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

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

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

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

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

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.

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

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

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.

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

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

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

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

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.

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

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

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.

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