1
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0answers
24 views

lattice symmetries portrayed in animation

Below is an animated gif by Dave Whyte: What is the orbifold/fibrifold notation for the symmetries of the lattice depicted below? (a la /The Symmetries of Things/? Are there related theta function ...
1
vote
1answer
48 views

Counting sum of lattice points

Assume a set $S$ with $|S|$ entries. Indeed, $S$ is the set of lattice points inside a $k$-sphere. Assume $V=S\oplus S$ where $\oplus$ is the Minkowski sum of two sets. Do you know any lower bound on ...
2
votes
1answer
48 views

Is this the subgroup lattice for $\Bbb{Z}_4 \times \Bbb{Z}_8$?

I have been attempting to create the subgroup lattice for $\Bbb{Z}_4 \times \Bbb{Z}_8$. I have, so far, this: http://www.scribd.com/doc/223680804/Subgroup-Lattice-of-Z-4-x-Z-8 While I have calculated ...
3
votes
1answer
156 views

Generators of Special Linear Groups

Linear algebra and special-linear group experts please help: I learn that in principle one can generate this $M$ matrix form the $B_1$ and $B_2$ matrix below. Here $$ M=\begin{pmatrix} 0& 1& ...
7
votes
1answer
58 views

Orbits of action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$

I'm considering the action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$: if $A\in SL_m(\mathbb{Z})$ and $v\in\mathbb{Z}^m$, then $Av\in\mathbb{Z}^m$. My question is: what are the orbits of this action? ...
1
vote
1answer
46 views

Orbits of action of $SL_2(\mathbb{Z})$ on lattice

I'm interested in the action of $SL_2(\mathbb{Z})$ on $\mathbb{Z}^2$: if $A\in SL_2(\mathbb{Z})$ and $v\in\mathbb{Z}^2$, then $Av\in\mathbb{Z}^2$. Specifically, what are the orbits of this action?
1
vote
1answer
92 views

Fundamental volume of quotient group

I came across this rule in my old notes, but I need an explanation to how it could possibly originate: The theorem says that for any lattice $L$ in $\mathbb{R}^n$, the order of the quotient group, ...
0
votes
1answer
424 views

Index of a sublattice in a lattice and a homomorphism between them

I am asked to show that if $\phi_A$ is the homomorphism from $\mathbb{Z}^k \rightarrow \mathbb{Z}^k$ given by $\phi_A(x)=xA$ then the index of $\phi(\mathbb{Z}^k)$ in $\mathbb{Z}^k$ is finite if ...
3
votes
1answer
86 views

Nonzero element of minimal length in a lattice?

Given two linearly independent vectors $a,b\in\mathbb{R}^2$ we form the lattice $L=\{ma+nb|m,n\in\mathbb{Z}\}$. Now, a proof starts with "choose a nonzero vector in $L$ of smallest length...". Why ...