mathieu dutour- c-types, a generalization of l-types

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


2/47

I. Delaunay polytopes

and

-type theory

p.2/4


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


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


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


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


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


8/47

II. -types

(by Ryshkov & Barano

p.7/4


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


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


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


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


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


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


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


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


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


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


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


20/47

III. Algorithms

p.19/4

General algorithms


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


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


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


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


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


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


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


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


29/47

IV. Generalization

p.28/4

j

k

l -spaces


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


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


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


33/47

V. First

generalization

p.32/4

Settings


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


35/47

q

If)


!

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)


!

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


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


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


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


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


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


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


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


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


44/47

VI. Second

generalization

p.43/4

-parity classes


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


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


47/47

THANK

YOU

p.46/4

mathieu dutour- c-types, a generalization of l-types

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