cohomolgy designs as building blocks in constructions of ... · cohomolgy designs as building...

Cohomolgy Designs as building blocks in

constructions of Weighing and Hadamard

Matrices

Radel Ben-Av1 Giora Dula4 Assaf Goldberger2 Yossi Strassler3

ArasuFest, Kalamata Aug 1 2019

1Holon Institue of Technology [email protected]

2Tel-Aviv University [email protected]

3 Dan Yishay [email protected]

4 Netanya Accademic College [email protected]

Introduction

Weighing matrices

Definition

A Weighing Matrix W (N,w) is a {0,±1}-matrix W s.t.

WW T = wIN .

Examples (q an odd prime power):

• Hadamard Matrices H(N), (w = N)

• Payley Conference Matrices W (q + 1, q)

• Projective Spaces W(qn+1−1q−1 , qn

)• Smallest unknown: W (35, 25).

1

Monomial Transformations and Automprphisms

Monomial Transformations

Fact: Weighing Matrices are preserved by monomial

transformations:

X = W (n,w) =⇒ MXNT = W (n,w),

where M,N are monomial.

Definition

A monomial pair (M,N) is an Automorphism of X is

MXNT = X .

They form a group Aut(X ).

2

A Motivating Example

This is a W (7, 4):

X =

1 1 1 1 0 0 0

1 −1 0 0 1 1 0

1 0 −1 0 −1 0 1

1 0 0 −1 0 −1 −1

0 1 −1 0 0 1 −1

0 1 0 −1 1 0 1

0 0 1 −1 −1 1 0

• Every (permutation) automorphism of |X | (entrywise abs

value) can be lifted to an automorphism of X .

• How general is this phenomena?

3

The ALP

The Automprphism Lifting Problem (ALP):

The ALP

Given a {0,1}-matrix |X |, and a subgroup G ⊆ Aut(|X |), find all

{0,−1, 1}-matrices X above |X | and a subgroup G ⊆ Aut(X ),

mapping onto G .

4

Cohomology Developed Matrices

We begin with the following data:

• Let G be a finite group.

• Let X and Y be finite sets with G -action.

• Let O ⊂ X × Y be G -stable.

• Let µ be an Abelian Group, with G action. Let µ+ := µ∪{0}.

Definition

An (µ-valued) X × Y matrix (with support O) is a matrix

F = (f (x , y)) for some function f : X × Y → µ+,

and supp(f ) = O.

• G acts on X × Y matrices:

gF =(gf (g−1x , g−1y)

).

5

Cohomology Developed Matrices

Definition

Two matrices A and B are D-equivalent if

A = D1BD2, Di diagonal.

Write A ∼D B.

Definition

An X × Y F is a Cohomology Developed Matrix=CDM if

∀g ∈ G , gF ∼D F .

• So every g ∈ G induces an automorphism of F .

• But Note: The notion of automorphisms includes twisting

coefficients in µ.

6

Examples

• Group Developed Matrices: Take X = Y = G with actions

(g , x) 7→ xg−1 and (g , y) = gy , and µ with trivial action.

Then F = (f (xy)) satisfies gF = F for all g ∈ G .

• Cocyclic Matrices: Take X ,Y , µ as above, and let

F = (f (xy)ω(x , y)), ω : G × G → µ is a 2-cocycle:

(∗ − cocycle condition) ω(y , z)ω(x , yz) = ω(xy , z)ω(x , y).

Verification

Using the cocycle condition (∗):

ω(xg−1, gy)f (xg−1gy) = ω(g−1, gy)ω(x , y)f (xy)ω(x , g−1)−1

∼D ω(x , y)f (xy).

7

Examples-The Fourier Matrix

• The Fourier Matrix: Let G = the Affine Group over Z/n:

G = {x 7→ ax + b | a ∈ (Z/n)×, b ∈ Z/n}.

• Let X = Y = Z/n with the affine G -action.

• Let µ = µn = {n th roots of 1}, with the following twist

action:

(a, b)ζ = ζa2.

• Then F = (exp(2πixy/n)) is a CDM:

(a, b)F = diag(ωb2/2−bx) · F · diag(ωb2/2−by ).

8

Examples-DetFourier

• Let X = Y = (Z/n)2, with the action of

G = {v 7→ Av + a|A ∈ GL2(Z/n), a ∈ (Z/n)2}.

• Again let µ = µn with the twist action

(v 7→ Av + a)ζ = ζdetA.

• Then the n2 × n2 DetFourier Matrix given by

DF (u, v) = exp(2πi det(u, v)/n)

is a CDM. It is also Hermitian and Orthogonal.

9

CDMs and other Mathematics

A list of topics and extensions related to CDM’s:

• Group Cohomology

• The theory of Brauer Groups

• Hecke Algebras and Representation Theory

• Association Schemes and Weights (as of D.G. Higman)

• Constructions of Weighing and Hadamard Matrices

• Simplices and MUB’s in complex projective spaces

• Higher dimensional Design Theory

• (Dual theory) Magic Squares

-Will (not) be slightly discussed.

10

Some Theory

CDM Groups

Definition

Given G ,X ,Y , µ and O ⊂ X × Y as above, let

CDM(G ,O) = {All CDM’s w.r.t. G ,O}.

CDM(G ,O)D = CDM(G ,O)/D-equivalence.

• Both sets are groups w.r.t. the Hadamard Multiplication.

• The group CDM(G ,O)D is best described in terms of a

spectral sequence (not discussed here).

• The elements of CDM(G ,O) have G as a (monomial)

automorphism subgroup, whose permutation data was given

in advance by X ,Y and µ.

11

Cohomology Basics

Let G be a group, acting on a (multiplicative) Abelian group µ.

For any function

f : Gn → µ,

let di f : Gn+1 → µ given by

di f (g0, g1, . . . , gn) =

f (g0, . . . , gi−1gi , . . . , gn) i > 0

g0f (g1, . . . , gn) i = 0.,

and let

df =n∏

i=0

(di f )(−1)i .

Definition

1. f is a cocycle if df = 1,

2. f is a coboundary if f = dh. Coboundaries ⊆ Cocycles.

12

Cohomology Basics

The n-Cohomology Group is

Hn(G , µ) =n − cocycles

n − coboundaries.

If H is is subgroup of G , then there is a homomorphism

res : Hn(G , µ)→ Hn(H, µ),

by restricting cocycles to H.

13

Construction of CDM’s

For the sake of exposition, assume the following:

1) X ,Y are G -transitive, with chosen basepoints x0 ∈ X and

y0 ∈ Y .

2) O is irreducible: If L,R are diagonal and LAR∗ = A for all

O-supported matrices A, then L = R = scalar .

The Data we shall need is:

• G ,X ,Y , µ,O as above

• The stabilizers HX of x0 and HY of y0

• Two 1-Cocycles ψX : HX → µ and ψY : HY → µ

• One 2-Cocycle ω : G × G → µ

It is required that res ω will vanish in H2(HX , µ) and H2(HY , µ).

14


• Compute trivializations λX and λY s.t. ω|HX×HX= dλX and

ω|HY×HY= dλY .

- For any x ∈ X , y ∈ Y , choose gx , gy ∈ G , s.t.

x = gxx0

y = gyy0.

- Define maps forX : G → HX and forY : G → HY by

forX (g) = g · gg−1x0 ,

forY (g) = g · gg−1y0 .

15


- For any g ∈ G , x ∈ X , let:

h = forX (g−1x g).

- Let

δg ,x := gx

(ψX (h)λX (h)ω(g−1

x , g)

ω(h, 1)

).

- Finally, let

DX (g) = diag(δg ,x)x

and define DY (g) similarly.

16


Theorem

- The map A 7→ DX (g)(gA)DY (g)−1 defines a monomial G -action

on O-matrices.

- An invariant matrix A is a CDM.

- All CDM’s arise this way.

Theorem

There is a filtration CDM0 ⊆ CDM1 ⊆ CDM2 := CDMD such that:

CDM0 = {A|gA = A}

CDM1/CDM0 = A subquotient of H1(HX , µ)⊕ H1(HY , µ),

CDM2/CDM1 = A subgroup of H2(G , µ).

17

Orientability

• Having constructed a monomial G -action, we may try toconstruct a matrix by Spreading:

• Choose basepoints (orbit heads) for each G -orbit in

O ⊆ X × Y .

• Fix a value in µ for orbit head.

• Use the G -action to spread the values along the orbits.

Problem

The G -action may give conflicting values to some entries (orbits).

These orbits are said to be non-orientable.

Otherwise the orbit is orientable.

18

Example G = B3

• Consider the symmetry group of the cube, B3.

• We think of B3 as the group of 3× 3 monomial matrices.

• Let (ei = the standard vectors)

F = Faces of the cube = {±e1,±e2,±e3}

D = Antipodal Edges = {±ei ± ej , i < j}/{±1}.

• There are two B3-orbits in F × D:

1 2 2 1 1 1

1 1 1 1 2 2

2 1 1 2 1 1

2 1 1 2 1 1

1 1 1 1 2 2

1 2 2 1 1 1

19

Example Cont’d

• There are independent characters χ1, χ2 : B3 → µ = {±1}:

χ1(g) = det(g)

χ2(g) = product of signs in g .

• We may form a G -action by A 7→ χ1(g)χ2(g)(gA).

• Then orbit 1 is orientable and orbit 2 isn’t.

• The resulting CDM is

±

1 0 0 1 −1 −1

−1 1 1 −1 0 0

0 −1 −1 0 1 1

0 −1 −1 0 1 1

−1 1 1 −1 0 0

1 0 0 1 −1 −1

.

20

Application to Constructions of

Weighing and Hadamard matrices

Projective Space Weighing Matrices

• The well known W = W(qn+1−1q−1 , q, µn

)weighing matrices (q

a prime power, n|q − 1) happen to be CDM’s.

• In this setting we have

X = The points of Pn(Fq),

Y = The hyperplanes of Pn(Fq),

G = PGL(n + 1,Fq).

• Two orbits in X × Y , by occurence relations.

• Only the non-occuring orbit is orientable.

• The associated 2-cocycle ω is the one associated with the

extension

1→ F×q → GL(n + 1,Fq)→ G → 1.

21

Projective Space Weighing Matrices

• Q: But why is W orthogonal?

• A1: It is balanced. The products Wi ,jW∗i ,k are equi-distributed

along µn.

• A2: WW ∗ is a CDM with two orbits. Only the diagonal is

orientable!

• So non-orientability can be good!

22

Grassmannian and Flag Weighing Matrices

• By using the power of CDM theory, we may construct:

Grassmannian Weighing Matrices

Under similar conditions

∃W

([d

k

]q

, qk(d−k), µn

)

([ dk

]q=Gaussian Binomial Coefficients).

Flag Variety Weighing Matrices

Let d =∑r

i=0 ki , ki > 0. Then

∃W

([d

k0, k1, . . . , kr

]q

, q∑

i<j kikj , µn

),

Provided that n ≥ r . 23

Some Constructions

The Single/Double Circulant Core Structure:

A Ba b...

...a b

C Dc d...

...c d

e · · · e f · · · f p q

g · · · g h · · · h r s

G = Z/nZ, X = Y = G ∪ G ∪ {∗} ∪ {∗}

24

Single Negacyclic Core of order 2n (n-odd)

a1 a2 a3 a4 a5 a6 a b

−a6 a1 a2 a3 a4 a5 b −a−a5 −a6 a1 a2 a3 a4 −a −b−a4 −a5 −a6 a1 a2 a3 −b a

−a3 −a4 −a5 −a6 a1 a2 a b

−a2 −a3 −a4 −a5 −a6 a1 b −ac d −c −d c −d e f

−d c d −c −d c −f e

G = Z/nZ, X = Y = Z/nZ ∪ Z/2,

Negacyclic action. Such matrices form an Algebra.

25

This is an OD(12, 6, 6)

a −b −b b b b −b −a −a a a a

−b a b −b b b −a −b a −a a a

−b b a b −b b −a a −b a −a a

b −b b a −b b a −a a −b −a a

b b −b −b a b a a −a −a −b a

b b b b b a a a a a a −b

b a a −a −a −a a −b −b b b b

a b −a a −a −a −b a b −b b b

a −a b −a a −a −b b a b −b b

−a a −a b a −a b −b b a −b b

−a −a a a b −a b b −b −b a b

−a −a −a −a −a b b b b b b a

G ' A5 × Z/2Z, X = Y = G/D5 × {0}

Such matrices generate over Q the field Q(√

5,√−1).

26

Orthogonal Pairs

Definition

1) Two (possibly rectangular) CDM’s A,B are an Orthogonal Pair

(OP) if AB∗ is nowhere orienatble.

2) A tuple of CDM’s (A1, . . . ,Am) is an Orthogonal Tuple if each

(Ai ,Aj) is an OP.

• If (A,B) is an OP, then necessarily AB∗ = 0.

• Orthogonality is a consequence of the group action. So one

obtains symbolic (A,B) which are formally orthogonal.

• One may substitute (rectangular) blocks in place of symbols.

No commutativity/amicability restrictions.

27

Here is an example of an OP:

A =

−a −b a −b −a −b b b −b a b b

−b a b b b −b −b b −a −b a −a−b b −b −a b a −a a −b b −b b

−b b −b a b −a a −a −b b −b b

−a b a b −a b −b −b b a −b −b−b −a b b b −b −b b a −b −a a

B =

c −d −c −d −c −d d d −d c d d

−d c d −d d −d d d c −d −c −cd d −d −c −d −c c c d d −d −d−d −d d c d c −c −c −d −d d d

−c d c d c d −d −d d −c −d −dd −c −d d −d d −d −d −c d c c

.

28

This is a W (16, 8) constructed from OT’s. Each 16× 8 column isan OT .

1 0 1 0 −1 −1 0 0 0 −1 0 −1 0 0 1 1

−1 0 −1 0 0 0 1 −1 0 −1 0 −1 1 1 0 0

0 −1 0 1 1 1 0 0 −1 0 −1 0 0 0 1 1

0 −1 0 1 0 0 1 −1 1 0 1 0 −1 −1 0 0

0 0 0 0 −1 −1 1 −1 −1 1 −1 1 0 0 0 0

−1 1 −1 −1 0 0 0 0 0 0 0 0 −1 −1 1 1

0 −1 0 −1 0 0 −1 −1 1 0 −1 0 −1 1 0 0

0 1 0 1 −1 1 0 0 1 0 −1 0 0 0 1 −1

1 0 −1 0 0 0 1 1 0 1 0 −1 −1 1 0 0

−1 0 1 0 1 −1 0 0 0 1 0 −1 0 0 1 −1

−1 1 1 1 0 0 0 0 0 0 0 0 −1 1 −1 1

0 0 0 0 1 −1 1 1 1 −1 −1 1 0 0 0 0

1 1 0 0 1 0 0 −1 1 1 0 0 1 0 0 1

0 0 1 −1 0 1 1 0 0 0 1 1 0 1 1 0

−1 −1 0 0 −1 0 0 1 1 1 0 0 1 0 0 1

0 0 −1 1 0 −1 −1 0 0 0 1 1 0 1 1 0

G = B3 (Symmetry Group of the Cube)

Sets: X1 = X2 = Edges/{±1}, X3 = X4 = Orientations

Y1 = Y2 =Vertices,

Cohomology Classes: trivial 2-cocycle, nontrivial 1-cocycles.

29

A similar Hadamard 16 example (Same Group, different Sets)

1 −1 −1 1 −1 1 −1 1 1 −1 1 1 −1 −1 1 −1

−1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 −1 −1 1 −1

1 1 −1 −1 −1 1 1 −1 −1 −1 −1 1 1 1 1 −1

1 1 −1 −1 −1 1 1 −1 1 1 1 −1 −1 −1 −1 1

−1 1 1 −1 1 −1 1 −1 1 −1 1 1 −1 −1 1 −1

−1 1 1 −1 −1 1 −1 1 1 −1 1 1 1 1 −1 1

1 −1 1 −1 −1 −1 1 1 1 −1 −1 −1 −1 1 −1 −1

−1 1 −1 1 −1 −1 1 1 −1 1 1 1 −1 1 −1 −1

1 1 1 1 −1 −1 −1 −1 −1 −1 1 −1 1 −1 −1 −1

−1 −1 −1 −1 1 1 1 1 −1 −1 1 −1 1 −1 −1 −1

1 −1 1 −1 −1 −1 1 1 −1 1 1 1 1 −1 1 1

1 −1 1 −1 1 1 −1 −1 −1 1 1 1 −1 1 −1 −1

1 1 1 1 1 1 1 1 1 1 1 −1 1 1 1 −1

−1 −1 1 1 −1 1 1 −1 1 1 −1 1 1 −1 −1 −1

1 1 −1 −1 1 −1 −1 1 1 1 −1 1 1 −1 −1 −1

1 1 1 1 1 1 1 1 −1 −1 −1 1 −1 −1 −1 1

30

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