modular surface $\...
user84899's user avatar
  • 241
1 vote
0 answers
65 views

Automorphism groups of which lattices act irreducibly on the ambient Euclidean space

(I asked this question on MSE a few days ago but it hasn't drawn any response yet.) Let $V$ be a finite-dimensional real inner product space and let $L \subset V$ be a lattice of full rank. Consider ...
Ivan Solonenko's user avatar
3 votes
1 answer
133 views

Covering radius of a lattice from relevant vectors

Let $L$ be an $n-$dimensional lattice. The Voronoi region of $L$ is given by $$ \mathcal{V}(L)=\big\{x\in\mathbb{R}^n~|~ \|x\|_2\leq \|x-v\|_2~\forall v\in L\setminus\{0\}\big\}. $$ Considering the ...
FermaX's user avatar
  • 33
2 votes
0 answers
69 views

Why is Schröder numbers equivalent to the number of perfect matchings for triangular grid of n squares and how the graph look like? [duplicate]

In the OEIS entry for the Schröder numbers is A006318. There is a comment which related the sequence to perfect matchings: The number of perfect matchings in a triangular grid of n squares (n = 1, 4, ...
Xuemei's user avatar
  • 141
2 votes
1 answer
202 views

Shifted lattices and the discriminant group

I'm studying a geometrical problem where an (even, negative-definite) lattice $L$ arises. Roughly, as an intersection pairing for curves on a surface. In fact, the problem naturally leads me to ...
Benighted's user avatar
  • 1,701
1 vote
0 answers
147 views

Proving that $\mathrm{SL}_2(\mathbb{Z}[i])$ is a lattice in $\mathrm{SL}_2(\mathbb{C})$

I need to show that $\mathrm{SL}_2(\mathbb{Z}[i])$ is a lattice in $\mathrm{SL}_2(\mathbb{C})$. I was thinking of applying Borel-Harish Chandra theorem, which says that: Let $G \subseteq \mathrm{GL}_{...
mahbubweb's user avatar
  • 111
2 votes
0 answers
80 views

Showing an action of a higher rank lattice on hyperbolic space has a fixed point

In the introduction to this paper, the author mentions that any action of a lattice $\Gamma < G$ on a rank one symmetric space $X$ has a fixed point, where $G$ is a higher rank semisimple algebraic ...
Mauro's user avatar
  • 191
3 votes
1 answer
140 views

Dense and decodable lattices in high dimensions

We are currently looking for both dense and decodable lattices. Precisely, we want a lattice which CVP can be solved in polynomial time like $O(n^2)$ or $O(n^3)$ where $n$ is the dimension like 128 or ...
Kaiyi Zhang's user avatar
-1 votes
1 answer
116 views

A complete Boolean algebra on a function space

If $(B_1, \cdot_1, +_1, -_1)$ is a complete atomic Boolean algebra (where the induced partial order is $\leq_A$), and $(B_2, \cdot_2, +_2, -_2)$ is a complete atomic algebra (where the induced partial ...
user65526's user avatar
  • 629
1 vote
0 answers
90 views

Is this equivariant function constant?

Let $G$ be a linear algebraic group (think of $SL_n(\mathbb{R})$), $B$ its Borel (standard minimal parabolic) subgroup (think of upper triangular subgroup), and let $\Gamma \leq G$ be a cocompact ...
BharatRam's user avatar
  • 939
4 votes
0 answers
231 views

Computing theta functions of lattices in practice

I am motivated by a problem in 2d CFT to compute "generalized theta functions," expressions of the form \begin{equation} \vartheta_{L,u}(\tau) := \sum_{\alpha \in L} u(\alpha) q^{{\langle\...
Justin Kulp's user avatar
8 votes
2 answers
547 views

Orthogonal Hamiltonian cycles in (n x n x n) grids

Let $C_n$ be a cubical $n \times n \times n$ subset of the integer lattice, so consisting of $n^3$ vertices. I am interested in special Hamiltonian cycles in $C_n$, special in the sense that (a) each ...
Joseph O'Rourke's user avatar
4 votes
1 answer
211 views

Classification of root lattice embeddings in $E_{10}$

There is a well-known classification result which gives a complete list of all root lattices which can embed into the lattice $E_8$. Moreover, each of these root lattices admits a unique embedding ...
Wendelin Lutz's user avatar
3 votes
0 answers
273 views

Torsion-free subgroups in arbitrary lattices

Let $\Gamma $ be a lattice in a semi-simple Lie group $G$. If the Lie group $G$ is linear (that is, it has a faithful finite-dimensional linear representation), then $\Gamma$ contains a torsion-free ...
Vladimir47 's user avatar

1 2
3
4 5
14