0
votes
0answers
40 views

Subgroup of spinor norm 1 has index 2

Let $L$ be a lattice (free $\mathbb{Z}$-module) with even unimodular quadratic form $q$. Let $L$ be of even rank $2m$ and let's write $L_{\mathbb{R}}$ for $L\otimes \mathbb{R}$. I know that each ...
7
votes
1answer
212 views

Main problems on lattice-basis reduction algorithms (such as LLL)?

What are the main open problems on lattice-basis reduction algorithms (such as LLL)? I am looking for problems satisfying the following two conditions: (a) their solution would likely be of some ...
3
votes
2answers
156 views

Minkowski successive minima inequality for a lattice base?

Let $\Lambda$ be a lattice of $\mathbb{R}^n$, and $\lambda_i$ be the radius of the smallest ball containing $i$ linearly independent lattice vectors. The Minkowski successive minima inequality says ...
1
vote
0answers
120 views

Determine lattice basis from given lattice points

I'm working on the Shortest Lattice Vector Problem (SVP) for a paper that I'm currently writing. I wish to verify whether a particular structural, namely the building block property ( refer to the ...
1
vote
1answer
188 views

Diagonalization of Quaternion Hermitian matrices

How do I go about diagonalizing such a matrix. I ask because I need to sort out the following problem: Let $D$ be the quaternion algebra over $\mathbb{Q}$ with $i^2 = -1, j^2 = -11, ij=-ji=k$. ...
0
votes
0answers
112 views

Lattice basis reductions and finding minimal values

While reading several articles about lattice basis reduction I am left with a few questions. For one, I came across this piece of text Let $\alpha$ and $\beta \in \mathbb{R}$. Also let $X>0$ and ...
1
vote
0answers
39 views

Tensor product with $\mathbb{R}$ of an even unimodular lattice

Let $\Lambda$ be an unimodular even lattice of signature $(m,n)$. By a classifying theorem by Milnor, $\Lambda$ must be of the form $U^k\oplus E_8(\pm 1)^l$, where $U$ is the hyperbolic plane. Now ...
1
vote
0answers
74 views

Orthogonal trasformations with trivial spinor norm as product of reflections $r_w$ with $(w,w)=-2$

I'm trying to prove that, for a standard unimodular even lattice $\Lambda$ (by standard i mean that it is direct sum of copies of the hyperbolic plane $U$ and $E_8$) every element of $O^+(\Lambda)$ ...
0
votes
1answer
92 views

orthogonality in a lattice

Let $\Lambda$ be a lattice with a quadratic form $q$ of signature (3,19). Let $\Lambda_{\mathbb{R}}:=\Lambda\otimes \mathbb{R}$ and $W\subset \Lambda_{\mathbb{R}}$ a positive subspace of dimention 3. ...
1
vote
1answer
107 views

orthogonal base in unimodular lattice

Let $\Lambda$ be an unimodular lattice with a quadratic form $(-,-)$ of signature $(m,n)$ , $m,n>0$. I know that, fixed a base $e_1,\cdots,e_{m+n}$ for $\Lambda$, the matrix which has entries ...
2
votes
2answers
238 views

Lattice reduction on an orthonormal lattice?

Suppose you are given an inner product on a vector space and given a set of linearly independent vectors, and that you have been promised that the lattice they span has an orthonormal basis. Can you ...
1
vote
0answers
74 views

can minimal volume rational subspaces in a lattice be arbitrarily 'close'.

Let $\Gamma$ be a cocompact lattice in $\mathbb{R}^n$, eg. $\Gamma = A \mathbb{Z}^n$ for some $A \in SL_n \mathbb{R}$. Then any $k$-dimensional subspace $P$ which is rational in $\Gamma$ has a volume: ...
1
vote
0answers
122 views

Comparing the volume of a rational lagrangian under a linear symplectomorphism.

Let's fix the standard symplectic structure $(\mathbb{R}^{2g}, \omega, J)$. A (marked) symplectic lattice then has the form $A\mathbb{Z}^{2g}$ for $A \in Sp_{2g}\mathbb{R}$. We say a vector subspace ...
2
votes
0answers
252 views

Integer relation detection for Subset Sum or NPP?

Is there a way to encode an instance of Subset Sum or the Number Partition Problem so that a (small) solution to an integer relation yields an answer? If not definitely, then in some probabilistic ...
2
votes
1answer
437 views

How do you construct a symplectic basis on a lattice?

Is this possible to do constructively? The only sources that I have for the possibility of this construction is an exercise in Lang's Algebra (on p. 598, I believe) which states that one can be ...
2
votes
1answer
566 views

Theta Functions and Cousins

So I am (barely) familiar with the construction of the theta function of an integral lattice L. The theta function, as I understand it, is defined as the function which takes a variable z and spits ...