Applied Mathematics
Vol.05 No.10(2014), Article ID:46516,8 pages
10.4236/am.2014.510137
Characterization of Self Dual Lattices in
and
Comlan de Souza1, David W. Kammler2
1Department of Mathematics, California State University at Fresno, Fresno, USA
2Department of Mathematics, Southern Illinois University at Carbondale, Carbondale, USA
Email: csouza@csufresno.edu, dkammler@siu.edu
Copyright © 2014 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
Received 19 March 2014; revised 19 April 2014; accepted 26 April 2014
ABSTRACT
This paper shows that the only self dual lattices in are rotations of
,
and
.
Keywords:
Self dual Lattice
1. Introduction
Let
be nonsingular real matrices with column vectors
and
, respectively. Let
be the lattices in that are generated by the columns of
. The lattice
will be a subset of the lattice
if and only if the generators
of
all lie in
, i.e.,
for suitably chosen integers. Equivalently,
i.e.,
is a matrix of integers. Analogously, the lattice is a subset of
if
is a matrix of integers. In this way we see that
if and only if both and
are matrices with integer elements. When this is the case, and
are both integers and since
this implies that
Such a matrix is said to be unimodular. The above analysis (that can be found in [1] ) is summarized as follows.
Theorem 1 The lattices are identical if and only if
is a matrix of integers with
Corollary 1 Lattices are preserved under integer column operations.
Proof 1 Let generate the lattice
, and let
be a strictly upper triangular matrix of integers. Then is an upper triangular matrix of integers with a unit diagonal, and we can write
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.
2. Dual lattices
Definition 1 Two linearly independent sets of real (column) vectors
and
are said to be biorthogonal if
where is the Kronecker’s delta,
denotes the transpose and
denotes the usual inner product. When the columns of
and
are biorthogonal, we find
so that
This being the case, given linearly independent vectors we can form
and then obtain the biorthogonal vectors
as the columns of
The lattice generated by vectors biorthogonal to
is said to be the dual of the lattice
. More generally,
is dual to
if and only if
generates the same lattice as
, i.e.,
is a matrix of integers with determinant.
Suppose now that generate the same lattice, i.e.,
Let
be the generators of lattices dual to
, respectively. Since
we see that will be a matrix of integers with determinant
if and only if the same is true of
. Thus
if and only if
.
We are interested in characterizing those lattices that are self dual, i.e.,
This will be the case if and only if
is a matrix of integers with determinant. Since
this will be the case only if
or equivalently
In this way we see that a lattice is self dual if and only if
is a matrix of integers with unit determinant. The parallelopiped in
with vertices
, i.e., the unit cell of the lattice has the volume
[2] [3] . Thus a lattice can be self dual only if each of its primitive cells, has unit volume.
Self dual lattices are preserved under orthogonal transformations. Indeed, let be an orthogonal transformation on
, i.e.,
and let be the lattices generated by the columns of a nonsingular
matrix
and
. The matrix
has columns
that generate the lattice. We can use such a matrix
to rotate
, to reflect one or more vectors of the set
, to permute
, etc. The lattice
which is dual to
is generated by the columns of
i.e., by
Thus the generators of the dual lattice are transformed in the same way as the generators of the lattice
. In this way we see that a lattice
is self dual if and only if the lattice
is self dual. Indeed,
so is a matrix of integers with unit determinant if and only if the same is true of
. Moreover, since
we see that the orthogonal transformation preserves the Euclidean lengths of a set of generators for the lattice
.
3. Main results
We will now show that the only self dual lattices in are rotations of
, and
, respectively.
The case n = 1
Let be a vector in
that generates the lattice
. We do not change the lattice if we assume that
. Let
be biorthogonal to
. The lattice
generated by
will be identical to the
lattice if and only if
i.e., if and only if
Thus the only self dual lattice in is the lattice
The case n = 2
Theorem 2 Every self dual lattice in is some rotation of
.
Proof 2 Let where
are linearly independent vectors in
and assume that
is self dual. Fix the origin at some lattice point of
and rotate the axes, if necessary, so that the nearest nonzero lattice point of
lies on the positive
-axis, i.e.
where and
(1.1)
The lattice does not change if
is replaced by
so we can and do assume that
. Likewise the lattice
does not change if
is replaced by
since this is the result of an integer column operation. Thus we can and do assume that
(1.2)
By hypothesis the lattice is self dual so the same is true of
. This implies that
and
Since is self dual, the first column of
can be expressed as an integral linear combination of the columns of
, i.e.,
where. In this way we see in turn that
(1.3)
for some
(1.4)
for some and
(1.5)
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 and
. In this way we prove that
, i.e., the columns of
and thus those of
are orthonormal. Thus
is some rotation of
.
A theorem of Minkowski [1] states that
where is the shortest nonzero vector in a lattice
in
. Within the present context, this leads to the bound
which implies that Our argument gives
from which we immediatly obtain
.
Another result in [4] states that if is a self-dual lattice in
then
which leads to
The case n = 3
Theorem 3 Every self dual lattice in is some rotation of
.
Proof 3 Let the self dual lattice in
be generated by the columns of
chosen so that
are as small as possible subject to the constraint
Following the analysis from the previous section, we set
where is an orthogonal matrix chosen so that
with
By hypothesis the lattice is self dual, and since
is orthogonal, the same is true of
. This being the case
Since the lengths of the generators of the lattice are preserved under the orthogonal transformation
, it follows that
(1.6)
The columns of (and thus the columns of
) have been chosen to be as small as possible subject to the above constraints, so we must have
(1.7)
It can be verified that has the inverse
and after using to simplify the components we obtain
Since is self dual, the columns of
generate the same lattice as the columns of
so we can write
and
for suitably chosen In this way we see in turn that
(1.8)
(1.9)
for some and
(1.10)
We also have
(1.11)
(1.12)
for some and
so that
(1.13)
Using (1.7) and (1.8)-(1.12) we find
(1.14)
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 we must have
or
In this way we see in turn that and
so that
Finally, we again use (1.6) with (1.13), (1.12), (1.9) to write
It follows that so we must have
and
In this way we see that the columns of
( and thus those of
) must be orthonormal. Thus
is some rotation of
.
Suppose now that are linearly independent vectors in
and that
where. We know that
where the biorthogonal vectors are the columns of
. In this way we see that
if and only if is self dual, where
. This proves the following.
Theorem 4 Let be linearly independent vectors in
. Then
if and only if
for some orthonormal choice of the vectors.
Analogously, we can prove the following 3-dimensional generalization.
Theorem 5 Let be linearly independent vectors in
. Then
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 slightly complicates the generalization of this result.
References
- Kannan, R. (1987) Minkowski’s Convex Body Theorem and Integer Programming. Mathematics of Operations Research, 12, 415-440.
- Strang, G. (1976) Linear Algebra and Its Applications. Academic Press, Inc., New York.
- Senechal, M. (1995) Quasicrystals and Geometry. Cambridge University Press, New York.
- Conway, J.H., Odlyzko, A.M. and Sloane, N.J.A. (1978) Extremal Self-Dual Lattices Exit Only in Dimensions 1 to 8, 12, 14, 15, 23, and 24.