Moments of Random Matrices and Weingarten Functions. Yinzheng Gu

Transcription

1 Moments of Random Matrices and Weingarten Functions by Yinzheng Gu A project submitted to the Department of Mathematics and Statistics in conformity with the requirements for the degree of Master of Science Queen s University Kingston, Ontario, Canada August 2013 Copyright c Yinzheng Gu, 2013

2 Abstract Let G be the unitary group, orthogonal group, or (compact) symplectic group, equipped with its Haar probability measure, and suppose that G is realized as a matrix group. Consider a random matrix X = (x i,j ) 1 i,j picked up from G. We would like to know how to compute the moments E [ x i1 j 1 x injn x i 1 j 1 x i nj n] or E [xi1 j 1 x i2n j 2n ]. In this report, we focus on the unitary group U and use the methods established in [5] and [9] which express the moments as sums in terms of Weingarten functions. The function Wg U (, ), called the unitary Weingarten function, has rich combinatorial structures involving Jucys-Murphy elements. We discuss and prove some of its properties. Finally, we consider some applications of the formula for integration with respect to the Haar measure over the unitary group U. We compute matrix-valued expectations with the goal of having a better understanding of the operator-valued Cauchy transform. i

3 Acknowledgments First and foremost, I would like to thank my supervisors Serban Belinschi and James A. Mingo for their guidance and continuous support. I would also like to thank all of my professors and fellow students. I have learned a great deal from their wisdom and helpfulness. I am also thankful to all members and staff in the Department of Mathematics and Statistics at Queen s University. Finally, I am grateful to my family, especially my parents for their unconditional love and encouragement. ii

4 Table of Contents Abstract Acknowledgments Table of Contents i ii iii 1 Introduction and Preliminaries Expectation of products of entries Integration formula for the unitary group Integration formula for the orthogonal group Partitions and pairings Weingarten Functions and Jucys-Murphy Elements Gram matrices and Weingarten matrices Jucys-Murphy elements The group algebra C[S n ] Group representation theory Basic definitions The regular representation Maschke s theorem Artin-Wedderburn theorem Young tableaux Basic definitions The irreducible representations of S n Some properties of unitary Weingarten functions The invertibility of G Proof of Proposition Asymptotics of Wg U Additional properties of unitary Weingarten functions Explicit formulas for G and Wg U Character expansion of Wg U Integration over the Unitary Group and Applications Unitarily invariant random matrices Expectation of products of matrices iii

5 3.3 Matricial cumulants Conclusion and Future Work The Cauchy transform The operator-valued Cauchy transform A Free Probability Theory 49 B Tables of Values 51 B.1 Unitary Weingarten functions B.2 Orthogonal Weingarten functions Bibliography 53 iv

6 Chapter 1 Introduction and Preliminaries Let U denote the group of unitary matrices, with the group operation that of matrix multiplication. In other words, an complex matrix U U if UU = U U = I, where I is the identity matrix and U is the conjugate transpose of U. Viewing it as a subset of M (C) the algebra of complex matrices with usual matrix multiplication, it is common to endow U with the corresponding subspace topology so that U is compact as a topological space. By Haar s theorem, every locally compact Hausdorff topological group has a unique (up to a positive multiplicative constant) lefttranslation-invariant measure and a unique (up to a positive multiplicative constant) righttranslation-invariant measure. When the group is compact, these two measures coincide and we call it the Haar measure. This Haar measure is finite, therefore we can normalize it to a probability measure the unique Haar probability measure on the compact group. Definition 1.1. A random matrix is a matrix of given type and size whose entries consist of random numbers from some specified distribution. In other words, a random matrix is a matrix-valued random variable. Definition 1.2. We equip the compact group U with its Haar probability measure a probability measure on U which is invariant under multiplication from the left and multiplication from the right by any arbitrary unitary matrix. Random matrices distributed according to this measure will be called Haar-distributed unitary random matrices. Thus the expectation E over this ensemble is given by integrating with respect to the Haar probability measure. 1.1 Expectation of products of entries Integration formula for the unitary group The expectation of products of entries (also called moments or matrix integrals) of Haardistributed unitary random matrices can be described in terms of a special function on the 1

7 permutation group. Since such considerations go back to Weingarten [23], Collins [5] calls this function the (unitary) Weingarten function and denotes it by Wg U. otation 1.3. Let S n be the symmetric group on [n] = {1, 2,..., n}. As we shall see later, for α S n and n, Wg U (α, ) = U 11 U nn U 1α(1) U nα(n) du U = E [ ] U 11 U nn U 1α(1) U nα(n), where U is an Haar-distributed unitary random matrix, du is the normalized Haar measure, and Wg U is called the (unitary) Weingarten function. A crucial property of the Weingarten function Wg U (α, ) is that it depends only on the conjugacy class (in other words, the cycle structure) of the permutation α. This will be explained in more details in the next chapter. First, we introduce some notations by an example. Example 1.4. As mentioned above, Wg U (α, ) depends only on the cycle structure of α. Therefore when a table of values is provided, the value of Wg U (α, ) is usually given according to the cycle structure of α. For example, Wg U ([2, 1], ) denotes the common value of every α S 3 whose cycle decomposition consists of a transposition and a fixed point. Suppose α, β, γ S 3 such that α = (1, 2) = (1, 2)(3), β = (1, 3) = (1, 3)(2), and γ = (1, 2, 3), then by the table of values (for some values) of the unitary Weingarten functions, provided in appendix B, we have Wg U (α, ) = Wg U (β, ) = Wg U ([2, 1], ) = Wg U (γ, ) = Wg U ([3], ) = 1 ( 2 1)( 2 4) and 2 ( 2 1)( 2 4). In the paper [9], a formula is given so that general matrix integrals over the unitary group U can be calculated as follows: Theorem 1.5. Let U be an Haar-distributed unitary random matrix and n, then E [ U i1 j 1 U injn U i 1 j 1 U i n j n = ] α,β S n δ i1 i α(1) δ i ni α(n) δ j 1 j β(1) δ j nj β(n) WgU (βα 1, ), (1.1) 2

8 where U ij denotes the ij th entry of U and δ ij = { 1, i = j 0, i j. Remark ote that otation 1.3 follows immediately from this theorem and thus E [ ] U 11 U nn U 1α(1) U nα(n) is sometimes referred to as the integral representation of Wg U (α, ). 2. If n n, then by the invariance of the Haar measure, we have ] ] E [U i1 j1 U injn U i 1 j 1 U i n j n = E [λu i1 j1 λu injn λu i 1 j 1 λu i n j n for every λ C with λ = 1. ] Without loss of generality, assume n > n, then E [U i1 j1 U injn U i 1 j 1 U i n j n = ] λ n n E [U i1 j1 U injn U i 1 j 1 U i n j n and thus the integral vanishes since it is impossible to have λ n n = 1 λ C with λ = 1. otation 1.7. Let A = (a ij ) i,j=1 M (C) be an complex matrix, we denote by Tr (or Tr ) the non-normalized trace and tr (or tr ) the normalized trace: Tr(A) := a ii and tr(a) := 1 i=1 a ii = 1 Tr(A). i=1 Example 1.8. (An application of Theorem 1.5) Let U be an Haar-distributed unitary random matrix and B be an complex matrix. We wish to compute E [UBU ], where E denotes expectation with respect to the Haar measure. ote that E [UBU ] denotes the matrix-valued expectation of the random matrix UBU. In other words, ( ) E [UBU ] := (UBU ) ij du, U where du is the normalized Haar measure. i,j=1 Therefore we first compute the ij th entry of E [UBU ] and obtain E [UBU ] ij = E[U ij1 B j1 j 2 (U ) j2 j] j 1,j 2 =1 3

9 = = j 1,j 2 =1 j 1,j 2 =1 B j1 j 2 E [ U ij1 U jj2 ] B j1 j 2 δ ij δ j1 j 2 Wg U (e, ) (by (1.1) with n = 1 and α = β = e) = δ ij B j1 j 1 Wg U ([1], ) j 1 =1 = δ ij 1 Tr(B) = δ ij tr(b). This allows us to conclude that E [UBU ] = tr(b)i, where I denotes the identity matrix Integration formula for the orthogonal group The real analogue of a unitary matrix is an orthogonal matrix. Definition 1.9. An orthogonal matrix is a square matrix with real entries whose rows and columns are orthogonal unit vectors (i.e. orthonormal vectors). Equivalently, a matrix O is orthogonal if its transpose is equal to its inverse: O T = O 1. The set of orthogonal matrices forms a group O, known as the orthogonal group. The main focus of this report will be on the unitary group U, but it is worth mentioning that in the paper [9], the authors also defined a function Wg O on S 2n they called the orthogonal Weingarten function. Then they proved the following formula for integration with respect to the Haar measure over the orthogonal group O when n : Theorem Let O be an Haar-distributed orthogonal random matrix, then E [O i1 j 1 O i2n j 2n ] = p,q P 2 (2n) δ i1 i p(1) δ i2n i p(2n) δ j1 j q(1) δ j2n j q(2n) Wg O (p), q, (1.2) where P 2 (2n) is the set of all pairings of [2n] = {1, 2,..., 2n} and Wg O (p), q denotes the pq th entry of the orthogonal Weingarten matrix (to be introduced in section 2.1). Observe that, similar to the unitary case, the moments of an odd number of factors vanish. In other words, we always have E [ O i1 j 1 O i2n+1 j 2n+1 ] = 0 as the Haar measure is invariant under the transformation 1 1. However, to fully understand (1.1) and (1.2), we need some preliminaries on the constituent parts of the formulas, to be discussed in the 4

10 following sections. It is also worth mentioning that the authors of [9] considered the case of integration over the symplectic group in their paper as well, but for our purposes we will not discuss it in this report. 1.2 Partitions and pairings Definition Let S be a finite totally ordered set. We call π = {V 1,..., V r } a partition of the set S if and only if the V i (1 i r) are pairwise disjoint, non-empty subsets of S such that V 1 V r = S. We call V 1,..., V r the blocks of π. Definition For any integer n 1, let P(n) be the set of all partitions of [n]. If n is even, then a partition π P(n) is called a pair partition or pairing if each block of π consists of exactly two elements. The set of all pairings of [n] is denoted by P 2 (n). For any partition π P(n), let #(π) denote the number of blocks of π (also counting singletons). Let S n be the symmetric group on [n]. Given a permutation π S n, it is often convenient to consider the cycles of π as a partition of [n], thus #(π) also denotes the number of cycles in the cycle decomposition of π (also counting fixed points) when π is a permutation in S n. This map S n P(n) forgets the order of elements in a cycle and so is not a bijection. Conversely, given a partition π P(n), we put the elements of each block into increasing order and consider this as a permutation. Restricted to pairings this is a bijection. Definition Let π, σ be in P(n). The set P(n) is a finite partially ordered set (poset) in which π σ means each block of π is completely contained in one of the blocks of σ (that is, if π can be obtained out of σ by refining the block structure). The partial order obtained in this way is called the reversed refinement order. Definition Let P be a finite partially ordered set and let π, σ be in P. If the set U = {τ P τ π and τ σ} is non-empty and has a minimum τ 0 (that is, an element τ 0 U which is smaller than all the other elements of U), then τ 0 is called the join of π and σ, and is denoted by π σ. To make use of formula (1.2), we need to address one more question: given two pairings p and q, what is the relationship between the cycles of pq and the blocks of p q? First we illustrate it with an example. 5

11 Example If p, q P 2 (8) such that p = (1, 2)(3, 5)(4, 8)(6, 7) and q = (1, 6)(2, 5)(3, 7)(4, 8). Then pq = (1, 7, 5)(2, 3, 6)(4)(8) and p q = {{1, 2, 3, 5, 6, 7}, {4, 8}}. We notice that the cycles of pq appear in pairs in the sense that if (i 1,..., i k ) is a cycle of pq, then (q(i k ),..., q(i 1 )) is also a cycle of pq. This is not a coincidence as shown by the following lemma. Lemma (Lemma 2 in [16]) Let p, q P 2 (n) be pairings and (i 1, i 2,..., i k ) a cycle of pq. Let j r = q(i r ). Then (j k, j k 1,..., j 1 ) is also a cycle of pq, and these two cycles are distinct; {i 1,..., i k, j 1,..., j k } is a block of p q and all are of this form; #(pq) = 2#(p q). Proof. Since (i 1, i 2,..., i k ) is a cycle of pq, we have pq(i r ) = i r+1, thus j r = q(i r ) = ppq(i r ) = p(i r+1 ). Hence pq(j r+1 ) = pqq(i r+1 ) = p(i r+1 ) = j r. If {i 1,..., i k } and {j 1,..., j k } were to have a non-empty intersection, then for some n, q(pq) n would have a fixed point, but this would in turn imply that either p or q had a fixed point, which is impossible. Since {q(i r )} k r=1 = {j s } k s=1 and {p(j s )} k s=1 = {i r } k r=1, {i 1,..., i k, j 1,..., j k } must be a block of p q. Since every point of [n] is in some cycle of pq, all blocks must be of this form. Since every block of p q is the union of two cycles of pq, we have #(pq) = 2#(p q). This gives us a way to obtain the values of Wg O (p), q (p and q are pairings) for given p and q from the table of values (for some values) of the orthogonal Weingarten functions, provided in appendix B: since the cycles of pq always appear in pairs, we choose one cycle from each pair of cycles (in other words, delete half of the cycles of pq), then the value of Wg O (p), q depends only on the cycle structure of the modified pq. In the previous example where p = (1, 2)(3, 5)(4, 8)(6, 7), q = (1, 6)(2, 5)(3, 7)(4, 8), and pq = (1, 7, 5)(2, 3, 6)(4)(8). We have Wg O (p), q = Wg O ([3, 1], ) = 2 ( 1)( 2)( 3)( + 1)( + 2)( + 6). ow let us see how formula (1.2) can be applied to calculate general matrix integrals over the orthogonal group O n. Example Suppose O is an Haar-distributed orthogonal random matrix where Integrands of type O i1 j 1 O i2n j 2n which do not have pairings integrate to zero. For example, E [O 3 11O 12 ] = E [O 11 O 11 O 11 O 12 ] = If O i1 j 1 O i2 j 2 O i3 j 3 O i4 j 4 = O 2 11O 2 22 = O 11 O 11 O 22 O 22, then the indexes only admit one pairing, namely p = {{1, 2}, {3, 4}} and q = {{1, 2}, {3, 4}}. 6

12 Thus E [ ] O11O = Wg O ({{1, 2}, {3, 4}}), {{1, 2}, {3, 4}} = Wg O ([1, 1], ) + 1 = ( 1)( + 2). 3. If O i1 j 1 O i2 j 2 O i3 j 3 O i4 j 4 = O 4 11 = O 11 O 11 O 11 O 11, then all pairings are admissible because all the indexes are the same, equal to 1. A similar computation shows that E [ ] O11 4 3( + 1) 6 = ( 1)( + 2) = 3 ( + 2). 7

13 Chapter 2 Weingarten Functions and Jucys-Murphy Elements In this chapter, we study some of the properties of Weingarten functions. 2.1 Gram matrices and Weingarten matrices Let be a positive integer. We define the unitary Gram matrix G U n by G U n (α, β) = #(α 1β), the uni- and let Wg U n = ( Wg U tary Weingarten matrix. n (α, β) ) α,β S n be the pseudo-inverse of G U n = ( G U n (α, β) ) α,β S n. We call Wg U n Proposition 2.1. For any m n matrix A, there exists a unique n m matrix A +, called the Moore-Penrose pseudo-inverse (or simply pseudo-inverse) of A, satisfying all of the following four criteria: 1. AA + A = A; 2. A + AA + = A + ; 3. (AA + ) = AA + ; 4. (A + A) = A + A. ote that if A is invertible, then A + = A 1. Proof. First, if D is an m n (rectangular) diagonal matrix, then we define an n m matrix D + whose entries are { (D + (D ii ) 1, if D ii 0 for i = 1,..., min{m, n} ) ij =. 0, otherwise 8

14 It is easy to check that D + is the pseudo-inverse of D. The existence of A + then follows from the singular value decomposition theorem which states that any m n matrix A has a factorization of the form A = UDV, where U is an m m unitary matrix, D is an m n (rectangular) diagonal matrix with non-negative real numbers on the diagonal, and V is the conjugate transpose of an n n unitary matrix V. Let A + = V D + U and we can show that A + is the pseudo-inverse of A: 1. AA + A = UDV V D + U UDV = UDD + DV = UDV = A; 2. A + AA + = V D + U UDV V D + U = V D + DD + U = V D + U = A + ; 3. (AA + ) = (UDV V D + U ) = (UDD + U ) = U(DD + ) U = UDD + U = UDV V D + U = AA + ; 4. (A + A) = (V D + U UDV ) = (V D + DV ) = V (D + D) V = V (D + D)V = V D + U UDV = A + A. To prove the uniqueness of A +, suppose both B and C are n m matrices satisfying all of the pseudo-inverse criteria, then AB = (AB) = B A = B (ACA) = B A C A = (AB) (AC) = ABAC = AC. Similarly, we have BA = CA and therefore B = BAB = BAC = CAC = C. Analogously, we define the orthogonal Gram matrix G O n = ( G O n (p, q) ) by p,q P 2 (2n) G O n (p, q) = #(p q), and the orthogonal Weingarten matrix Wg O n as the pseudo-inverse of G O n. Example 2.2. Suppose n = 3 and n, then G U 3 = () (1, 2) (1, 3) (2, 3) (1, 2, 3) (1, 3, 2) () (1, 2) (1, 3) (2, 3) (1, 2, 3) (1, 3, 2) Since the determinant of G U 3 is is invertible and we have Wg U 3 = ( G U 3 ) ( 2 1) 5 ( 2 4) and 3 by assumption, GU 3 9

15 = 1 ( 2 1)( 2 4) Therefore σ S 3 Wg U (σ, ) is the sum of entries of any row or column of Wg U 3. In particular, choosing the first row gives us Wg U (σ, ) = Wg U 3 (e, β) σ S 3 β S 3 { } = ( 2 1)( 2 4) 1 = ( + 1)( + 2). Remark 2.3. ote that we implicitly used the fact that the elements in the first row of the unitary Weingarten matrix Wg U n define the unitary Weingarten function Wg U by Wg U (π, ) = Wg U n (e, π), where π S n. This will be explained in more details in the following sections. Moreover, the calculation above leads to the following known fact. We will state it as a proposition here, to be proved later. Proposition 2.4. For all k 1, σ S k Wg U (σ, ) = 1 ( + 1) ( + k 1). 2.2 Jucys-Murphy elements Let n be a positive integer, then the group algebra C[S n ] is an algebra (over C) with the symmetric group S n as a basis, with multiplication defined by extending the group multiplication linearly, and an involution defined by σ = σ 1, σ S n, and extended conjugate linearly. 10

16 Definition 2.5. Let n be a positive integer. Consider the natural embedding C[S 1 ] C[S 2 ] C[S n ], where elements of C[S k ] act trivially on numbers greater than k. The Jucys-Murphy elements J 1, J 2,..., J n C[S n ] are transposition sums defined by: J 1 = 0 and J k = (1, k) + (2, k) + + (k 1, k) for 2 k n. Remark The definition of J 1 is a convenient convention. 2. For n 2, J n commutes with C[S n 1 ]. Indeed, for every σ S n 1, we have σj n σ 1 = n 1 n 1 σ(i, n)σ 1 = (σ(i), n) = J n. Therefore J m J n = J n J m for all m and n. This i=1 i=1 implies that C[J 2,..., J n ] is a commutative subalgebra of C[S n ]. It is in fact a maximal commutative subalgebra known as the Gelfand-Zetlin subalgebra as shown by Okounkov and Vershik in [20]. ext, we need the classical identity (see [13]). Theorem 2.7. (Jucys) Let be a positive integer, then n ( + J k ) = #(σ) σ. (2.1) σ S n k=1 There are many ways to prove this identity. We will follow the one provided by Zinn- Justin as Proposition 1 in [24], based on a standard inductive construction of permutations. Proof. First, note that the right-hand side has n! terms and the left-hand side is equal to ( + J 1 )( + J 2 ) ( + J n ), which also has n! terms after expanding the product. Our proof will consist of a term by term identification by induction on n. If n = 1, then the statement is trivial as both the left-hand side and the right-hand side are equal to. Assume the statement holds for all natural numbers less than or equal to n 1 and consider a permutation σ S n. The goal is to identify #(σ) σ with one of the terms in the expansion of ( + J 1 ) ( + J n ). There are two cases: 1. If n is a fixed point of σ, then we apply the induction hypothesis to σ {1,...,n 1} S n 1, and we know that #(σ {1,...,n 1}) σ {1,...,n 1} corresponds to one term in ( +J 1 ) ( + J n 1 ). Furthermore, since σ has one more cycle than σ {1,...,n 1}, we can identify #(σ) σ with the term in ( +J 1 ) ( +J n ) with the same choice as #(σ {1,...,n 1}) σ {1,...,n 1} in the first n 1 factors, and we further pick the multiplication by in + J n. 11

17 2. Suppose n is not a fixed point. First, we notice that if σ = (other cycles of σ)(σ 1 (n), n, σ(n),... ), then σ(σ 1 (n), n) = (other cycles of σ)(σ 1 (n), σ(n),... )(n), and thus we can apply the induction hypothesis to (σ(σ 1 (n), n)) {1,...,n 1} S n 1. Since σ has as many cycles as (σ(σ 1 (n), n)) {1,...,n 1}, we can identify #(σ) σ with the term in ( + J 1 ) ( + J n ) with the same choice as #((σ(σ 1 (n),n)) {1,...,n 1}) (σ(σ 1 (n), n)) {1,...,n 1} in the first n 1 factors, and we further pick the transposition (σ 1 (n), n) inside + J n. The proof is completed. This theorem will play an important role in our study of the unitary Weingarten functions. First, we review some relevant concepts. 2.3 The group algebra C[S n ] Let C[S n ] be the group algebra with the symmetric group S n as a basis. If A(S n ) is the algebra of all complex-valued functions on S n with multiplication given by the convolution (f g)(π) = σ S n f(σ)g(σ 1 π), (f, g A(S n ), π S n ), then C[S n ] is isomorphic as a complex algebra to A(S n ), via the map ϕ : C[S n ] A(S n ), σ δ σ, { 1, σ = π where δ σ : S n C is the function defined by δ σ (π) = 0, σ π. It is easily checked that for any σ 1, σ 2 S n, δ σ1 σ 2 = δ σ1 δ σ2 (δ σ1 δ σ2 )(π) = τ S n δ σ1 (τ)δ σ2 (τ 1 π) = δ σ1 (σ 1 )δ σ2 (σ 1 1 π) = δ σ2 (σ 1 1 π) = δ σ1 σ 2 (π), π S n. under this map: In general, if a = σ S n α σ σ C[S n ] where α σ C, then a a = σ S n α σ δ σ under ϕ. Since a (π) = ( σ S n α σ δ σ ) (π) = α π, a is sent to the function a that maps π to α π. 12

18 Moreover, if b = β τ τ C[S n ] where β τ C, then b b = β τ δ τ under ϕ and so τ S n τ S n ( ) ( ) ab = α σ σ β τ τ σ S n τ S n = α σ β τ στ σ,τ S n = ( ) α τ β τ 1 σ σ. σ S n τ S n Therefore ab (ab) = ( ) α τ β τ 1 σ δ σ as an element in A(S n ) and thus σ S n τ S n { ( ) } (ab) (π) = α τ β τ 1 σ δ σ (π) τ S n σ S n = τ S n α τ β τ 1 π = τ S n a (τ)b (τ 1 π) = (a b )(π), π S n. Under this isomorphism, we may pass easily between C[S n ] and A(S n ). For simplicity, we will denote both C[S n ] and A(S n ) by C[S n ] in the following sections and it should be clear from context which algebra we refer to. ow, note that the right-hand side of (2.1) is σ S n #(σ) σ C[S n ]. It is sent to ( ) #(σ) δ σ A(S n ) under the map ϕ. Since #(σ) δ σ (π) = #(π), this means σ S n σ S n #(σ) σ is sent to the function that maps π to #(π). If we let G A(S n ) be the σ S n n function G(σ) = #(σ), then (2.1) becomes ( + J k ) = G. k=1 2.4 Group representation theory In this section, we briefly review some concepts from group representation theory that will be used later in this report. There might be more information than actually needed, it will be a good introduction of the subject nevertheless. 13

19 2.4.1 Basic definitions We begin with some basic definitions. Definition 2.8. A representation of a group G on a vector space V over a field K is a group homomorphism from G to GL(V ), the general linear group on V. In other words, a representation is a map ρ : G GL(V ) such that ρ(g 1 g 2 ) = ρ(g 1 )ρ(g 2 ), for all g 1, g 2 G. Definition 2.9. Let ρ : G GL(V ) be a representation. 1. Let V be an n-dimensional vector space and K = C. If we pick a basis of V, then every linear map corresponds to a matrix, so we get an isomorphism GL(V ) = GL n (C) (dependent on the basis). The representation ρ now becomes a homomorphism ρ : G GL n (C), and the number n is called the dimension (or the degree) of the representation. 2. The trivial representation of G assigns to each element of G the identity map on V. 3. If ρ is injective, then we say ρ is a faithful representation. In other words, different elements g of G are represented by distinct linear mappings ρ(g) and thus ker(ρ) = {e}. 4. A subrepresentation of ρ is a vector space W V such that ρ(g)(x) W g G and x W. This means that every ρ(g) defines a linear map from W to W, i.e. we have a representation of G on the subspace W and we call W an invariant subspace of ρ. If we pick a basis ( of a subrepresentation ) W and extend it to V, then for all g G, ρ(g) takes the form where the top-left block is a dim(w ) dim(w ) matrix. 0 Thus, this block gives the matrix for a representation of G on W. 5. If ρ has exactly two subrepresentations, namely the 0-dimensional subspace and V itself, then the representation ρ is said to be irreducible. If it has a proper subrepresentation of non-zero dimension, the representation is said to be reducible. ote that the representation of dimension 0 is considered to be neither reducible nor irreducible and any 1-dimensional representation is irreducible. 6. Let G = S n, if ρ : σ 1 τ 1 for all even permutations σ and odd permutations τ, then ρ is called the sign representation, i.e. ρ(σ) is simply multiplication by sgn(σ) for each σ S n. 14

20 7. Let G = S n and V be an n-dimensional vector space with standard basis {e 1,..., e n }. Then we have the permutation representation ρ(σ)(e i ) = e σ(i), σ S n. For example, suppose G = S 3, then we have, in matrix forms, ρ((1, 2)) = and ρ((2, 3)) = In other words, ρ is a permutation representation if ρ(g) is a permutation matrix for all g G. 8. The character of the representation ρ is the function χ ρ : G K g Tr(ρ(g)). ote that if g 1 and g 2 are conjugate in G, then χ ρ (g 1 ) = χ ρ (g 2 ) since trace is constant under conjugation, so we can view χ ρ as a function on the conjugacy classes of G The regular representation Recall that C[S n ] is the algebra of all complex-valued functions on S n. It has a basis {δ σ } σ Sn, where δ σ is the function sending σ to 1 C and all other group elements to 0 C. Given a function f : S n C and an element σ S n, the left regular representation λ(δ σ ) is defined by λ(δ σ ) : f λ(δ σ )f on the basis element δ σ, where (λ(δ σ )f)(π) = f(σ 1 π) for all π S n. Similarly, the right regular representation ρ(δ σ ) sends the function f to ρ(δ σ )f on the basis element δ σ, where (ρ(δ σ )f)(π) = f(πσ) for all π S n. ow, if G C[S n ] is the function G(σ) = #(σ), it can be seen that the unitary Gram matrix G U n is the matrix of G acting in either the left or right regular representation of C[S n ], with standard basis {δ σ } σ Sn. If G U n is invertible, then G is invertible in C[S n ] and we let the inverse of G be Wg U, the unitary Weingarten function Maschke s theorem Let V and W be two vector spaces over C. Recall that the direct sum V W is the vector space of all pairs (x, y) such that x V and y W. Its dimension is dim(v ) + dim(w ). 15

21 Suppose G is a group and we have representations ρ V : G GL(V ), ρ W : G GL(W ). Then there is a representation ρ V W of G on V W given by ρ V W : G GL(V W ), ρ V W (g) : (x, y) (ρ V (g)(x), ρ W (g)(y)). Moreover, suppose dim(v ) = n and dim(w ) = m. We can choose a basis {a 1,..., a n } for V, and {b 1,..., b m } for W, then the set {(a 1, 0),..., (a n, 0), (0, b 1 ),..., (0, b m )} is a basis for V W. For all g G, ( ρ V (g) GL n (C),) ρ W (g) GL m (C), and ρ V W is given ρv (g) 0 by the (n + m) (n + m) matrix under the induced basis on V W. 0 ρ W (g) A matrix of this form is called block-diagonal. Remark Suppose ρ : G GL(V ) is a representation. ote that 1. If V = W U, then the subspace {(x, 0) x W } W U is a subrepresentation and it is isomorphic to W. Similarly, the subspace {(0, y) y U} is isomorphic to U. The intersection of these two subrepresentations is {0}. 2. If W V and U V are subrepresentations such that W U = {0} and dim(w ) + dim(u) = dim(v ), then V = W U. This raises the following question: suppose ρ : G GL(V ) is a representation and W V is a subrepresentation, is there another subrepresentation U V such that V = W U? It turns out the answer to this question is always yes, and U is called a complementary subrepresentation to W. This is known as Maschke s theorem. It is an important theorem in group representation theory and the proof can be found in most of the graduate level algebra texts (see, e.g. [10]). We will state the theorem here and omit the proof. Theorem (Maschke s theorem) Let ρ : G GL(V ) be a representation and W V be a subrepresentation. Then there exists a complementary subrepresentation U V to W. This leads to the following result, called the complete reducibility theorem. 16

22 Corollary Every complex representation of a finite group can be written as a direct sum W 1 W 2 W r of subrepresentations, where each W i is irreducible. Proof. Let G be a finite group and V be a representation of G of dimension n. If V is irreducible, then the statement holds trivially. If not, V contains a proper subrepresentation W V, and by Maschke s theorem, V = W U for some other subrepresentation U. Both W and U have dimension less than n. If they are both irreducible, the proof is complete. If not, at least one of them contains a proper subrepresentation, so it splits as a direct sum of smaller representations. Since n is finite, this process will terminate in a finite number of steps. Definition If a representation can be decomposed as a direct sum of irreducible subrepresentations, then it is said to be completely reducible Artin-Wedderburn theorem The Artin-Wedderburn theorem is a classification theorem for semisimple rings and semisimple algebras. For our purposes, we only need the following corollary. Corollary Let G be a finite group, then C[G] = i M ni (C), where the direct sum is indexed by (all of) the irreducible representations of G and n i is the dimension of the i th irreducible representation. 2.5 Young tableaux In this section, we explore a connection between representations of the symmetric group S n and combinatorial objects called Young tableaux Basic definitions Definition A partition of a positive integer n is a sequence of positive integers λ = (λ 1, λ 2,..., λ r ) satisfying λ 1 λ 2 λ r > 0 and n = λ 1 + λ λ r. We write λ n to denote that λ is a partition of n. 17

23 Definition A Young diagram is an array of boxes arranged in left-justified rows, with the row sizes weakly decreasing (i.e. non-increasing). The Young diagram associated to the partition λ = (λ 1, λ 2,..., λ r ) is the one that has r rows, with λ i boxes in the i th row. Example The number 4 has five partitions: (4), (3, 1), (2, 2), (2, 1, 1), (1, 1, 1, 1) and the associated Young diagrams are. It is clear that there is a one-to-one correspondence between partitions and Young diagrams. Definition Suppose λ n. A Young tableau T of shape λ is an assignment of the numbers {1, 2,..., n} to the n boxes of the Young diagram associated to λ such that each number occurs exactly once. For example, here are all the Young tableaux corresponding to the partition (2, 1): Definition A standard Young tableau T of shape λ, denoted by SYT(λ), is a Young tableau whose entries are increasing along each row (to the right) and each column (downwards). The only two standard Young tableaux for (2, 1) are and Definition A skew shape is a pair of partitions (λ, µ) such that the Young diagram of λ contains the Young diagram of µ, it is denoted by λ/µ. 18

24 2. The skew diagram of a shew shape λ/µ is the set-theoretic difference of the Young diagrams λ and µ: the set of squares that belong to the diagram of λ but not to that of µ. 3. A skew tableau of shape λ/µ is obtained by filling the squares of the corresponding skew diagram. For example, the following is a skew tableau of shape (5, 4, 2, 2)/(3, 1): The irreducible representations of S n Representations are often used to characterize finite groups. In this report, we are mostly interested in the symmetric group S n. By Corollary 2.12, we know that every representation of S n can be decomposed as a direct sum of a finite number of irreducible representations, but the corollary does not tell us the number of distinct irreducible representations of a given group and their dimensions. The answer to the first question is given by the following theorem, sometimes called the completeness of irreducible characters. Theorem The number of irreducible representations of a finite group is equal to the number of conjugacy classes of that group. The proof of this theorem is beyond the scope of this report and we refer to [11] for those who are interested. Since we focus on S n, we want to know the number of conjugacy classes and then classify all representations of S n. Let us take a look at S 3 first. Definition The standard representation of a symmetric group on [n] = {1, 2,..., n} is an irreducible representation of dimension n 1 (over a field whose characteristic does not divide n!) defined in the following way: Take the permutation representation of S n as defined in Definition 2.9 and look at the (n 1)-dimensional subspace of vectors whose sum of coordinates in the basis is zero. The representation restricts to an irreducible representation of dimension n 1 on this subspace. This is the standard representation. 19

25 Example Since S 3 has three conjugacy classes, it has three irreducible representations: 1. One of them is the trivial representation which is on C and acts by σ(α) = α for σ S 3 and α C. 2. Another one is the alternating representation (or sign representation) which is also on C, but acts by σ(α) = (sgn(σ))α for σ S 3 and α C. 3. The third one is the standard representation and it is on the subspace { (z1, z 2, z 3 ) C 3 z 1 + z 2 + z 3 = 0 } acting by σ((z 1, z 2, z 3 )) = (z σ(1), z σ(2), z σ(3) ) for σ S 3 and (z 1, z 2, z 3 ) C 3. These are the only irreducible representations of S 3. Let ρ be a representation of S 3 and ρ triv, ρ alt, and ρ std denote the trivial, alternating, and standard representations respectively. Then for any σ S 3, ρ(σ) = aρ triv (σ) bρ alt (σ) cρ std (σ), where a, b, and c are values determined by ρ. The dimensions of ρ triv (σ), ρ alt (σ), and ρ std (σ) are 1, 1, and 2 respectively. ote that for n > 3, there are more irreducible representations than just these three and it turns out that the number of conjugacy classes of S n is the number of ways of writing n as a sum of a sequence of non-increasing positive integers. Proposition The conjugacy classes of S n correspond to the partitions of n. Proof. The conjugacy classes of S n are uniquely determined by the cycle type of their elements and each class contains only one cycle type. Thus, there is a bijection between each conjugacy class of S n and a partition of n into the cycle type of that class. Therefore the number of irreducible representations of S n is same as the number of Young diagrams corresponding to the partitions of n. For example, there are five irreducible representations of S 4 according to Example The dimension of each irreducible representation of S n corresponding to a partition λ of n is equal to the number of different standard Young tableaux that can be obtained from the Young diagram of the representation. Example One of the partitions of 4 is λ = (2, 1, 1) with associated Young diagram. 20

26 There are three standard Young tableaux of shape λ, namely , , and Therefore the irreducible representation of S 4 that corresponds to the partition (2, 1, 1) is 3-dimensional. Alternatively, we have a direct formula for the dimension of an irreducible representation of S n, known as the hook-length formula: To each box in the Young diagram associated to λ, we assign a number called the hooklength. The hook-length for a box is calculated by taking the number of boxes in the same row to the right of it plus the number of boxes in the same column below it plus one (for the box itself). Let h be the product of all hook-lengths in the Young diagram, then the dimension of the irreducible representation corresponding to λ is n! h. In the previous example where n = 4 and λ = (2, 1, 1), if we label each box of the associated Young diagram with its hook-length, we have = 3 as ex- Therefore the corresponding irreducible representation of S 4 has dimension 4! 8 pected. 2.6 Some properties of unitary Weingarten functions We are now ready to discuss and prove some of the properties of unitary Weingarten functions The invertibility of G Recall that n (+J k ) = G, where J k C[S n ] are the Jucys-Murphy elements and G C[S n ] k=1 is the function G(σ) = #(σ). Since every permutation in S n can be written as a permutation matrix an n n matrix created by rearranging the rows and/or columns of the n n 21

27 identity matrix, and this extends linearly to C[S n ], we can define a matrix norm on C[S n ]. Let A be an n n matrix, there are many ways to assign a matrix norm to A. We will use the spectral norm where A = max{λ λ is an eigenvalue of A A}. Under this norm, σ = A σ = 1 for every permutation matrix A σ corresponding to the permutation σ S n. In particular, (r, s) = 1 for every transposition (r, s) S n and thus J k = (1, k) + + (k 1, k) (1, k) + + (k 1, k) = k 1. It is known that if x < 1, then (1 + x) is invertible with (1 + x) 1 = ( 1) k x k. Therefore if we assume n, then k if k n and so k 1 <, which implies + J k is invertible since + J k = (1 + 1 J k ) and 1 J k = J k k 1 < 1. n This holds for all k = 1,..., n and thus G = (+J k ) is invertible in C[S n ] when n. k=1 k= Proof of Proposition 2.4 When n, we have G 1 = Wg U C[S n ]. Let ρ triv be the trivial representation of S n, i.e. ρ triv (σ) = 1 for all σ S n, then ρ triv (J k ) = ρ triv ((1, k) + + (k 1, k)) = k 1. If we apply ρ triv to both sides of (2.1), we have ( n ) ρ triv ( + J k ) = ρ triv (G) k=1 n ( + k 1) = ρ triv (G) k=1 n ( ( + k 1) 1 = (ρ triv (G)) 1 = ρ triv (G 1 ) = ρ ) triv Wg U. k=1 ote that every f C[S n ] can be written as f = σ S n f(σ)δ σ. Therefore ρ triv (f) = ρ triv ( σ S n f(σ)δ σ ) = σ S n f(σ)ρ triv (δ σ ) = σ S n f(σ), 22

28 n and thus ( + k 1) 1 = Wg U (σ, ). σ S n k=1 This proves part of Proposition 2.4, i.e. when n. Wg U (σ, ) = σ S n 1 ( + 1) ( + n 1) If n >, then Wg U is the pseudo-inverse of G U and as functions in C[S n ], we have GWg U G = G. Therefore ( ρ triv (G) = ρ triv GWg U G ) = ρ triv (G)ρ triv ( Wg U ) ρ triv (G) (since ρ triv is a representation) = (ρ triv (G)) 2 ρ triv ( Wg U ) (since ρ triv is 1-dimensional) ( ρ ) triv Wg U = (ρ triv (G)) 1 (since ρ triv (G) 0) n = ( + k 1) 1, k=1 which is same as before when n. This proves Proposition 2.4 completely Asymptotics of Wg U Consider the length function on S n, where σ (σ S n ) is the minimal non-negative integer l such that σ can be written as a product of l transpositions. In the paper [9], the authors showed the asymptotic estimate Wg U (σ, ) = O( n σ ). We will show this result using Theorem 2.7 (Jucys). When n, G is invertible with G 1 = Wg U, therefore n Wg U = ( + J k ) 1 k=1 = n (1 + 1 J 1 ) 1 (1 + 1 J n ) 1. Since 1 J k < 1 for all k n, ( ) ( ) n Wg U = ( 1) k 1 ( 1 J 1 ) k 1 ( 1) kn ( 1 J n ) kn = k 1 =0 ( ) l l=0 k 1,...,k n 0 k 1 + +k n=l k n=0 J k 1 1 J kn n. 23

29 ote that J k 1 1 Jn kn is a linear combination of permutations of length at most k 1 + +k n. Therefore for any σ S n, σ does not appear in any of the sums J k 1 1 Jn kn k 1,...,k n 0 k 1 + +k n=l when l < σ. This implies n Wg U (σ, ) = O( σ ). ow for any σ S n, one knows that σ = n #(σ) (see, e.g. Proposition in [18]), thus Wg U (σ, ) = O( 2n+#(σ) ), which implies the asymptotic decay Wg U (σ, ) 1 as. 2n #(σ) 2.7 Additional properties of unitary Weingarten functions In this section, we discuss some additional properties and examples involving unitary Weingarten functions. The proofs will be omitted but references will be given for those who are interested Explicit formulas for G and Wg U By the isotypic decomposition, the left (or right) regular representation of C[S n ] can be decomposed as (dimλ)v λ, λ n where V λ is the irreducible representation associated with λ and dimλ denotes its dimension. Therefore when the unitary Weingarten function Wg U was first introduced as a function on S n (see, e.g. Theorem 2.1 in [5] or Proposition 2.3 in [9]), it was defined as a sum in terms of irreducible characters of S n over the partitions of n. We shall briefly review the definition here. First, recall that if f and g are two functions in C[S n ], we denote by the classical convolution operation (f g)(π) = σ S n f(σ)g(σ 1 π) = τ S n f(πτ 1 )g(τ). 24

30 { 1, π = e Let δ e C[S n ] be the function defined by δ e (π) =. It can be easily checked 0, π e that f δ e = δ e f = f for all f C[S n ]. The inverse function of f with respect to, if it exists, is denoted by f ( 1) and it satisfies f f ( 1) = f ( 1) f = δ e. We have the following definition. Let Z (C[S n ]) be the center of C[S n ]: Z (C[S n ]) = {h C[S n ] h f = f h (f C[S n ])}. For the unitary group U, we define the element G U (, ) in Z (C[S n ]) by G U (σ, ) = #(σ), σ S n. ote that we previously denoted G U (, ) simply by G. ow if λ = (λ 1, λ 2,..., λ r ) is a partition of n, we write l(λ) for the length r of λ, then G can be expanded in terms of irreducible characters χ λ of S n as follows: G = 1 f λ C λ ()χ λ, (2.2) n! λ n where f λ is the dimension of the irreducible representation associated with λ and C λ () is the polynomial in given by l(λ) λ i C λ () = ( + j i). i=1 j=1 The unitary Weingarten function Wg U (, ) on S n is defined by Wg U (, ) = Wg U = 1 n! λ n f λ C λ () χλ, (2.3) summed over all partitions λ of n. It is the pseudo-inverse element of G, i.e. the unique element in Z (C[S n ]) satisfying G Wg U G = G. 25

31 In particular, unless {0, ±1, ±2,..., ±(n 1)}, functions G and Wg U are inverses of each other and satisfy G Wg U = Wg U G = δ e. Example Consider S 3. First note that for each partition λ, the irreducible character χ λ depends only on the conjugacy class, thus the unitary Weingarten function Wg U has this property as well. Since S 3 has three conjugacy classes: the class of the identity e, the class of the transposition (1, 2), and the class of the single cycle (1, 2, 3). These three conjugacy classes correspond to the three partitions (1, 1, 1), (2, 1), and (3) of the number 3, with associated Young diagrams Therefore,, and. Wg U ([1, 1, 1], ) = Wg U (e, ), Wg U ([2, 1], ) = Wg U ((1, 2), ) = Wg U ((1, 3), ) = Wg U ((2, 3), ), and Wg U ([3], ) = Wg U ((1, 2, 3), ) = Wg U ((1, 3, 2), ). To compute these values, it is enough to evaluate Wg U at the representative elements e, (1, 2), and (1, 2, 3) using equation (2.3). First, we compute the character table of S 3 using the Murnaghan-akayama Rule (see, e.g. [21] for more details). The method gives a combinatorial way of computing the character table of any symmetric group S n. It has the following steps: 1. Since the characters of a group are constant on its conjugacy classes, we index the columns of the character table by the three partitions of 3. Moreover, by Theorem 2.21, there are precisely as many irreducible characters as conjugacy classes, so we can also index the irreducible characters by the partitions. We index the rows of the character table by the associated Young diagrams: (1, 1, 1) (2, 1) (3). 26

32 2. We now calculate the entry in row λ (λ denotes a Young diagram) and column µ (µ denotes a partition). Let µ 1, µ 2,... be the parts of µ in decreasing order. Drawing λ as a Young diagram, define a filling of λ with content µ to be a way of writing a number in each square of λ such that the numbers are weakly increasing along each row (to the right) and each column (downwards) and there are exactly µ i squares labeled i for each i. 3. Consider all fillings of λ with content µ such that for each label i, the squares labeled i form a connected skew tableau that does not contain a 2 2 square. (A skew tableau is connected if the graph formed by connecting horizontally or vertically adjacent squares is connected.) Such a tableau is called a border-strip tableau, with each label representing a border-strip. For the purpose of illustrating how this method works, suppose we are trying to calculate the entry for λ = (3, 2) and µ = (2, 2, 1) in the character table of S 5, then , , and are the only three border-strip tableaux that are fillings of λ = (3, 2) with content µ = (2, 2, 1). 4. For each label in the border-strip tableau, define the height of the corresponding borderstrip to be one less than the number of rows of the border-strip. We then define the weight of the border-strip tableau to be ( 1) s where s is the sum of the heights of the border-strips that compose the tableau. Finally, the entry in the character table is the sum of all the weights of the possible border-strip tableaux. For example, the three border-strip tableaux of λ = (3, 2) with µ = (2, 2, 1), as shown above, have weights 1, 1, and -1 respectively, for a total weight of 1. Therefore the corresponding entry in the character table of S 5 is 1. Using this method, we can easily compute the character table of S 3 : (1, 1, 1) (2, 1) (3) ow for S 3, the three irreducible representations, as shown earlier, are λ 1 = (1, 1, 1) =, λ 2 = (2, 1) =, and λ 3 = (3) =. 27

33 The corresponding dimensions can be calculated using the hook-length formula. We have f λ 1 = 1, f λ 2 = 2, and f λ 3 = 1. Moreover, Therefore by equation (2.3), we have C λ1 () = ( 1)( 2), C λ2 () = ( + 1)( 1), C λ3 () = ( + 1)( + 2). 1. Wg U ([1, 1, 1], ) = Wg U (e, ) = 1 { } 1 6 ( 1)( 2) + 4 ( + 1)( 1) + 1 ( + 1)( + 2) 2 1 = ( 2 1)( 2 4), 2. Wg U ([2, 1], ) = Wg U ((1, 2), ) = 1 { } 1 6 ( 1)( 2) + 1 ( + 1)( + 2) 1 = ( 2 1)( 2 4), and 3. Wg U ([3], ) = Wg U ((1, 2, 3), ) = 1 { } 1 6 ( 1)( 2) + 2 ( + 1)( 1) + 1 ( + 1)( + 2) 2 = ( 2 1)( 2 4) Character expansion of Wg U It is known that the set { χ λ} λ n of irreducible characters of S n forms a basis of the center Z (C[S n ]) of the group algebra C[S n ]. Definition If f Z (C[S n ]) is a central function, then the expression f = λ n f(λ)χ λ of f with respect to the character basis of Z (C[S n ]) is called the character expansion of f (see, e.g. [19] for more details). 28

34 Definition Let be a square in a Young diagram λ n. The content of, denoted c( ), is defined to be the column index of minus the row index of. 2. Given a Young diagram λ n, let H λ denote the product of hook-lengths of λ. 3. Let s λ (1 ) = 1 ( + c( )). H λ λ In the paper [19], the author proved the following character expansion of Wg U C[S n ]. We will only state the theorem. The proof, and some other properties, can be found in section 3 of [19]. Theorem (Theorem 3.2 in [19]) The character expansion of Wg U C[S n ] is Wg U = λ n 1 H 2 λ s λ(1 ) χλ. (2.4) It can be easily checked that, using this theorem, evaluating Wg U at any σ S 3 would produce the same result as the previous example. 29

35 Chapter 3 Integration over the Unitary Group and Applications The main goal of this chapter is to discuss how Theorem 1.5 can be used to solve various problems in random matrix theory. Recall that in Theorem 1.5, a formula is given so that general matrix integrals E [ U i1 j 1 U injn U i 1 j 1 U ] i nj n (3.1) can be calculated as a sum of unitary Weingarten functions over the symmetric group S n, where U is an Haar-distributed unitary random matrix, E denotes expectation with respect to the Haar measure, and n. Expressions of the form (3.1) appear very naturally in random matrix theory. The reason for this is that quite many random matrix ensembles are invariant under unitary conjugation, i.e. X is unitarily invariant if the joint distribution of its entries is unchanged when we conjugate X by an independent unitary matrix. As a result, expressions similar to E [Tr (X 1 U ε 1 X 2 U ε2 X n U εn )] (3.2) are quite common in random matrix theory, where X 1,..., X n are random matrices and ɛ 1,..., ɛ n {1, }. We will show by examples that the calculation of (3.2) can be reduced to the calculation of (3.1). First, we introduce some notations. otation 3.1. Let S n be the symmetric group on {1,..., n}, n Let π be a permutation in S n, then for any n-tuple X = (X 1,..., X n ) of complex matrices Tr π (X ) = Tr π (X 1,..., X n ) := ( ) Tr X j, C C(π) j C where C(π) is the set of all the disjoint cycles of π (including fixed points). 30

36 2. Let γ n denote the cyclic permutation of order n. γ n = (1, 2,..., n) S n 3. Let M (C) be the algebra of complex matrices, we use {E a,b } 1 a,b as a basis where (E a,b ) ij = δ (a,b),(i,j) = δ a,i δ b,j. In other words, E a,b is the basis matrix whose entries are all 0 except for the entry at row a and column b, which is 1. It has the property that Tr(XE a,b ) = (XE a,b ) ii = (XE a,b ) bb = X ba i=1 for every X M (C). 3.1 Unitarily invariant random matrices Definition 3.2. Let X = (X 1,..., X n ) be an n-tuple of random matrices, we say X is invariant under unitary conjugation if (X 1,..., X n ) and (UX 1 U,..., UX n U ) are identically distributed for any independent unitary matrix U. For two sequences i = (i 1,..., i n ) and i = (i 1,..., i n) of positive integers and for a permutation σ S n, we put n δ σ (i, i ) = δ ik i. σ(k) k=1 Under this notation, Theorem 1.5 becomes E [ U i1 j 1 U injn U i 1 j 1 U ] i nj n = δ α (i, i )δ β (j, j )Wg U (βα 1, ), (3.3) α,β S n where U is an Haar-distributed unitary random matrix and n. ow, suppose W is an Hermitian (i.e. W = W ) random matrix that is invariant under unitary conjugation. For expressions of the form E [W i1 j 1 W i2 j 2 W inj n ] where n, we expect to have a formula that is closely related to (3.3). This is shown in the following theorem. 31