5
votes
1answer
134 views

The lattice points of a f.g. rational cone form a f.g. monoid.

In their book "Polytopes, Rings, and K-Theory" Bruns and Gubeladze sketch an alternative approach to Gordan's Lemma, which is stated in the headline (f.g. = finitely generated). I don't understand the ...
3
votes
1answer
184 views

Algebraic proof of Ehrhart's theorem

Let $P \subset \mathbb{R}^d$ be a $d$-dimensional polytope, where all vertices lie on integral coordinates, and let $L(P,n)$ denote the number of integral lattice points contained in the scaled ...
0
votes
0answers
19 views

smallest integer contained in sublattice $\Rightarrow$ $L'=[q,r\tau+s]$

Let $L'$ be a sublattice of the lattice $[1,\tau]$ in a imaginary quadratic field. Reminder: a lattice $L$ consists of the $\mathbb{Z}$-linear combinations of $1$ and $\tau$, with $\{1,\tau \}$ linear ...
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
115 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& ...
1
vote
1answer
75 views

Obtaining a new basis for a lattice with one of the new basis vectors fixed

Suppose that a lattice $L$ in $\mathbb{R}^3$ is given with a basis $B = \mathbf{ \{ v_1, v_2, v_3 \} }$. Is there an algorithm that would help me obtain a new basis $B' = \mathbf{ \{ v_1', v_2', v_3' ...
1
vote
0answers
20 views

Non zero Unimodular matrices

I have a question about the lattice reduction algorithms like LLL algorithm. Lattice reduction algorithms like LLL generate a unimodular matrix which makes more orthogonal basises for a given matrix. ...
1
vote
0answers
30 views

Inclusion-minimality of a lattice basis

An integer lattice is a subgroup of $\mathbb{Z}^n$. Since $\mathbb{Z}$ is PID, each lattice has a well-defined rank and a generating set of rank many elements is a basis. I wonder if there is a way ...
3
votes
1answer
141 views

Classifying all ideals of a lattice $\mathbb{Z}[\sqrt{-d}]$

In Artin's Algebra he presents a method (that I am sure I am butchering) for classifying ideals of a given lattice $\mathbb{Z}[\sqrt{-d}]$ by taking any ideal $I$, choosing an element of minimum norm ...
2
votes
0answers
47 views

Automorphisms of a lattice and changing to a nicer $\mathbb{Z}$-base

Suppose I have an integral lattice $L$ with an arbitrary $\mathbb{Z}$-base, equipped with a positive-definite nondegenerate symmetric bilinear form $\langle\cdot,\cdot\rangle$, and an isometry $\nu$ ...
0
votes
2answers
189 views

Finding a basis for a complex lattice given a nondivisible vector in the lattice

If I am given some lattice defined as, say $$L=\{Az_1+Bz_2\ |\ A,B \in\mathbb{Z}\}$$ and a vector $v=az_1+bz_2$ , where $\gcd(a,b)=1$, I would like to find another vector $\,w\in L\,$ such that ...
0
votes
1answer
418 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 ...
8
votes
2answers
452 views

Elementary proof that if $A$ is a matrix map from $\mathbb{Z}^m$ to $\mathbb Z^n$, then the map is surjective iff the gcd of maximal minors is $1$

I am trying to find an elementary proof that if $\phi$ is a linear map from $\mathbb{Z}^n\rightarrow \mathbb{Z}^m$ represented by an $m \times n$ matrix $A$, then the map is surjective iff the gcd ...
0
votes
2answers
83 views

Relating lattice bases in $\mathbb R^2$

this is a homework question but I am pretty confused on it--just don't know where to start. We're given a lattice basis $(a, b)$ for a lattice $L$ in $\mathbb{R}^2$, and are supposed to show that ...