3
votes
1answer
72 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
51 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? ...
0
votes
0answers
33 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
71 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
371 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
82 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 ...