1. GROUPS & THEIR REPRESENTATIONS: a card shuffling approach
Wayne Lawton
Department of Mathematics
National University of Singapore
S ,

2. CRYSTALLOGRAPHY
as 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
is the
standard basis for the module
over the ring
over the field
just like it is a basis for the vector space
thus
For integers
is a subgroup of
isomorphic to

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

4. SMITH FORM FOR INTEGRAL MATRICES
Theorem 1. (Smith Form) Every integral matrix M with d-rows, can be reduced, using elementary row and column operations
with integer coefficients, to the unique diagonal form
where
and
where
and for
Remark The three examples on the preceding page are examples of reduction to Smith Form of integral matrices

5. ELEMENTARY INTEGRAL MATRICES
Integral row / column operations can be described by left / right multiplication by elementary integral matrices
Examples Left multiplication of a matrix with d-rows by
subtracts 3 times the third
row from the second row
interchanges the second and third row the third

6. IMPLICATIONS OF THE SMITH FORM
Corollary 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 elementary
matrices are elementary matrices and
Corollary 2. For every subgroup
there exists an
automorphism
with the amazing property:
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.

7. IMPLICATIONS OF THE SMITH FORM
Corollary 3. Let F be a finitely generated abelian group. Then
where
and
Remark.
is a cyclic group (generated by one element) and
Proof. Choose generators
define by
and
be the kernel of
Choose
and
as in Corollary 2, and then observe that,
by a standard result in group theory

8. IMPLICATIONS OF THE SMITH FORM
Corollary 4. (Chinese Remainder Theorem) If integers
are pairwise relatively prime
then
Proof. Clearly it since it suffices to prove this result when d = 2,
Assume
are relatively prime positive integers.
be its
and let
let
Smith Form, hence by Corollary
Since

9. PROOF OF SMITH FORM
Proof 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
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
To 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.

10. EXAMPLES OF ABELIAN GROUPS
For each positive integer, the multiplicative group
Amazing Fact: If N is prime then
Useful Fact: Rivest, Shamir, Adelman Public Key Cryptography

11. EXAMPLES OF ABELIAN GROUPS
Define the homomorphism by
The circle group is defined by
therefore, since
it follows that

12. GROUP REPRESENTATIONS
Let V, W be vector space over a field of dimension m, n.
mn dimensional F-vector space
F-linear maps from V into W
group of F-linear isomorphisms
from V onto V
Definition
A representation of a group G is a homomorphism
This means
therefore

13. UNITARY REPRESENTATION
Let V be a Euclidean, Hermitian space over R, C.
Definition
A representation of G over V is orthogonal, unitary if
Lemma
If G is finite and F = R, C and
is a representation then there exists a Euclidean, Hermitian structure ( - , - ) : V x V  F such that is orthogonal, unitary.
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

14. MASCHKE’S THEOREM Definition
A subspace W of V is invariant wrt a representation if
is irreducible if {0}, V are the only invariant subspaces.
Theorem 2 (Maschke)
Every representation over R, C of a finite
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.
Proof
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 satisfies
and 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.

15. ABELIAN GROUPS Theorem 3
Every representation of a abelian group over C
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.
Corollary
We can introduce the natural Hermitian structure on C
and then every irreducible representation of a finite group over C
is described by
where
is a homomorphism or character of G.
Remark
(Gauss 1801) If
then char. is

16. EXAMPLES Example The representation defined by is not semisimple.
(Persi Diaconis) For
let
denote the
permutation (or symmetric) group on n letters. Then using cycle notation the permutation group on 3 letters has 6 elements
and the following representation on
is irreducible

17. EXAMPLES Example The permutation representation
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.

18. To Be Continued (Added in Future)
Card shuffling and transition matrices constructed from the permutation representation
Convergence to the uniform distribution of the location of a single card as the number of shuffles increases
Generation of all permutations from a single transposition and a single cyclic permutation
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