of 47
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
1/47
-types,
a generalization of
-typesMathieu Dutour
Institut Rudjer Boskovic
Institut of Statistical Mathematics
p.1/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
2/47
I. Delaunay polytopes
and
-type theory
p.2/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
3/47
Empty sphere and Delaunay polytopes
A sphere
of radius and center in an -dimensionallattice
is said to be an empty sphere if:
(i)
for all
,
(ii) the set
contains
affinely independentpoints.
A Delaunay polytope
in a lattice
is a polytope, whosevertex-set is
.
cr
p.3/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
4/47
Gram matrix and lattices
Take an isometry of
!
."
is a Delaunay polytope of alattice
if and only if
"
is a Delaunay polytope of
. We want to study isometry classes of lattices.
Denote by
!
the vector space of real symmetric #
matrices and by
!
$
& the convex cone of positive definiteones.
Lattice
generated by
'
( ( (
! corresponds to
)
0
2
3
4
5
6
'
7
4
8
5
7
!
!
$
&
(
)
0 depends only on the isometry class of
.
Given
!
$
& , one can find vectors ' ( ( ( ! such that2
)
0 .
p.4/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
5/47
Gram matrix and lattices
Two matrices A
are arithmetically equivalent ifthere exist
)
!
B
such that
A
2
C
(
For any two basis D, DA
of a lattice
,)
E and)
E
F arearithmetically equivalent.
Lattices up to isometric equivalence correspond to
!
$
&
up to arithmetic equivalence.
In practice it is preferable to think and draw in terms oflattices, but to compute in terms of matrices in
!
$
& .
In the following, the Delaunay decomposition of a matrix
!
$
& is the Delaunay decomposition ofB
!
with
respect to the scalar product GC
H. p.4/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
6/47
-type domains
A
-type domain is the set of matrices
!
$
&
with thesame Delaunay decomposition.
Geometrically this means that the Gram matrices)
E ,)
E
F
of following lattices
and
A
1v
2v
2v
1v
are part of the same
-type domain.Specifying Delaunay polytopes, means putting somelinear equalities and inequalities on the Gram matrix
)
E .
A priori, infinity of inequalities but a finite numbersuffices.
p.5/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
7/47
Equivalence and enumeration
If there is no equalities, i.e. if all Delaunays aresimplices, then the
-type is called primitive.
The group)
!
B
acts on
!
$
& by arithmetic
equivalence and preserve the primitive
-type domains.Voronoi proved that there is a finite number of primitive
-type domains up to arithmetic equivalence.
Bistellar flipping creates new triangulations. In dim.I
:
Enumerating them is done classically:
Find one primitive
-type domain.
Find the adjacent ones by bistellar flipping andreduce by arithmetic equivalence. p.6/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
8/47
II. -types
(by Ryshkov & Barano
p.7/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
9/47
-primitivity
If"
is a Delaunay polytope, an edge P 2Q
'
R
S
of"
between two vertices ' and R of"
is a face of"
.
The edge P is encoded by its middle vectorT
P
2
'
R
'
R
. Up to translation, one can assume
that T
P
U
V
'
R
W
!
.
A parity class is a vector
U
V
'
R
W
!
U
V
W
; we denote byX Ythe set of all parity classes.
The matrix
!
$
& is said to be`
-primitive if for every
X
Y
, there exist an edgeP
2
Q
'
R
S
of the Delaunaydecomposition of such that T
P
2
.
p.8/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
10/47
-rigidity index
If
X
Y
, denote bya
the vectors, which are closestto .
a
is a centrally symmetric face of a Delaunaypolytope of
B!
. is at equal distance from all points in
a
so there isb
c
V
such that
C
2
b
for all
a
(
This makes linear equalities on .
The`
-rigidity index is defined as the dimension of the
space defined by those equalities.
p.9/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
11/47
-type
Denotee
f
B!
the family of all finite subsets ofB
!
.
A`
-type is
a functiona
g
X
Y
i
e
f
B
!
witha
being a collection of vertices inB
!
, which isinvariant by the action Gp
i
I
G.
A
`
-type is called primitive if for every
X
Y
, one hasa 2 U '
R
W
.
A primitive`
-type can be encoded by the family
U
R
'
q
X
Y
W
Primitive`
-types can be reconstructed from this
information.
p.10/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
12/47
-type domain
A`
-type is called realizable if there exists a matrix
!
$
& having centrally symmetric faces of Delaunay
being in this`
-type.
We will consider only realizable`
-types. Associated toa realizable
`
-type, there is its`
-type domain, i.e. theset of matrices
r
!
$
& whose centrally symmetric
faces are this`
-type.A
`
-type domain is primitive if and only if its centrallysymmetric faces are simply edges.
p.11/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
13/47
Matrix expression
Take a`
-typeY
and in the`
-type domain. For any
X
Y
, one should have
Take &
a
; for any
a
:
C
2
&
C
&
For any
B
!
a
:
C
c
&
C
&
The first part makes linear equalities, the second partmakes linear inequalities.
Hence`
-type domains are convex cone in !
$
& .
p.12/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
14/47
Dimension example
Take a lattice
2
B
'
B
R
:
v2
cv1
v +v1 2
The condition that '
R is outside of the edgeQ
'
I
'
S
of center 2s
R
' yields
q q
'
R
s
R
'
q q
c
q q
'
s
R
'
q q
i.e.q q
R
q q
R
3
'
R
6
V
In fact in dimensionI
,`
-types coincide with
-types.
p.13/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
15/47
General theorem
The group)
!
B
acts on the set of`
-type domains byarithmetic action
)
!
B
#
!
$
&
i
!
$
&
p
i
C
Thm.(Ryshkov & Baranovskii)`
-type domains are polyhedral cones.`
-type domains realize a face-to-face tesselation of
!
$
&
The
-type domain tesselation of !
$
& is a finite
refinement of the`
-type domain tesselation.
In a fixed dimension, there are a finite number of`
-type domains up to equivalence.
p.14/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
16/47
Results
All results on`
-type were obtained by Ryskov &Baranovskii.
They used it as technical tool for enumerating the
-types in dimensiont
.
dim Primitive Authors Primitiveu
-typesv
-types
2 1 Dirichlet (1860) 1
3 1 Fedorov (1885) 1
4 3 Voronoi (1908) 3
5 222 BaRy (1976), Engel & Gr (2002) 76
p.15/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
17/47
Proofs
If
!
satisfy all the condition of a`
-type then
!
w& .
proof: for
B!
andb
B
one has
b
C
b
&
C
&
with &
a
passing to the limitb
i
y , one obtains C
V
.
Every
-type domain
is contained in a unique`
-type domainY
denoted by
.proof: The
-type domain
defines all the Delaunay
polytopes. Computing their centrally symmetric faces,one obtains a
`
-type domain.
A`
-typeY
is the union of
-type domains.
proof: if
Y
, then
with
a
-typedomain. By the above
Y
. p.16/4
P f
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
18/47
Proofs
A`
-typeY
contains a finite number of
-type domains.
proof: There are a finite number of primitive
-typedomains up to equivalence. Take
' , . . . ,
some
representatives.
4
4
2
Y
4 .
Now it suffices to prove that for only a finite number of ) !
B
one has
C
4
Y
4 .
Take a Delaunay polytope"
of
4 and find a basis
P
'
( ( (
P
!
of
!
made of edges of"
If
C
4
Y
4 then
P
'
( ( (
P
!
is a family of edgesof
Y
4 . So, there is a finite number of possible
.
p.17/4
P f
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
19/47
Proofs
`
-type domains are polyhedral cone.proof: We know that
`
-type domains are finite union of
-type domains. Since`
-type domains are convex,
they are necessarily polyhedral.`
-type domains realize a face-to-face tesselation of
!
$
& .
proof:
!
$
&
is an union of
`
-type domains. They aredefined by linear inequalities, so automatically, thismakes a face-to-face tiling.
p.18/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
20/47
III. Algorithms
p.19/4
General algorithms
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
21/47
General algorithms
We want to enumerate primitive`
-type domains, thestrategy used is
Find a primitive`
-type domain and insert it into the
list of primitive`
-type domains.For every undone primitive
`
-type domain,Compute the non-redundant inequalities defining
it.For every facet, find the adjacent`
-type domain.For every adjacent
`
-type domain, do anisomorphism test with the elements in the existing
list and insert them if they are new.
p.20/4
Obtaining primitive type domain
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
22/47
Obtaining primitive -type domain
The algorithm is similar to the one for
-types.
Iterate the following
Find a random integral matrix, compute its Delaunaydecomposition.
If one of the Delaunay has a centrally symmetricface, which is not an edge, then we know that the
`
-rigidity index is less than!
!
'
R and we restart the
computation.
Otherwise, return the corresponding`
-type.
This algorithm is of Las Vegas type, i.e. it always returna correct answer but the running time is not known.
p.21/4
The geometrical picture
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
23/47
The geometrical picture
Geometrically the flipping consists in dim.I
of:
c c c
If one puts the three parity classes in dim.
:
ccc
p.22/4
Finding non redundant inequalities
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
24/47
Finding non-redundant inequalities
Find all doubles
'
R
such that
Q
V
'
S
and
Q
V
R
S
areedges of the Delaunay decomposition.
v
v2
1
0c
For any double
'
R
, define the linear inequality
R
C
R
'
C
'
with 2
I
'
Denote by
0
8
0
V
the corresponding inequality.
This form a finite setf
of inequalities. We extract a
non-redundant set fromf
by linear programming.
p.23/4
The flipping
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
25/47
The -flipping
Take a primitive`
-type domainY
and anon-redundant inequality
V
. We want to flipY
along the facet defining equality
2
V
.
Find all double
'
R
such that there is
c
V
with
0
8
0
2
.
For every such double replace the edgeQ
V
'
S
by the
edge
Q
R
'
R
S
.
v
v2
1
0c 0
v2
c
v v1 2
v v1 2
1v
We then get the adjacent primitive
`
-type domain.
p.24/4
Testing equivalence
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
26/47
Testing equivalence
Associate toY
with edge vectors
'
( ( (
R
'
thevector family
Y
2
'
'
( ( (
R
'
R
'
Two`
-type domainsY
andY
A
are equivalent if there
exist a matrix
)
!
B
such thatY
A
2
C
Y
.
In other wordsY
andY
A
are equivalent if and only ifthere exist a matrix
)
!
B
such that
Y
2
Y
A
.
The automorphism group question is expressedsimilarly.
p.25/4
Lemma
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
27/47
Lemma
IfY
is a`
-type, then its edgesB
-generatesB
!
.
proof: Take
a
-types such that
2
Y
. If
and A
are two vertices ofB
!
, then we can find a
sequence of vertices
2
&
( ( (
2
A
such that
4
and 4
'
belong to the same Delaunay polytope.
For any two vertices , A
of a Delaunay polytope"
one
can find a sequence of vertices 2 &
( ( (
2
A
such that 4
and 4
'
form an edge of"
.
Every edge of"
corresponds to an edge of the`
-type.
Hence,
A
2
b
P withb
B
p.26/4
Algorithm
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
28/47
Algorithm
To the`
-type with vector family
Y
one associates
2
0
C
Associates toY
the edge colored graph)
Y
on
Y
with edge colors
0
8
0
F
2
C
'
A
for any A
Y
There exist a matrix
)
!
such that Y 2 YA
if and only if the edge-colored graph)
Y
and)
Y
A
are isomorphic.
Y
and
Y
A
are
B
-generating, so
)
!
B
.
p.27/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
29/47
IV. Generalization
p.28/4
j
k
l -spaces
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
30/47
spaces
A
!
$
& -spacef
X
is a vector space of
!
, which intersect
!
$
& .
We want to study the centrally symmetric faces of
matrices
f
X
!
$
& .
Example of possible spaces are
f
X
)
2
U
m
! q
n
C
m
n
2
m
for alln
)
W
with)
a finite subgroup of)
!
B
.
p.29/4
-invariant faces
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
31/47
-invariant faces
Centrally symmetric faces are faces, which are invariantby a transformation Gp
i
G with
B!
.
If)
is a finite subgroup of)
!
B
, why not consider the
faces that are invariant under)
?
p.30/4
-faces
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
32/47
-faces
-types are the specification of all Delaunay, i.e. of-dimensional faces.
`
-types are the specification of centrally symmetric
faces but in primitive case, it is the specification of
-dimensional faces.
Would it be possible to extend the theory to the case ofo
-dimensional faces with
o
?After that one would want a subspace version of it
p.31/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
33/47
V. First
generalization
p.32/4
Settings
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
34/47
Settings
Takef
X
a
!
$
& -space.
We want to describe the centrally symmetric faces ofDelaunay decomposition of matrices in
f
X
!
$
& .
A
f
X
`
-type is defined as the assignation of centrallysymmetric faces of the Delaunay tesselation. A
f
X
`
-type domain is the corresponding convex cone.
A
f
X
`
-type domain is obtained as intersection of a`
-type domain (in !
$
& ) withf
X
. They are thus
polyhedral domains.
Two
f
X
`
-type domainsY
' andY
R are calledequivalent if there exist
)
!
B
such that
C
Y
'
2
Y
R .
p.33/4
Equivariance and finiteness
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
35/47
q
If)
is a finite subgroup of)
!
B
, then
f
X
)
2
U
!q
n
C
n
2 for all n
)
W
Thm.(Zassenhaus): One has the equality
U
n
)
!
B
|nf
X
)
n
2
f
X
)
W
2
a
)
Thm.(DSV): Take
a polyhedral face-to-face tiling of !
$
& , which is invariant under)
!
B
and has a finite
number of classes. If)
is a finite subgroup of)
!
B
then
f
X
)
has a finite number of classes underaction of
a
)
.
Thm. For a given finite group)
)
!
B
, there are afinite number of
`
-types under the action ofa
)
. p.34/4
Finiteness
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
36/47
Finiteness
Supposef
X
is an
!
$
& , define
{
f
X
2
n
)
!
B
such thatn
f
X
n
2
f
X
We know some examples wheref
X
is irrational such
that { f X 2
|
}
!
f
X
contains an infinite number of
f
X
`
-typedomains.
And so contain an infinite number of`
-types afteraction of
{
f
X
.
But we know no example withf
X
rational and an infinitenumber of
f
X
`
-types after action of
{
f
X
. p.35/4
General algorithms
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
37/47
g
A
f
X
`
-type domain is called primitive if it hasmaximal dimension in
f
X
.
We fix a !
$
& -spacef
X
and we want to enumerate
primitive
f
X
`
-type domains, the strategy used isFind a primitive
f
X
`
-type domain and insert itinto the list of primitive
f
X
`
-type domains
For every undone primitive
f
X
`
-type,Compute the non-redundant inequalities defining itFor every facet, find the adjacent
`
-type domain.
For every adjacent
f
X
`
-type domain, Do anisomorphy test with elements in the existing listand insert them if they are new.
Finding primitive
f
X
`
-type domain is easy: takeelement at random and finish when it is ok.
p.36/4
Linear inequalities
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
38/47
q
We have the family ofa
and we want to find thecorresponding inequality.
The first step consists in finding the facets ofa
. For
every such facet~
, find all centers
A
, such thata
A
anda
share~
. Saying that vertices ofa
A
areoutside the sphere around
a
makes one linear
inequality. Denote this inequality by
8
F
V
.There is a finite number of such inequalities.
We extract the set of non-redundant inequalities from
this finite set.
p.37/4
Lifted Delaunay decomposition
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
39/47
y p
The Delaunay polytopes of a lattice
correspond to thefacets of the convex cone
Y
with vertex-set:
U
G
q q
G
q q
R
with
G
W
'
(
Faces of Delaunay polytopes
faces of
p.38/4
Oriented graph
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
40/47
g p
Take a
f
X
`
-type and suppose we know all thenon-redundant inequalities of the
f
X
`
-type domain.Take
V
one such inequality.
Construct an oriented graph)
onX
Y
by
i
A
if and only if there is c
V
with
8
F
2
Take an oriented graph)
, the directed component"
`
of a vertex is the set of vertices A
of)
suchthat there exist a path
2
&
i
'
i
i
2
A
p.39/4
-repartitionning polytope
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
41/47
For every directed component"
`
of this graph, the
f
X
`
-repartitioning polytope
is the polytopewith vertex-set
with a vertex of a Delaunay of"
`
Every face of a Delaunay of the forma
A
with
A
"
`
correspond to a face of
.
c
c
p.40/4
Faces of
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
42/47
If
A
"
`
, then define the affine line
A
P
!
'
in
!
'
and create the intersection
A
P
!
'
2
Q
A
b
'
P
!
'
A
b
R
P
!
'
S
a
A
is the smallest face containing A
b
'
P
!
' .a
A
A
is the smallest face containing A
b
R
P
!
' .
Considerb
such that A
b
P
!
'
. Then thereexist G 0 , such that
A
b
P
!
'
2
0
G
0
withG
0
V
2
0
G
0
b
'
,
b
R
and
a
A
A
are found by linear programming. p.41/4
The -flipping
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
43/47
Take a
f
X
`
-type domain and
V
a relevantinequality of
Y
.
The
f
X
`
-flipping ofY
along
2
V
is realized in
the following way:Find all oriented directed component
"
`
For every A
"
`
, ifb
'
2
b
R , do linear
programming and changea
A
bya
A
A
.We then get the new
f
X
`
-type domain.
p.42/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
44/47
VI. Second
generalization
p.43/4
-parity classes
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
45/47
We take)
a finite subgroup of)
!
B
and consider thespace
f
X
)
. We do not assume that }
!
)
.
The)
-parity classes are the vectors
!
such that
for alln
)
one hasn
B!
.
We want a finite number of)
-parity classes
This means that we want the system of equationn G 2 G
withn
)
impliesG
2
V
.
For all G
!
one has n G 2V
.
p.44/4
Nearest neighbors
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
46/47
We assume that the solution of the equationn G
2
G
foralln
)
is onlyV
.
The seta
of nearest neighbors to a)
-parity class is)
-invariant.By above property one will have
q
a
q
0
2
All the preceding theory generalizes by replacing parityclasses by
)
-parity classes. Also, one can take a linearsubspace
f
X
off
X
)
.
p.45/4
8/3/2019 Mathieu Dutour- C-types, a generalization of L-types
47/47
THANK
YOU
p.46/4