Index of /ag/schulze/Hermitian-lattices
explanation of the hermitian lists
These tables have been computed by
Alexander Schiemann when he was a member of my group. He also wrote the
explaining text below.
If you use the results available here please quote him (not me) as the
author, along with the article mentioned at the bottom of this page
that explains the algorithm used.
This Webpage is currently maintained by me.
Rainer Schulze-Pillot (for email
adress go up one level to the
homepage of my group).
The subdirectories contain complete classifications
of some genera of unimodular hermitian lattices in a hermitian space
over a complex quadratic field K=Q(sqrt(d)) .
Each subdirectory D-x contains genera in spaces over one
field with discriminant -x.
In these directories each file holds one genus of unimodular lattices.
All classes (with respect to
the unitary group U(V,h) ) are listed.
Files >50K are zipped with gzip. Unimodular genera of the
same dimension in different hermitian spaces are distinguished by
extensions .1, .2,...
Even and odd genera in the same space have an additional extension
(only spaces of even dimension over a field with even discriminant may
even unimodular lattices).
In each subdirectory w denotes a member of the "standard"
Z-base of the ring of integers of K=Q(sqrt(d))
where w=sqrt(d) if d=2,3 (mod 4) or w=(1+sqrt(d))
/2 if d=1 (mod 4).
The lattices are not necessarily free modules over the ring of
integers. Nevertheless, a lattice L in an n-dimensional
hermitian space admits a representation
where a_i are ideals in the ring of integers and x_i
vectors of a K-base of V. The lattice
L is given as a list of the ideals a_1... a_n and the
triangle of the Gram matrix with respect to x_1... x_n . The
ideals is omitted if all ideals are the whole ring of integers.
The K-base x_1... x_n is called a pseudo-base
of L and the ideal a_i the coefficient ideal
&Hlattice (#1 <-- #8) the
1st lattice of a list
13 lower triangle of its Gram matrix
3-6w 16 with respect to
a base x_1,..,x_n
|Aut| = 2^2*3 order of the group of automorphisms
0 0 12 12 numbers of integral vectors representing 1,2,3...
&Hlattice (#7 <-- #9) the
7-th lattice of a list
<1> <2,-1+w> list of ideals a_1...a_n
ideals are given by their Z-generators or <1>
for the whole ring of integers
2 lower triangle of Gram matrix
1/2w 1 note: 1/2w = (1/2)w
|Aut| = 2^2*3
0 6 0 6
note: lattices that are free modules over the ring of integers of
K may also be given in a "non-free representation" if such a
pseudo-base leads to
a Gram matrix with smaller coefficients.
For more information see
Classification of hermitian forms with the neighbour method,
J. Symbolic Comput. 26 (1998), no. 4, 487--508.
email: aschiem@math.***.de (with
Verantwortlich: Rainer Schulze-Pillot-Ziemen, Universität des Saarlandes, Saarbrücken