^{1}

^{*}

^{2}

^{*}

**This paper shows that the only self dual lattices in R, R^{2}, R^{3}^{} **

**are rotations of Z**

**, Z×Z**

**and Z×Z×Z**

**.**

Let

be nonsingular

be the lattices in

for suitably chosen integers

i.e.,

is a matrix of integers. Analogously, the lattice

if and only if both

are matrices with integer elements. When this is the case,

this implies that

Such a matrix is said to be unimodular. The above analysis (that can be found in [

Theorem 1 The lattices

is a matrix of integers with

Corollary 1 Lattices are preserved under integer column operations.

Proof 1 Let

be a strictly upper triangular matrix of integers. Then

where

is a strictly upper triangular matrix of integers. The columns of

i.e.,

generate the same lattice as the columns of A. To see this we observe that

is a matrix of integers with unit determinant.

Definition 1 Two linearly independent sets of real

where

and

are biorthogonal, we find

so that

This being the case, given linearly independent vectors

The lattice

is a matrix of integers with determinant

Suppose now that

Let

be the generators of lattices

we see that

We are interested in characterizing those lattices

This will be the case if and only if

is a matrix of integers with determinant

this will be the case only if

or equivalently

In this way we see that a lattice

[

Self dual lattices are preserved under orthogonal transformations. Indeed, let

and let

has columns

that generate the lattice

i.e., by

Thus the generators of the dual lattice

so

we see that the orthogonal transformation

We will now show that the only self dual lattices in

The case n = 1

Let

lattice

i.e., if and only if

Thus the only self dual lattice in

The case n = 2

Theorem 2 Every self dual lattice in

Proof 2 Let

where

The lattice

By hypothesis the lattice

and

Since

where

for some

for some

Using these expressions with (1.2) we find

so

Using these expressions with (1.1) we find

and since

this implies that

It follows that

A theorem of Minkowski [

where

which implies that

Another result in [

which leads to

The case n = 3

Theorem 3 Every self dual lattice in

Proof 3 Let the self dual lattice

Following the analysis from the previous section, we set

where

with

By hypothesis the lattice

Since the lengths of the generators of the lattice

The columns of

It can be verified that

and after using

Since

and

for suitably chosen

for some

We also have

for some

so that

Using (1.7) and (1.8)-(1.12) we find

Using (1.6) and (1.7) we see that,

which implies that

Again using (1.6) and (1.7) we see that,

which implies that

so that

Since

or

In this way we see in turn that

It follows that

Suppose now that

where

where the biorthogonal vectors

if and only if

Theorem 4 Let

if and only if

for some orthonormal choice of the vectors

Analogously, we can prove the following 3-dimensional generalization.

Theorem 5 Let

if and only if

for some orthonormal choice of the vectors

These results correspond to the familiar identity

III ^{˄}= III

from univariate Fourier analysis. The possibility of rotations (other than reflections) in