# Questions tagged [euclidean-lattices]

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### Would anybody be able to summarise Minkowski Successive Minima in slightly simpler terms?

I came across this definition in Lenny Fukshansky's paper "Revisiting the hexagonal lattice: on optimal lattice circle packing" and I can't seem to fully grasp the concept of successive ...
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### Under which assumptions on generating matrices $A,B$ is the lattice $A \mathbb Z^n$ contained in the lattice $B \mathbb Z^n$? [closed]

Let $A,B \in \mathbb R^{n \times n}$ be invertible matrices and let $L_1 = A \mathbb Z^n, L_2 = B \mathbb Z^n$ be the lattices in $\mathbb R^n$ generated by $A$ and $B$ (i.e. $L_1,L_2$ are the image ...
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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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### 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 $S$ ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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.
1 vote
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### Minimal volume of fundamental domains of lattices

Consider a full rank integer lattice in $\mathbb{R}^n$. Let $v_1$ be the shortest non-zero vector in the lattice, $v_2$ be the shortest one among those not parallel to $v_1$, $v_3$ be the shortest one ...
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### Deep Holes of the tensor product of two lattices

Let $L_0, L_1$ be Euclidean lattices (say full rank) of dimension $n_i$. Let $\lambda_1(L_i)$ denote the length of the shortest vector of $L_i$, and let $\rho(L_i)$ denote the covering radius of $L_i$:...
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### what is the number of paths returning to 0 on the hexagonal lattice

I am looking for an estimation of the number of paths of length $n$ going from 0 to 0 on the hexagonal (or honeycomb) lattice. I can find plenty on references on self avoiding paths, but I am looking ...
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### The Monster Moonshine Module from the engineering or algorithmic point of view

From what I understand (see, e.g., this question), the Monster Moonshine Module is a kind if "third generation" (or "second quantization"?) after the Golay code (with automorphism group $M_{24}$) and ...
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### Tiling with incommensurate triangles

Say that two triangles are incommensurate if they do not share an edge length or a vertex angle, and their areas differ. Suppose you'd like to tile the plane with pairwise incommensurate triangles. I ...
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### Periodicity of density of laminated lattices

In Sphere Packings, Lattices and Groups, Conway and Sloane explore laminated lattices. If we let $X_d$ be the set of $d$-dimensional Euclidean lattices where every pair of points are separated by ...
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### Lattice basis reduction over rings of number fields

Can one use lattice basis reduction algorithms, such as LLL over (low-rank) module lattices over rings of number fields of degree greater than 1? Is there any work on lattice reductions over Euclidean ...
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### On necessary condition for no integer points in polytope

For a convex polytope $\mathcal K$ in $\Bbb R^n$ presented by $O(n^c)$ linear inequalities is it true that for $|\mathcal K\cap \Bbb Z^n|=0$ it is necessary that at least one axis of John's ellipsoid ...
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### Lattice question

Consider a lattice $\mathcal{L} = \mathbb{Z}v_1 \oplus \ldots \oplus \mathbb{Z}v_l$ in $\mathbb{R}^n$ and let $S_0$ be the set of edges of the fundamental unit of $\mathcal{L}$. We call a region $X$ ...
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### Lattice paths in polytopes

Let $P$ be a polytope in $\mathbb{R}^n$. Let $A_ix = b_i$ be the defining equations of its codimension $1$ faces. Is there an algorithm or some kind of criterion to decide if the lattice points inside ...
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### lattice with Voronoi cell inside a circle

This considers real-valued lattices in two dimensions. I need to find the densest lattice $\Lambda$, i.e., the one with the smallest determinant of its generator matrix, such that the Voronoi cell of ...
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### What is the spinor genus of the Leech lattice?

The Leech lattice and the 23 Niemeier lattices make a single genus. How does it break up into spinor genera?
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### Bound on the determinant of a quadratic form restricted to a subspace

Let $Q\colon \mathbb{Z}^{n}\oplus\mathbb{Z}^m\to\mathbb{R}$ be a real quadratic form, which we denote $Q(x,y)$, $x\in\mathbb{Z}^n$, $y\in\mathbb{Z}^m$. Suppose: The minimum of $Q(x,y)$ as $y$ varies ...
Does anybody have a reference answering the following (at least for me surprisingly non trivial) question? Given an $n \times n$ integer grid, what is the minimum angle between any two distinct lines,...
### Does $E_8$ know $Spin(7)$?
One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For example,...