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])

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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 ...
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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 ...
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5 votes
3 answers
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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 ...
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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 ...
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8 votes
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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/...
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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 \...
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Explicit family of Euclidean lattices which maximize $\lambda_1(L)\lambda_1(L^\vee)$?

For a lattice $L\subseteq\mathbb{R}^n$ (normalized to have determinant 1), one consequence of Minkowski's first theorem is that $$\lambda_1(L) \leq \sqrt{n},$$ where $\lambda_1(L) = \min_{x\in L\...
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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 ...
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Embedding into odd unimodular lattices

Let $L$ be a unimodular even lattice of signature $(l+,l-)$, let $S$ be an even lattice of signature $(t+,t-)$, we know that when $(t+)+(t-)<\min(l+,l-)$, there exists a primitive embedding of $S$ ...
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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 ...
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9 votes
1 answer
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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 ...
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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 ...
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1 answer
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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 ...
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1 vote
1 answer
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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(...
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Orders of Niemeier lattice stabilizers

What is the order of the stabilizer of each Niemeier lattice, scaled so that its shortest nonzero vectors form a subset of those of the Leech lattice, within the latter’s automorphism group?
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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 $i = 1,2$?
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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 ...
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6 votes
1 answer
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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-...
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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 ...
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1 vote
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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 ...
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Weak compactness is to trees as [?] is to lattices?

Let $\kappa$ be an inaccessible cardinal. Recall that $\kappa$ is weakly compact if every tree of height $\kappa$ has either a level of size $\kappa$ or a branch of size $\kappa$. So if $\kappa$ is a ...
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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 ...
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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} ...
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6 votes
2 answers
393 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 ...
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0 votes
1 answer
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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 ...
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4 votes
0 answers
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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 $\operatorname{Rep}(\mathbb{Z}_2)$) ...
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1 vote
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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 ...
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5 votes
1 answer
170 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 ...
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1 answer
117 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 ...
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8 votes
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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 ...
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Enumerating lattice paths with various diagonals

Consider the set of integer lattice paths starting on the non-negative y-axis and ending on the non-negative x-axis with the following moves: Right by $1$, Down by $1$, Diagonals going from $(a,b)$ ...
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Relation between integral lattice and an associated unimodular lattice construction

$M\in\mathbb Z^{n\times n}$ is an arbitrary integral matrix whose determinant is $det(M)$. We pick a random integral $T$ of size $O(1)\times O(1)$ having determinant $det(M)-1$. We construct using the ...
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On lattice generated by integral vectors close to eigenvectors

$M\in\mathbb Z^{n\times n}$ is an unimodular symmetric matrix whose eigenvectors are $v_1$ to $v_n$ and the lattice $\mathcal L$ generated by $M$ have the shortest vectors $u_1$ to $u_n$ where $u_i$ ...
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3 votes
2 answers
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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.
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4 votes
1 answer
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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 ...
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2 votes
1 answer
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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 \...
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0 votes
1 answer
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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, ...
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9 votes
1 answer
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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 $\...
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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 ...
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3 votes
1 answer
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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 ...
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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, ...
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2 votes
1 answer
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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 ...
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1 vote
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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}_{...
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2 votes
0 answers
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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 ...
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1 answer
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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 ...
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