GROUPS & THEIR REPRESENTATIONS: a card shuffling approach

Wayne Lawton

Department of Mathematics

National University of Singapore

S14-04-04, matwml@nus.edu.sg

http://math.nus.edu.sg/~matwml

CRYSTALLOGRAPHYas well as the theories of numbers and equations motivated the study of groups. Consider a lattice group, it is isomorphic (equal structure, denoted by ) with the group of d-dimensional column vectors with integer entries under addition

The set

)(:],...,[ 1 folddZZZnnnZ iT

dd

TT ]1,0,...,0[e,...,]0,...,0,1[e d1 standard basis for the module

dZ

is the

over the ring Z

just like it is a basis for the vector space dR Rover the field

thus d22111 eee],...,[ dT

d nnnnn

.1 ZmZmG d

,,,1 dmm For integers Znmnmn iT

dd :],...,[ 11

is a subgroup of Z isomorphic to

ROW AND COLUMN OPERATIONSHere are three examples of elementary row and column operations on integral matrices using integer coefficients.

40

01

40

01

04

10

04

12

20

12

60

01

06

10

30

12

30

32

30

02

040

002

020

004

32900

004

329016

004

16900

408

12408

203024

SMITH FORM FOR INTEGRAL MATRICESTheorem 1. (Smith Form) Every integral matrix M with d-rows, can be reduced, using elementary row and column operationswith integer coefficients, to the unique diagonal form

0000

0000

0000

2

1

dm

m

m

S

where and0||| 21 dmmm 1 jjj ggm

where 10 g and for

} :){det(gcd M ofsubmatrix j x j a isjjj MMg Remark The three examples on the preceding page are examples of reduction to Smith Form of integral matrices

d11

ELEMENTARY INTEGRAL MATRICES

1000

0100

0310

0001

Examples Left multiplication of a matrix with d-rows by

Integral row / column operations can be described by left / right multiplication by elementary integral matrices

1000

0010

0100

0001

subtracts 3 times the third row from the second row

interchanges the second and third row the third

IMPLICATIONS OF THE SMITH FORM1Corollary 1. Every unimodular integral d x d matrix (det )

is the product of elementary integral matrices.

Proof. Let U be an unimodular d x d matrix. Clearly its Smith Form, obtained by left / right multiplication by elementary matrices, is the identity matrix since the product of its diagonal entries equals 1. The result follow since inverses of elementarymatrices are elementary matrices and

11

111111

FFEEUIFUFEE hkhk Corollary 2. For every subgroup

dZK dd ZZ :A

there exists an

automorphism with the amazing property:

ZmZmGAK d 1 0||| 21 dmmm and

Proof. Apply the Smith reduction to the integral matrix M whose columns are the elements of K and note that column operations do not change the group generated by the columns.

IMPLICATIONS OF THE SMITH FORM

dmm ZZF 1

., ZmmZZZm Remark.

and

Corollary 3. Let F be a finitely generated abelian group. Then

Proof. Choose generators

where

and

is a cyclic group (generated by one element) and

0||| 21 dmmm

mZ

}0{/1,}0/{0 10 ZZZZZZZZZZ

,,,1 Fff d define

by

FZ d :

,,],...,[ 111 Znfnfnnn iddT

d

by a standard result in group theory

0)(:)ker( xZxK d be the kernel of .Choose A Gand as in Corollary 2, and then observe that,

.//1 dmm

ddd ZZGZAKAZKZF

IMPLICATIONS OF THE SMITH FORM

Proof. Clearly it since it suffices to prove this result when d = 2,

let

21,aa are relatively prime positive integers.

and let

Smith Form, hence by Corollary Since

2

1

0

0

a

aA

Assume

21122011 ,1 aaggmggm

.2121 mmaa ZZZZ

Corollary 4. (Chinese Remainder Theorem) If integers

daa ,...,1 are pairwise relatively prime jijaia ,1),gcd(

thendd aaaa ZZZ

11

2

1

0

0

m

mS

2122110 ,1),gcd(,1 aagaagg

be its

2121 aaZmZmZ

PROOF OF SMITH FORMProof of Theorem 1. Since the result is obvious for d = 1,

we use induction on d. Let M be an integral matrix with d rows.

We can assume M has at least one nonzero entry and perform R&C Ops until the upper left element a is the smallest positive integer obtainable by R&C Ops. Hence a divides all the elements in the first row and first column (else it can be replaced by a smaller positive number through R & C Ops) and we can use

R & C Ops to make all the other elements in the first row and the first column equal to 0. We can transform the matrix to the form

),...,,( 2211 ddaaaadiag using R & C Ops on the last d-1 rows and columns. If a does not divide some diagonal entry then for some integers j, b, c

acabaaaA jjjj ),gcd(0To replace a by A : add b times 1-st column to j-column, add c times j-th row to 1-st row, then interchange 1 and j columns.

EXAMPLES OF ABELIAN GROUPS

Useful Fact: Rivest, Shamir, Adelman Public Key Cryptography

Amazing Fact: If N is prime then

5mod3prime ,;5)1)1)(1((; qpqpKpqN

1)34 (2Z1),(2*2mod5*2mod5*2mod5

4*5

*2mod3

2*3 ZZZ

Example 1. For each positive integer, the multiplicative group

1),gcd(,1:* NkNkkZN

as encryptedmessage 5 NxN

1*

NN ZZ

1)5462 (3Z*3mod7*3mod7*3mod7*3mod7*3mod7

6*7 Z

Nxy mod5

Nyx K mod messagerecover decrypt tothen

EXAMPLES OF ABELIAN GROUPS

byDefine the homomorphism

Rssies ,)(φ 2

Example 2.

The circle group is defined by

Z)(kernel

*:φ CR

1||:)(1 zCzRS therefore, since

/1 ZRS

it follows that

GROUP REPRESENTATIONS

group of F-linear isomorphismsfrom V onto V

W),(VHomF

Definition

)(: VAutG F

,),()()( Gvuvuuv

mn dimensional F-vector space F-linear maps from V into W

Let V, W be vector space over a field of dimension m, n.

(V)A Fut

A representation of a group G is a homomorphism

This means

)( VIid ,)()( 11 Guuu

therefore

UNITARY REPRESENTATION

Definition

)(: VAutG F

Gg ,,),,()))((),)((( Vvuvuvgug

Let V be a Euclidean, Hermitian space over R, C.

A representation of G over V is orthogonal, unitary if

If G is finite and F = R, C and

VvuvgugvuGg

,,)( ,)(),(

is a representation then there exists a Euclidean, Hermitian structure ( - , - ) : V x V F such that is orthogonal, unitary.

Lemma

Proof Choose a basis B for V over F and construct a Euclidean, Hermitian structure < , > : V x V F so that B is an < - , - > - orthonormal basis. Then define ( - , _ ) by

Remark Holds for compact groups – integrate wrt Haar Measure

MASCHKE’S THEOREMDefinition

)(: VAutG F

WwwuVuW ,0),(:

A subspace W of V is invariant wrt a representation

if

WWV

is irreducible if {0}, V are the only invariant subspaces.

Proof

Theorem 2 (Maschke) Every representation over R, C of a finite

GgWwWwg ,,)(

group is semisimple (or completely reducible) – that is V can be expressed as a direct sum of subspaces such that the restriction of the representation to each subspace is irreducible.

Construct an invariant Euclidean, Hermitian structure( - , - ) : F on V. Then V is irreducible or it has an invariant subspace W other than {0} and V. Then the complement of W

is clearly invariant and satisfiesand the result follows by induction on the dimension of V.Remark Maschke’s theorem holds for every finite field F

whose characteristic does not divide the order of G.

ABELIAN GROUPSTheorem 3

.w(g)w: W(g)),spec(, VwGg

Every representation of a abelian group over C

VvGgvgvg ,,)()(

can be decomposed into one dimensional representations. The converse is true for faithful representations.

Proof We can to consider an irreducible representation on V. Let

The eigenspace is clearly invariant hence it equals V.

We can introduce the natural Hermitian structure on C

and then every irreducible representation of a finite group over C

Corollary

is described by

where 1: SG is a homomorphism or character of G.

Remark (Gauss 1801) If prime ,* pZG p then char. is

p mod res.non residue, quadratic a gfor 1,1)( g

EXAMPLES

Znn

n

,

10

1)(

Example The representation

is not semisimple.defined by)(: 2CAutZ C

Example (Persi Diaconis) For Zn letn denote the

permutation (or symmetric) group on n letters. Then using cycle notation the permutation group on 3 letters has 6 elements

)231(),321(),31(),32(),21(,3 idand the following representation on 2C is irreducible

,

01

11))132((,

11

10))123((,

10

01)(

id

.01

10))13((,

11

01))23((,

10

11))12((

EXAMPLESExample )(: 3

3CAut

CThe permutation representation

,

001

200

010

))132((,

010

001

100

))123((,

100

010

001

)(

id

001

010

100

))13((,

010

201

001

))23((,

100

011

010

))12((

is not irreducible. It is isomorphic to the sum of the trivial representation on C and the 2-dimensional irreducible representation on the previous page.

To Be Continued (Added in Future)Card shuffling and transition matrices constructed from the permutation representation

Generation of all permutations from a single transposition and a single cyclic permutation

Convergence to the uniform distribution of the location of a single card as the number of shuffles increases

Regular representation and convolution

Convergence to the uniform distribution of the permutation of the entire deck as the number of shuffles increases

Fourier transform on the permutation group and the general theory of group representations

Estimates for the rate of convergence based on eigenvalues of irreducible representations