Gu_Yinzheng_201308_MSc-Weingarten Function and Random Matrices

8/9/2019 Gu_Yinzheng_201308_MSc-Weingarten Function and Random Matrices

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Moments of Random Matrices and

Weingarten Functions

by

Yinzheng Gu

A project submitted to theDepartment of Mathematics and Statistics

in conformity with the requirements forthe degree of Master of Science

Queens UniversityKingston, Ontario, Canada

August 2013

Copyright cYinzheng Gu, 2013


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Abstract

LetGbe 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 matrixX= (xi,j)1i,jNpicked up from G. We would like to know how to compute

the moments

E xi1j1 xinjnxi1j1 xinjn or E [xi1j1 xi2nj2n] .In this report, we focus on the unitary group UNand use the methods established in [5]

and[9] which express the moments as sums in terms of Weingarten functions. The func-

tion WgU(, N), 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 UN. We compute matrix-valued expectations with the

goal of having a better understanding of the operator-valued Cauchy transform.

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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 Queens University.

Finally, I am grateful to my family, especially my parents for their unconditional love

and encouragement.

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Table of Contents

Abstract i

Acknowledgments ii

Table of Contents iii

1 Introduction and Preliminaries 11.1 Expectation of products of entries . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Integration formula for the unitary group. . . . . . . . . . . . . . . . 11.1.2 Integration formula for the orthogonal group . . . . . . . . . . . . . . 4

1.2 Partitions and pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Weingarten Functions and Jucys-Murphy Elements 8

2.1 Gram matrices and Weingarten matrices . . . . . . . . . . . . . . . . . . . . 82.2 Jucys-Murphy elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 The group algebra C[Sn] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Group representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 Basic definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.2 The regular representation . . . . . . . . . . . . . . . . . . . . . . . . 152.4.3 Maschkes theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.4 Artin-Wedderburn theorem . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Young tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5.1 Basic definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5.2 The irreducible representations ofSn . . . . . . . . . . . . . . . . . . 19

2.6 Some properties of unitary Weingarten functions. . . . . . . . . . . . . . . . 212.6.1 The invertibility ofG . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6.2 Proof of Proposition2.4 . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.3 Asymptotics of Wg

U

. . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7 Additional properties of unitary Weingarten functions . . . . . . . . . . . . . 242.7.1 Explicit formulas forG and WgU . . . . . . . . . . . . . . . . . . . . 242.7.2 Character expansion of WgU . . . . . . . . . . . . . . . . . . . . . . . 28

3 Integration over the Unitary Group and Applications 30

3.1 Unitarily invariant random matrices. . . . . . . . . . . . . . . . . . . . . . . 313.2 Expectation of products of matrices . . . . . . . . . . . . . . . . . . . . . . . 34

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3.3 Matricial cumulants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Conclusion and Future Work 44

4.1 The Cauchy transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 The operator-valued Cauchy transform . . . . . . . . . . . . . . . . . . . . . 45

A Free Probability Theory 49

B Tables of Values 51

B.1 Unitary Weingarten functions . . . . . . . . . . . . . . . . . . . . . . . . . . 51B.2 Orthogonal Weingarten functions . . . . . . . . . . . . . . . . . . . . . . . . 52

Bibliography 53

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Chapter 1

Introduction and Preliminaries

LetUNdenote the group ofNNunitary matrices, with the group operation that of matrixmultiplication. In other words, an NNcomplex matrix U UN ifU U

= UU = IN,whereIN is the N N identity matrix and U

is the conjugate transpose ofU.

Viewing it as a subset ofMN(C) the algebra ofN Ncomplex matrices with usualmatrix multiplication, it is common to endow UNwith the corresponding subspace topologyso that UN is compact as a topological space. By Haars theorem, every locally compactHausdorff topological group has a unique (up to a positive multiplicative constant) left-translation-invariant measure and a unique (up to a positive multiplicative constant) right-translation-invariant measure. When the group is compact, these two measures coincide andwe call it the Haar measure. This Haar measure is finite, therefore we can normalize it to aprobability measure the unique Haar probability measureon the compact group.

Definition 1.1. A random matrix is a matrix of given type and size whose entries consistof random numbers from some specified distribution. In other words, a random matrix is amatrix-valued random variable.

Definition 1.2. We equip the compact group UNwith its Haar probability measure aprobability measure on UNwhich is invariant under multiplication from the left and multi-plication from the right by any arbitraryNNunitary matrix. Random matrices distributedaccording to this measure will be called Haar-distributed unitary random matrices. Thus theexpectation E over this ensemble is given by integrating with respect to the Haar probabilitymeasure.

1.1 Expectation of products of entries

1.1.1 Integration formula for the unitary group

The expectation of products of entries (also called moments or matrix integrals) of Haar-distributed unitary random matrices can be described in terms of a special function on the

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permutation group. Since such considerations go back to Weingarten[23], Collins [5]callsthis function the (unitary) Weingarten function and denotes it by WgU.

Notation 1.3. LetSn be the symmetric group on [n] ={1, 2, . . . , n}. As we shall see later,

for Sn andn N,

WgU(, N) =

UN

U11 UnnU1(1) Un(n)dU

= E

U11 UnnU1(1) Un(n)

,

whereU is an N N Haar-distributed unitary random matrix, dU is the normalized Haarmeasure, and WgU is called the (unitary) Weingarten function.

A crucial property of the Weingarten function WgU(, N) is that it depends only onthe conjugacy class (in other words, the cycle structure) of the permutation . This will

be explained in more details in the next chapter. First, we introduce some notations by anexample.

Example 1.4. As mentioned above, WgU(, N) depends only on the cycle structure of.Therefore when a table of values is provided, the value of WgU(, N) is usually given ac-cording to the cycle structure of. For example, WgU([2, 1], N) denotes the common valueof every S3 whose cycle decomposition consists of a transposition and a fixed point.

Suppose , , S3 such that = (1, 2) = (1, 2)(3), = (1, 3) = (1, 3)(2), and= (1, 2, 3), then by the table of values (for some values) of the unitary Weingarten functions,

provided in appendixB, we have

WgU(, N) = WgU(, N) = WgU([2, 1], N) = 1

(N2 1)(N2 4) and

WgU(, N) = WgU([3], N) = 2

N(N2 1)(N2 4).

In the paper [9], a formula is given so that general matrix integrals over the unitary groupUNcan be calculated as follows:

Theorem 1.5. Let U be an NN Haar-distributed unitary random matrix and n N,then

E

Ui1j1 UinjnUi1j1 Uinjn

=

,Sn

i1i(1) ini(n)j1j(1) jnj(n)WgU(1, N), (1.1)

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whereUij denotes theijth entry ofU andij =

1, i= j

0, i=j.

Remark 1.6. 1. Note that Notation1.3follows immediately from this theorem and thusE U11 UnnU1(1) Un(n)is sometimes referred to as theintegral representationofWgU(, N).

2. Ifn =n, then by the invariance of the Haar measure, we have

E

Ui1j1 UinjnUi1j1 Uinj

n

= E

Ui1j1 UinjnUi1j1 Uinj

n

for every C with ||= 1.

Without loss of generality, assume n > n, then E


=

nnE Ui1j1 UinjnUi1j1 Uinjn and thus the integral vanishes since it is impossi-ble to have nn = 1 C with||= 1.

Notation 1.7. LetA = (aij)Ni,j=1 MN(C) be an N Ncomplex matrix, we denote by Tr

(or TrN) the non-normalized trace and tr (or trN) the normalized trace:

Tr(A):=Ni=1

aii and tr(A):= 1

N

Ni=1

aii= 1

NTr(A).

Example 1.8. (An application of Theorem1.5)Let Ube an N N Haar-distributed unitary random matrix and B be an N Ncomplexmatrix. We wish to compute E [U BU], where E denotes expectation with respect to theHaar measure.

Note that E [U BU] denotes the matrix-valued expectation of the random matrix U BU.In other words,

E [UBU]:=

UN

(U BU)ijdU

Ni,j=1

,

wheredUis the normalized Haar measure.

Therefore we first compute the ij th entry ofE [U BU] and obtain

E [U BU]ij

=N

j1,j2=1

E[Uij1Bj1j2(U)j2j]

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=N

j1,j2=1

Bj1j2E

Uij1Ujj2

=N

j1,j2=1Bj1j2ijj1j2Wg

U(e, N) (by (1.1) withn = 1 and= = e)

=ij

Nj1=1

Bj1j1WgU([1], N)

=ij1

NTr(B)

=ijtr(B).

This allows us to conclude that E [U BU] = tr(B)IN, whereINdenotes the N N iden-tity matrix.

1.1.2 Integration formula for the orthogonal group

The real analogue of a unitary matrix is an orthogonal matrix.

Definition 1.9. Anorthogonal matrixis a square matrix with real entries whose rows andcolumns are orthogonal unit vectors (i.e. orthonormal vectors). Equivalently, a matrix O isorthogonal if its transpose is equal to its inverse: OT =O1. The set ofN N orthogonalmatrices forms a group ON, known as the orthogonal group.

The main focus of this report will be on the unitary group UN, but it is worth mentioningthat in the paper [9], the authors also defined a function WgO on S2n they called the or-thogonal Weingarten function. Then they proved the following formula for integration withrespect to the Haar measure over the orthogonal group ON when n N:

Theorem 1.10. LetO be anN NHaar-distributed orthogonal random matrix, then

E [Oi1j1 Oi2nj2n] =

p,qP2(2n)

i1ip(1) i2nip(2n)j1jq(1) j2njq(2n)

WgO(p), q

, (1.2)

whereP2(2n) is the set of all pairings of [2n] ={1, 2, . . . , 2n} and WgO(p), q denotes thepqth entry of the orthogonal Weingarten matrix (to be introduced in section2.1).

Observe that, similar to the unitary case, the moments of an odd number of factorsvanish. In other words, we always have E

Oi1j1 Oi2n+1j2n+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

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following sections.

It is also worth mentioning that the authors of[9] considered the case of integration overthe symplectic group in their paper as well, but for our purposes we will not discuss it inthis report.

1.2 Partitions and pairings

Definition 1.11. LetSbe a finite totally ordered set. We call = {V1, . . . , V r}a partitionof the set S if and only if the Vi (1ir) are pairwise disjoint, non-empty subsets ofSsuch that V1 Vr =S. We call V1, . . . , V r the blocks of .

Definition 1.12. For any integer n 1, let P(n) be the set of all partitions of [n]. If

n is even, then a partition P(n) is called a pair partition or pairing if each block of consists of exactly two elements. The set of all pairings of [n] is denoted by P2(n). Forany partition P(n), let #() denote the number of blocks of (also counting singletons).

Let Sn be the symmetric group on [n]. Given a permutation Sn, it is often conve-nient to consider the cycles of as a partition of [n], thus #() also denotes the numberof cycles in the cycle decomposition of (also counting fixed points) when is a permuta-tion in Sn. This mapSn P(n) forgets the order of elements in a cycle and so is not abijection. Conversely, given a partition P(n), we put the elements of each block intoincreasing order and consider this as a permutation. Restricted to pairings this is a bijection.

Definition 1.13. Let, be inP(n). The set P(n) is a finite partially ordered set (poset)in which means each block of is completely contained in one of the blocks of (thatis, if can be obtained out of by refining the block structure). The partial order obtainedin this way is called the reversed refinement order.

Definition 1.14. Let Pbe a finite partially ordered set and let , be in P. If the setU ={ P | and } is non-empty and has a minimum 0 (that is, an element0 Uwhich is smaller than all the other elements ofU), then0 is called thejoinof and

, and is denoted by .

To make use of formula (1.2), we need to address one more question: given two pairingsp andq, what is the relationship between the cycles ofpqand the blocks ofp q? First weillustrate it with an example.

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Example 1.15. Ifp, q P2(8) such thatp= (1, 2)(3, 5)(4, 8)(6, 7) and q= (1, 6)(2, 5)(3, 7)(4, 8).Thenpq= (1, 7, 5)(2, 3, 6)(4)(8) andp q= {{1, 2, 3, 5, 6, 7}, {4, 8}}. We notice that the cy-cles ofpqappear in pairs in the sense that if (i1, . . . , ik) is a cycle ofpq, then (q(ik), . . . , q (i1))is also a cycle ofpq. This is not a coincidence as shown by the following lemma.

Lemma 1.16. (Lemma 2 in [16])Letp, q P2(n)be pairings and(i1, i2, . . . , ik)a cycle ofpq. Letjr =q(ir). Then(jk, jk1, . . . , j1)is also a cycle of pq, and these two cycles are distinct; {i1, . . . , ik, j1, . . . , jk} is a block of

p qand all are of this form; #(pq) = 2#(p q).

Proof. Since (i1, i2, . . . , ik) is a cycle ofpq, we havepq(ir) =ir+1, thusjr =q(ir) =ppq(ir) =p(ir+1). Hencepq(jr+1) = pqq(ir+1) = p(ir+1) = jr. If{i1, . . . , ik} and {j1, . . . , jk} were tohave a non-empty intersection, then for some n, q(pq)n would have a fixed point, but thiswould in turn imply that eitherpor qhad a fixed point, which is impossible. Since {q(ir)}

kr=1

= {js}k

s=1 and {p(js)}k

s=1 = {ir}k

r=1, {i1, . . . , ik, j1, . . . , jk} must be a block ofp q. Sinceevery point of [n] is in some cycle ofpq, all blocks must be of this form. Since every blockofp qis the union of two cycles ofpq, we have #(pq) = 2#(p q).

This gives us a way to obtain the values of

WgO(p), q

(p and qare pairings) for givenp and qfrom the table of values (for some values) of the orthogonal Weingarten functions,provided in appendixB: since the cycles ofpqalways appear in pairs, we choose one cyclefrom each pair of cycles (in other words, delete half of the cycles ofpq), then the value of

WgO(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), andpq= (1, 7, 5)(2, 3, 6)(4)(8). We haveWgO(p), q

= WgO([3, 1], N)

= 2

(N 1)(N 2)(N 3)(N+ 1)(N+ 2)(N+ 6).

Now let us see how formula (1.2) can be applied to calculate general matrix integrals overthe orthogonal group On.

Example 1.17. SupposeO is anN NHaar-distributed orthogonal random matrix whereN2.

1. Integrands of type Oi1j1 Oi2nj2n which do not have pairings integrate to zero. Forexample, E [O311O12] =E[O11O11O11O12] = 0.

2. If Oi1j1Oi2j2Oi3j3Oi4j4 = O211O

222 = O11O11O22O22, then the indexes only admit one

pairing, namelyp= {{1, 2}, {3, 4}}andq= {{1, 2}, {3, 4}}.

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Thus

E

O211O222

=

WgO({{1, 2}, {3, 4}}), {{1, 2}, {3, 4}}

= WgO([1, 1], N)

=

N+ 1

N(N 1)(N+ 2) .

3. IfOi1j1Oi2j2Oi3j3Oi4j4 =O411= O11O11O11O11, then all pairings are admissible because

all the indexes are the same, equal to 1. A similar computation shows that

E

O411

= 3(N+ 1) 6

N(N 1)(N+ 2)=

3

N(N+ 2).

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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 Nbe a positive integer. We define theunitary Gram matrix GUNn =

GUNn (, ),Sn

byGUNn (, ) =N

#(1),

and let WgUNn =

WgUNn (, ),Sn

be the pseudo-inverse ofGUNn . We call WgUNn the uni-

tary Weingarten matrix.

Proposition 2.1. For anymnmatrixA, there exists a uniquen mmatrixA+, called theMoore-Penrose pseudo-inverse (or simply pseudo-inverse) ofA, satisfying all of the following

four criteria:

1. AA+A= A;

2. A+AA+ =A+;

3. (AA+) =AA+;

4. (A+A) =A+A.

Note that ifA is invertible, thenA+ =A1.

Proof. First, ifD is anm n(rectangular) diagonal matrix, then we define an n mmatrixD+ whose entries are

(D+)ij =

(Dii)

1, ifDii= 0 for i= 1, . . . , min{m, n}

0, otherwise.

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It is easy to check that D+ is the pseudo-inverse ofD.

The existence ofA+ then follows from the singular value decomposition theorem whichstates that any m n matrix A has a factorization of the form A= U DV, where U is anm munitary 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 nunitary matrix V.LetA+ =V D+U and we can show that A+ is the pseudo-inverse ofA:

1. AA+A= U DVV D+UUDV =U DD+DV =U DV =A;

2. A+AA+ =V D+UUDVV D+U =V D+DD+U =V D+U =A+;

3. (AA+) = (U DVV D+U) = (U DD+U) =U(DD+)U =U DD+U =U DVV D+U

=AA+;

4. (A+A) = (V D+UU DV) = (V D+DV) =V(D+D)V =V(D+D)V =V D+UU DV

=A+A.

To prove the uniqueness ofA+, suppose both B and C are n m matrices satisfyingall of the pseudo-inverse criteria, then AB = (AB) = BA = B(ACA) = BACA =(AB)(AC) = ABAC = AC. Similarly, we haveBA = CA and therefore B = BAB =BAC=CAC=C.

Analogously, we define the orthogonal Gram matrix GONn =

GONn (p, q)p,qP2(2n)

by

GONn (p, q) =N#(pq),

and the orthogonal Weingarten matrix WgONn as the pseudo-inverse ofG

ONn .

Example 2.2. Suppose n = 3 andNn, then

GUN3 =

() (1, 2) (1, 3) (2, 3) (1, 2, 3) (1, 3, 2)

() N3 N2 N2 N2 N N(1, 2) N2 N3 N N N2 N2

(1, 3) N2 N N3 N N2 N2

(2, 3) N2 N N N3 N2 N2

(1, 2, 3) N N2 N2 N2 N3 N

(1, 3, 2) N N2

N2

N2

N N3

.

Since the determinant ofGUN3 is 1

N6(N2 1)5(N2 4) and N 3 by assumption, GUN3

is invertible and we have WgUN3 =

GUN31

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= 1(N2 1)(N2 4)

N2 2

N 1 1 1

2

N

2

N

1 N2 2

N

2

N

2

N 1 1

1

2

N

N2 2

N

2

N 1 1

1 2

N

2

N

N2 2

N 1 1

2

N 1 1 1

N2 2

N

2

N2

N 1 1 1

2

N

N2 2

N

.

ThereforeS3

WgU(, N) is the sum of entries of any row or column of WgUN3 . In

particular, choosing the first row gives usS3

WgU(, N) =S3

WgUN3 (e, )

= 1

(N2 1)(N2 4)

N2 2 3N+ 4

N

=

1

N(N+ 1)(N+ 2).

Remark 2.3. Note that we implicitly used the fact that the elements in the first row of theunitary Weingarten matrix WgUN

n

define the unitary Weingarten function WgU by

WgU(, N) = WgUNn (e, ),

where Sn. 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 aproposition here, to be proved later.

Proposition 2.4.For allk 1,Sk

WgU(, N) = 1

N(N+ 1) (N+ k 1).

2.2 Jucys-Murphy elements

Let n be a positive integer, then the group algebra C[Sn] is an algebra (over C) with thesymmetric groupSn as a basis, with multiplication defined by extending the group multipli-cation linearly, and an involution defined by = 1, Sn, and extended conjugatelinearly.

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Definition 2.5. Let n be a positive integer. Consider the natural embedding C[S1] C[S2] C[Sn] , where elements ofC[Sk] act trivially on numbers greater than k.TheJucys-Murphy elements J1, J2, . . . , J n C[Sn] are transposition sums defined by:

J1= 0 andJk = (1, k) + (2, k) + + (k 1, k) for 2 k n.

Remark 2.6. 1. The definition ofJ1 is a convenient convention.

2. For n 2, Jn commutes with C[Sn1]. Indeed, for every Sn1, we have Jn1 =

n1i=1

(i, n)1 =n1i=1

((i), n) = Jn. Therefore JmJn = JnJm for all m and n. This

implies that C[J2, . . . , J n] is a commutative subalgebra ofC[Sn]. It is in fact a max-imal commutative subalgebra known as the Gelfand-Zetlin subalgebra as shown by

Okounkov and Vershik in [20].

Next, we need the classical identity (see [13]).

Theorem 2.7. (Jucys) LetNbe a positive integer, then

nk=1

(N+ Jk) =Sn

N#(). (2.1)

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(N+J1)(N+J2) (N+Jn), which also has n! terms after expanding the product. Ourproof will consist of a term by term identification by induction on n.

Ifn= 1, then the statement is trivial as both the left-hand side and the right-hand sideare equal to N. Assume the statement holds for all natural numbers less than or equal to

n 1 and consider a permutation Sn. The goal is to identify N#() with one of theterms in the expansion of (N+J1) (N+ Jn). There are two cases:

1. Ifn is a fixed point of, then we apply the induction hypothesis to |{1,...,n1} Sn1,

and we know thatN#(|{1,...,n1})|{1,...,n1}corresponds to one term in (N+J1) (N+Jn1). Furthermore, sincehas one more cycle than|{1,...,n1}, we can identifyN

#()

with the term in (N+ J1) (N+ Jn) with the same choice as N#(|{1,...,n1})|{1,...,n1}

in the first n 1 factors, and we further pick the multiplication by N in N+ Jn.

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2. Suppose n is not a fixed point.First, we notice that if= (other cycles of)(1(n), n , (n), . . . ), then (1(n), n) =(other cycles of)(1(n), (n), . . . )(n), and thus we can apply the induction hypothe-sis to ((1(n), n))|{1,...,n1} Sn1. Since has as many cycles as ((

1(n), n))|{1,...,n1},we can identifyN#() with the term in (N+ J1) (N+ Jn) with the same choice as

N#(((1

(n),n))|{1,...,n1})((1(n), n))|{1,...,n1}in the firstn 1 factors, and we furtherpick the transposition (1(n), n) inside N+ Jn.

The proof is completed.

This theorem will play an important role in our study of the unitary Weingarten func-tions. First, we review some relevant concepts.

2.3 The group algebraC

[Sn]Let C[Sn] be the group algebra with the symmetric group Sn as a basis.

IfA(Sn) is the algebra of all complex-valued functions on Sn with multiplication givenby the convolution

(f g)() =Sn

f()g(1), (f, g A(Sn), Sn),

then C[Sn] is isomorphic as a complex algebra to A(Sn), via the map

: C[Sn] A(Sn), ,

where :Sn C is the function defined by () =

1, =

0, =.

It is easily checked that for any 1, 2 Sn, 12 =1 2 under this map:

(1 2)() =Sn

1()2(1)

=1(1)2(11 )

=2

(11

)

=12(), Sn.

In general, ifa =Sn

C[Sn] where C, then a a =

Sn

under .

Sincea () =

Sn

() =,a is sent to the function a

that maps to .

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Moreover, ifb =Sn

C[Sn] where C, then b b =

Sn

under and so

ab=

Sn

Sn

=

,Sn

=Sn

Sn

1

.

Thereforeab (ab) =Sn

Sn

1

as an element inA(Sn) and thus

(ab)() = SnSn 1 ()=Sn

1

=Sn

a()b(1)

= (a b)(), Sn.

Under this isomorphism, we may pass easily between C[Sn] and A(Sn). For simplicity,we will denote both C[Sn] and A(Sn) by C[Sn] in the following sections and it should be

clear from context which algebra we refer to.

Now, note that the right-hand side of (2.1) isSn

N#() C[Sn]. It is sent to

Sn

N#() A(Sn) under the map . Since

Sn

N#()

() = N#(), this means

Sn

N#() is sent to the function that maps to N#(). If we let G A(Sn) be the

functionG() =N#(), then (2.1) becomesn

k=1(N+ Jk) =G.

2.4 Group representation theory

In this section, we briefly review some concepts from group representation theory that willbe used later in this report. There might be more information than actually needed, it willbe a good introduction of the subject nevertheless.

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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, arepresentation is a map : G GL(V) such that (g1g2) =(g1)(g2), for all g1, g2 G.

Definition 2.9. Let : 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, thenevery linear map corresponds to a matrix, so we get an isomorphism GL(V)=GLn(C)(dependent on the basis). The representation now becomes a homomorphism

: G GLn(C),

and the number n is called the dimension(or the degree) of the representation.

2. The trivial representation ofG assigns to each element ofG the identity map on V.

3. If is injective, then we say is a faithful representation. In other words, differentelementsg ofGare represented by distinct linear mappings(g) and thus ker() ={e}.

4. Asubrepresentationof is a vector space W V such that

(g)(x) W g G and x W.

This means that every (g) defines a linear map from W to W, i.e. we have a repre-

sentation ofG on the subspace Wand we callW an invariant subspaceof.

If we pick a basis of a subrepresentation Wand extend it to V, then for all g G,(g)

takes the form

0

where the top-left block is a dim(W) dim(W) matrix.

Thus, this block gives the matrix for a representation ofG onW.

5. If has exactly two subrepresentations, namely the 0-dimensional subspace and Vitself, then the representation is said to be irreducible. If it has a proper subrepre-sentation of non-zero dimension, the representation is said to be reducible. Note thatthe representation of dimension 0 is considered to be neither reducible nor irreducible

and any 1-dimensional representation is irreducible.

6. Let G = Sn, if

: 1

1

for all even permutations and odd permutations , then is called the sign repre-sentation, i.e. () is simply multiplication by sgn() for each Sn.

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7. Let G = Sn andV be an n-dimensional vector space with standard basis {e1, . . . , en}.Then we have the permutation representation()(ei) =e(i), Sn.For example, suppose G= S3, then we have, in matrix forms,

((1, 2)) = 0 1 0

1 0 00 0 1 and((2, 3)) = 1 0 0

0 0 10 1 0 .In other words, is a permutation representation if(g) is a permutation matrix forallg G.

8. The characterof the representationis the function

: GK

gTr((g)).

Note that ifg1

andg2

are conjugate in G, then

(g1) =

(g

2) since trace is constant

under conjugation, so we can view as a function on the conjugacy classes ofG.

2.4.2 The regular representation

Recall that C[Sn] is the algebra of all complex-valued functions on Sn. It has a basis {}Sn,where is the function sending to 1 C and all other group elements to 0 C.

Given a functionf :Sn C and an element Sn, the left regular representation()is defined by

(): f ()fon the basis element , where (()f)() =f(

1) for all Sn.

Similarly, the right regular representation () sends the function f to ()f on thebasis element , where (()f)() =f() for all Sn.

Now, ifG C[Sn] is the function G() =N#(), it can be seen that the unitary Gram

matrix GUNn is the matrix of G acting in either the left or right regular representation ofC[Sn], with standard basis {}Sn . IfG

UNn is invertible, thenG is invertible in C[Sn] and

we let the inverse ofG be WgU, the unitary Weingarten function.

2.4.3 Maschkes theorem

Let V andWbe two vector spaces over C. Recall that the direct sum V W is the vectorspace of all pairs (x, y) such that x V and y W. Its dimension is dim(V) + dim(W).

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SupposeG is a group and we have representations

V :G GL(V),

W :G GL(W).

Then there is a representation VW ofG on V Wgiven by

VW :G GL(V W),

VW(g):(x, y)(V(g)(x), W(g)(y)) .

Moreover, suppose dim(V) = n and dim(W) = m. We can choose a basis{a1, . . . , an} forV, and {b1, . . . , bm} for W, then the set

{(a1, 0), . . . , (an, 0), (0, b1), . . . , (0, bm)}

is a basis for V W. For all g G, V(g) GLn(C),W(g) GLm(C), andVW is given

by the (n+m) (n+m) matrix V(g) 00 W(g) under the induced basis on V W.A matrix of this form is called block-diagonal.

Remark 2.10. Suppose : G GL(V) is a representation. Note that

1. IfV = W U, then the subspace {(x, 0) | x W} W U is a subrepresentationand it is isomorphic to W. Similarly, the subspace {(0, y)| y U}is isomorphic toU.The intersection of these two subrepresentations is {0}.

2. IfW 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 : G GL(V) is a representation andW V is a subrepresentation, is there another subrepresentationUV such that V =W U?

It turns out the answer to this question is always yes, and U is called a complementarysubrepresentationtoW. This is known as Maschkes theorem. It is an important theorem ingroup representation theory and the proof can be found in most of the graduate level algebratexts (see, e.g. [10]). We will state the theorem here and omit the proof.

Theorem 2.11. (Maschkes theorem)Let : G GL(V) be a representation and W V be a subrepresentation. Then thereexists a complementary subrepresentationUV to W.

This leads to the following result, called the complete reducibility theorem.

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Corollary 2.12. Every complex representation of a finite group can be written as a directsum

W1 W2 Wr

of subrepresentations, where eachWi is irreducible.

Proof. Let G be a finite group and V be a representation of G of dimension n. If V isirreducible, then the statement holds trivially. If not,Vcontains a proper subrepresentationW V, and by Maschkes 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 iscomplete. If not, at least one of them contains a proper subrepresentation, so it splits as adirect sum of smaller representations. Since n is finite, this process will terminate in a finitenumber of steps.

Definition 2.13. If a representation can be decomposed as a direct sum of irreducible sub-representations, then it is said to be completely reducible.

2.4.4 Artin-Wedderburn theorem

TheArtin-Wedderburn theoremis a classification theorem for semisimple rings and semisim-ple algebras. For our purposes, we only need the following corollary.

Corollary 2.14. LetG be a finite group, then

C[G]= i

Mni(C),

where the direct sum is indexed by (all of) the irreducible representations ofG andni is thedimension of the ith irreducible representation.

2.5 Young tableaux

In this section, we explore a connection between representations of the symmetric groupSnand combinatorial objects called Young tableaux.

2.5.1 Basic definitions

Definition 2.15. A partition of a positive integer n is a sequence of positive integers = (1, 2, . . . , r) satisfying 1 2 r > 0 and n = 1 + 2 + + r.We write n to denote that is a partition ofn.

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Definition 2.16. AYoung diagramis an array of boxes arranged in left-justified rows, withthe row sizes weakly decreasing (i.e. non-increasing). The Young diagram associated to thepartition= (1, 2, . . . , r) is the one that has r rows, with i boxes in the i

th row.

Example 2.17. 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 dia-grams.

Definition 2.18. Suppose n. A Young tableau T of shape is an assignment of thenumbers {1, 2, . . . , n} to the n boxes of the Young diagram associated to such that eachnumber occurs exactly once.

For example, here are all the Young tableaux corresponding to the partition (2, 1):

1 2

3

1 3

2

2 1

3

2 3

1

3 1

2

3 2

1 .

Definition 2.19. A standard Young tableau T of shape , denoted by SYT(), is a Youngtableau whose entries are increasing along each row (to the right) and each column (down-wards).

The only two standard Young tableaux for (2, 1) are

1 23 and

1 32 .

Definition 2.20. 1. A skew shape is a pair of partitions (, ) such that the Youngdiagram of contains the Young diagram of, it is denoted by /.

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2. The skew diagram of a shew shape / is the set-theoretic difference of the Youngdiagramsand: the set of squares that belong to the diagram of but not to thatof.

3. A skew tableau of shape / 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):

1 79 6 2

5 43 8 .

2.5.2 The irreducible representations ofSn

Representations are often used to characterize finite groups. In this report, we are mostlyinterested in the symmetric group Sn. By Corollary2.12, we know that every representationofSn 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 agiven group and their dimensions. The answer to the first question is given by the followingtheorem, sometimes called the completeness of irreducible characters.

Theorem 2.21. The number of irreducible representations of a finite group is equal to thenumber of conjugacy classes of that group.

The proof of this theorem is beyond the scope of this report and we refer to [11]for thosewho are interested.

Since we focus onSn, we want to know the number of conjugacy classes and then classifyall representations ofSn. Let us take a look at S3 first.

Definition 2.22. The standard representationof a symmetric group on [n] ={1, 2, . . . , n}is an irreducible representation of dimension n 1 (over a field whose characteristic does notdividen!) defined in the following way:

Take the permutation representation ofSn as defined in Definition2.9and look at the(n 1)-dimensional subspace of vectors whose sum of coordinates in the basis is zero. Therepresentation restricts to an irreducible representation of dimensionn 1 on this subspace.This is the standard representation.

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Example 2.23. Since S3has three conjugacy classes, it has three irreducible representations:

1. One of them is the trivial representation which is on C and acts by() = for S3and C.

2. Another one is the alternating representation (or sign representation) which is also onC, but acts by () = (sgn()) for S3 and C.

3. The third one is the standard representation and it is on the subspace(z1, z2, z3) C

3 | z1+z2+z3 = 0

acting by((z1, z2, z3)) = (z(1), z(2), z(3)) for S3 and (z1, z2, z3) C3.

These are the only irreducible representations ofS3. Let be a representation ofS3 andtriv,alt, andstd denote the trivial, alternating, and standard representations respectively.Then for any S3, () = atriv() balt() cstd(), where a, b, and c are values

determined by. The dimensions oftriv(),alt(), andstd() are 1, 1, and 2 respectively.

Note that for n >3, there are more irreducible representations than just these three andit turns out that the number of conjugacy classes ofSn is the number of ways of writing nas a sum of a sequence of non-increasing positive integers.

Proposition 2.24. The conjugacy classes ofSn correspond to the partitions ofn.

Proof. The conjugacy classes ofSn are uniquely determined by the cycle type of their ele-

ments and each class contains only one cycle type. Thus, there is a bijection between eachconjugacy class ofSn and a partition ofn into the cycle type of that class.

Therefore the number of irreducible representations ofSnis same as the number of Youngdiagrams corresponding to the partitions ofn. For example, there are five irreducible repre-sentations ofS4 according to Example2.17.

The dimension of each irreducible representation ofSn corresponding to a partition ofn is equal to the number of different standard Young tableaux that can be obtained fromthe Young diagram of the representation.

Example 2.25. One of the partitions of 4 is = (2, 1, 1) with associated Young diagram

.

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There are three standard Young tableaux of shape , namely

1 234 ,

1 324 , and

1 423 .

Therefore the irreducible representation ofS4 that corresponds to the partition (2, 1, 1) is3-dimensional.

Alternatively, we have a direct formula for the dimension of an irreducible representationofSn, known as the hook-length formula:

To each box in the Young diagram associated to , we assign a number called the hook-length. The hook-length for a box is calculated by taking the number of boxes in the samerow 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 is n!

h.

In the previous example where n = 4 and = (2, 1, 1), if we label each box of theassociated Young diagram with its hook-length, we have

4 121 .

Therefore the corresponding irreducible representation ofS4 has dimension 4!

8 = 3 as ex-

pected.

2.6 Some properties of unitary Weingarten functions

We are now ready to discuss and prove some of the properties of unitary Weingarten func-tions.

2.6.1 The invertibility ofG

Recall thatn

k=1

(N+Jk) =G, whereJk C[Sn] are the Jucys-Murphy elements andG C[Sn]

is the function G() =N#(). Since every permutation in Sn can be written as a permuta-tion matrix ann nmatrix created by rearranging the rows and/or columns of the n n

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identity matrix, and this extends linearly to C[Sn], we can define a matrix norm on C[Sn].

Let A be an n n matrix, there are many ways to assign a matrix norm to A. Wewill use the spectral normwhere A=

max{| is an eigenvalue ofAA}. Under this

norm, = A= 1 for every permutation matrix A corresponding to the permutation

Sn. In particular,(r, s)= 1 for every transposition (r, s) Sn and thus

Jk= (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 =k=0

(1)kxk.

Therefore if we assume n N, thenk N ifk n and sok 1< N, which impliesN+ Jk

is invertible since N+ Jk =N(1 +N1Jk) andN

1Jk=Jk

N

k 1

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and thusn

k=1

(N+ k 1)1 =Sn

WgU(, N).

This proves part of Proposition 2.4, i.e.Sn

WgU(, N) = 1

N(N+ 1) (N+n 1)

whenn N.

Ifn > N, then WgUN is the pseudo-inverse ofGUN and as functions in C[Sn], we haveGWgUG= G.Therefore

triv(G) =triv

GWgUG

=triv(G)triv

WgU

triv(G) (since triv is a representation)

= (triv(G))2triv

WgU

(since triv is 1-dimensional)

triv

WgU

= (triv(G))

1 (sincetriv(G)= 0)

=

nk=1

(N+ k 1)1,

which is same as before when n N. This proves Proposition2.4 completely.

2.6.3 Asymptotics ofWgU

Consider the length function | | on Sn, where || ( Sn) is the minimal non-negativeinteger l such that can be written as a product of l transpositions. In the paper[9], theauthors showed the asymptotic estimate

WgU(, N) = O(Nn||).We will show this result using Theorem 2.7 (Jucys).

Whenn N, G is invertible with G1 = WgU, therefore

WgU =n

k=1

(N+Jk)1

=Nn(1 +N1J1)1 (1 +N1Jn)

1.

SinceN1Jk< 1 for allk n,

NnWgU =

k1=0

(1)k1(N1J1)k1

kn=0

(1)kn(N1Jn)kn

=l=0

(N)l

k1,...,kn0k1++kn=l

Jk11 Jknn .

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Note thatJk11 Jknn is a linear combination of permutations of length at mostk1+ +kn.

Therefore for any Sn, does not appear in any of the sumsk1,...,kn0k1++kn=l

Jk11 Jknn

whenl


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Let e C[Sn] be the function defined by e() =

1, = e

0, =e. It can be easily checked

that f e =e f =f for all f C[Sn]. The inverse function offwith respect to , if it

exists, is denoted by f(1) and it satisfies f f(1) =f(1) f =e. We have the followingdefinition.

LetZ (C[Sn]) be the center ofC[Sn]:

Z(C[Sn]) ={h C[Sn]| h f=f h(f C[Sn])} .

For the unitary group UN, we define the element GU(, N) inZ (C[Sn]) by

GU(, N) =N#(), Sn.

Note that we previously denoted GU(, N) simply by G. Now if = (1, 2, . . . , r) isa partition ofn, we write l() for the length r of, then G can be expanded in terms ofirreducible characters ofSn as follows:

G= 1

n!

n

fC(N), (2.2)

wheref is the dimension of the irreducible representation associated with and C(N) isthe polynomial in Ngiven by

C(N) =

l()i=1

ij=1

(N+ j i).

The unitary Weingarten function WgU(, N) onSn is defined by

Wg

U

(, N) = Wg

U

=

1

n! n f

C(N)

, (2.3)

summed over all partitions of n. It is the pseudo-inverse element of G, i.e. the uniqueelement inZ (C[Sn]) satisfying

G WgU G= G.

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In particular, unless N {0, 1, 2, . . . , (n 1)}, functions G and WgU are inversesof each other and satisfy

G WgU = WgU G= e.

Example 2.26. ConsiderS3. First note that for each partition , the irreducible character depends only on the conjugacy class, thus the unitary Weingarten function WgU has thisproperty as well.

Since S3 has three conjugacy classes: the class of the identitye, the class of the trans-position (1, 2), and the class of the single cycle (1, 2, 3). These three conjugacy classescorrespond to the three partitions (1, 1, 1), (2, 1), and (3) of the number 3, with associatedYoung diagrams

, , and .

Therefore

WgU([1, 1, 1], N) = WgU(e, N),

WgU([2, 1], N) = WgU((1, 2), N) = WgU((1, 3), N) = WgU((2, 3), N), and

WgU([3], N) = WgU((1, 2, 3), N) = WgU((1, 3, 2), N).

To compute these values, it is enough to evaluate WgU at the representative elements

e, (1, 2), and (1, 2, 3) using equation (2.3). First, we compute the character table of S3using the Murnaghan-Nakayama Rule (see, e.g. [21]for more details). The method gives acombinatorial way of computing the character table of any symmetric group Sn. It has thefollowing steps:

1. Since the characters of a group are constant on its conjugacy classes, we index thecolumns 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 alsoindex the irreducible characters by the partitions. We index the rows of the charactertable by the associated Young diagrams:

(1, 1, 1) (2, 1) (3)

.

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2. We now calculate the entry in row ( denotes a Young diagram) and column (denotes a partition). Let1, 2, . . . be the parts of in decreasing order. Drawingasa Young diagram, define a filling ofwith contentto be a way of writing a numberin each square of such that the numbers are weakly increasing along each row (tothe right) and each column (downwards) and there are exactlyi squares labeledi for

eachi.

3. Consider all fillings ofwith content such that for each label i, the squares labelediform a connected skew tableauthat does not contain a 2 2 square. (A skew tableauis connected if the graph formed by connecting horizontally or vertically adjacentsquares is connected.) Such a tableau is called a border-strip tableau, with each labelrepresenting a border-strip. For the purpose of illustrating how this method works,suppose we are trying to calculate the entry for = (3, 2) and = (2, 2, 1) in thecharacter table ofS5, then

1 1 3

2 2 ,

1 2 3

1 2 , and

1 2 2

1 3

are the only three border-strip tableaux that are fillings of = (3, 2) with content= (2, 2, 1).

4. For each label in the border-strip tableau, define theheightof the corresponding border-strip to be one less than the number of rows of the border-strip. We then define theweightof the border-strip tableau to be (1)s wheres is the sum of the heights of theborder-strips that compose the tableau. Finally, the entry in the character table is thesum of all the weights of the possible border-strip tableaux. For example, the threeborder-strip tableaux of= (3, 2) with = (2, 2, 1), as shown above, have weights 1,1, and -1 respectively, for a total weight of 1. Therefore the corresponding entry in thecharacter table ofS5 is 1.

Using this method, we can easily compute the character table ofS3:

(1, 1, 1) (2, 1) (3)

1 -1 1

2 0 -1

1 1 1

.

Now forS3, the three irreducible representations, as shown earlier, are

1 = (1, 1, 1) = , 2= (2, 1) = , and 3 = (3) = .

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The corresponding dimensions can be calculated using the hook-length formula. We havef1 = 1,f2 = 2, and f3 = 1. Moreover,

C1(N) =N(N 1)(N 2),C2(N) =N(N+ 1)(N 1),

C3(N) =N(N+ 1)(N+ 2).

Therefore by equation (2.3), we have

1. WgU ([1, 1, 1], N)= WgU(e, N)

=1

6

1

N(N 1)(N 2)+

4

N(N+ 1)(N 1)+

1

N(N+ 1)(N+ 2)

=

N2 1

N(N2 1)(N2 4),

2. WgU ([2, 1], N)= WgU((1, 2), N)

=1

6

1

N(N 1)(N 2)+

1

N(N+ 1)(N+ 2)

=

1

(N2 1)(N2 4), and

3. WgU ([3], N)

= Wg

U

((1, 2, 3), N)=

1

6

1N(N 1)(N 2)

+ 2

N(N+ 1)(N 1)+

1

N(N+ 1)(N+ 2)

=

2

N(N2 1)(N2 4).

2.7.2 Character expansion ofWgU

It is known that the set

n

of irreducible characters ofSn forms a basis of the centerZ(C[Sn]) of the group algebra C[Sn].

Definition 2.27. Iff Z(C[Sn]) is a central function, then the expression

f=n

f()

offwith respect to the character basis ofZ(C[Sn]) is called the character expansion off(see, e.g. [19] for more details).

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Definition 2.28. 1. Let be a square in a Young diagram n. The content of,denotedc(), is defined to be the column index ofminus the row index of.

2. Given a Young diagram n, let H denote the product of hook-lengths of.

3. Let s(1N

) =

1

H

(N+c(

)).

In the paper[19], the author proved the following character expansion of WgU C[Sn].We will only state the theorem. The proof, and some other properties, can be found insection 3 of [19].

Theorem 2.29. (Theorem 3.2 in [19])The character expansion ofWgU C[Sn] is

WgU

= n

1

H2s(1N)

. (2.4)

It can be easily checked that, using this theorem, evaluating WgU at any S3 wouldproduce the same result as the previous example.

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Chapter 3

Integration over the Unitary Group

and Applications

The main goal of this chapter is to discuss how Theorem 1.5can be used to solve various

problems in random matrix theory.

Recall that in Theorem1.5,a formula is given so that general matrix integrals

E


(3.1)

can be calculated as a sum of unitary Weingarten functions over the symmetric group Sn,whereU is anN NHaar-distributed unitary random matrix, Edenotes expectation withrespect to the Haar measure, and n 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 weconjugateXby an independent unitary matrix. As a result, expressions similar to

E [Tr (X1U1X2U

2 XnUn)] (3.2)

are quite common in random matrix theory, where X1, . . . , X n areN N random matricesand 1, . . . , n {1, }.

We will show by examples that the calculation of (3.2) can be reduced to the calculationof (3.1). First, we introduce some notations.

Notation 3.1. LetSn be the symmetric group on {1, . . . , n},n 1.

1. Let be a permutation in Sn, then for any n-tuple X = (X1, . . . , X n) of N Ncomplex matrices

Tr(X) = Tr(X1, . . . , X n):=

CC()

Tr

jC

Xj

,

whereC() is the set of all the disjoint cycles of (including fixed points).

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2. Let n denote the cyclic permutation

n= (1, 2, . . . , n) Sn

of order n.

3. LetMN(C) be the algebra ofN Ncomplex matrices, we use{Ea,b}1a,bNas a basiswhere

(Ea,b)ij =(a,b),(i,j) = a,ib,j.

In other words, Ea,b is the N Nbasis matrix whose entries are all 0 except for theentry at row a and column b, which is 1. It has the property that

Tr(XEa,b) =Ni=1

(XEa,b)ii= (XEa,b)bb= Xba

for everyX MN(C).

3.1 Unitarily invariant random matrices

Definition 3.2. LetX = (X1, . . . , X n) be ann-tuple ofN Nrandom matrices, we say Xisinvariant under unitary conjugation if (X1, . . . , X n) and (UX1U

, . . . , U X nU) are identi-

cally distributed for any independent N N unitary matrixU.

For two sequences i = (i1, . . . , in) and i = (i1, . . . , i

n) of positive integers and for a

permutation Sn, we put(i, i

) =n

k=1

iki(k) .

Under this notation, Theorem1.5 becomes

E


=

,Sn

(i, i)(j,j

)WgU(1, N), (3.3)

whereU is an N N Haar-distributed unitary random matrix and n N.

Now, supposeW is an N NHermitian (i.e. W =W) random matrix that is invariantunder unitary conjugation. For expressions of the form

E [Wi1j1Wi2j2 Winjn]

wheren N, we expect to have a formula that is closely related to ( 3.3). This is shown inthe following theorem.

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Theorem 3.3. (Theorem 3.1 in[8])LetWbe as announced andn N, then

E [Wi1j1Wi2j2 Winjn] =

,Sn

(i,j)WgU(1, N)E [Tr(W)] . (3.4)

Proof. For the remainder of this section, unless otherwise specified, all matrices are assumedto have size N N.

We will give two proofs. The first proof is a direct computation using the assumptionthat W is invariant under unitary conjugation.

LetU be a Haar-distributed unitary random matrix that is independent fromWand letV =U W U, then

E [Wi1j1Wi2j2 Winjn] = E [Vi1j1Vi2j2 Vinjn] .

Note that for each k where 1 k n, we have

E [Vikjk ] = E [(U W U)ikjk ]

=N

lk,mk=1

E

UiklkWlkmkUjkmk

.

Therefore

E [Vi1j1Vi2j2 Vinjn]

=N

l1,...,ln,m1,...,mn=1

E Ui1l1 UinlnUj1m1 UjnmnE [Wl1m1 Wlnmn]=

Nl1,...,ln,

m1,...,mn=1

,Sn

(i,j)(l,m)WgU(1, N)E [Wl1m1 Wlnmn ]

=

,Sn

(i,j)WgU(1, N)

Nl1,...,ln=1

E

Wl1l1(1) Wlnl1(n)

= ,Sn (i,j)Wg

U(1, N)E [Tr1(W)] , where W = (W , . . . , W n )=

,Sn

(i,j)WgU(1, N)E [Tr(W)] (see Proposition3.4below).

Alternatively, we can apply the spectral theorem for unitarily invariant random matriceswhich states that every Hermitian random matrix Wthat is unitarily invariant has the samedistribution as U DU, where

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1. Uis a Haar-distributed unitary random matrix,

2. D is a diagonal random matrix whose eigenvalues have the same distribution as thoseofW, and

3. U and D are independent.

This is a useful theorem in random matrix theory and we refer to the appendix sectionof [6] for a proof.

Now, every matrix entry Wij has the same distribution as

Nr=1

UirDrrUjr ,

whereU andD are as described above. It follows that

E [Wi1j1Wi2j2. . . W injn]

= E

Nr1=1

Ui1r1Dr1r1Uj1r1

Nr2=1

Ui2r2Dr2r2Uj2r2

Nrn=1

UinrnDrnrnUjnrn

=N

r1,r2,...,rn=1

E [Dr1r1Dr2r2 Drnrn]E

Ui1r1Ui2r2 UinrnUj1r1Uj2r2 Ujnrn

=N

r1,r2,...,rn=1

E [Dr1r1Dr2r2 Drnrn]

,Sn

(i,j)(r, r)WgU(1, N)

= ,Sn

(i,j)WgU(1, N)

Nr1,r2,...,rn=1

(r, r)E [Dr1r1Dr2r2 Drnrn]

=

,Sn

(i,j)WgU(1, N)E

Tr(D , . . . , D n

)

=

,Sn

(i,j)WgU(1, N)E

Tr(UDU, . . . , U D U n

)

= ,Sn (i,j)Wg

U(1, N)E [Tr(W)] , where W = (W , . . . , W n ).

Proposition 3.4.Note that

1. WgU is in the center ofC[Sn] (see, e.g. section2.7or [5]), i.e. it commutes with anyfunctionfdefined onSn, and

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2. whenW= (W , . . . , W n

) (i.e. W1 = W2 = = Wn = W), Tr(W) = Tr1(W) for

all Sn.

Therefore formula (3.4) can be written as a double convolution overSn

as

E [Wi1j1Wi2j2. . . W injn] =

,Sn

(i,j)WgU(1, N)E [Tr(W)]

=

,Sn

E [Tr1(W)]WgU(1, N)(i,j)

=E [Tr(W)] WgU (i,j)

(e).

Example 3.5. For each 1 i N and n 1, suppose i1 =i2 = = in =i = j1 =j2 = = jn, then(i,j) = 1 for all Sn and thus

E [Wnii ] =

,Sn

WgU(1, N)E [Tr(W)]

=Sn

WgU(, N)Sn

E [Tr(W)]

= 1

N(N+ 1) (N+ n 1)

Sn

E [Tr(W)] (by Proposition2.4).

1. Whenn = 1, we have the trivial identity E[Wii] = 1

NE [Tr(W)] = E[tr(W)].

2. Whenn = 2, E[W2ii] = E[(Tr(W))2] + E[Tr(W2)]

N(N+ 1) .

3. Whenn = 3, E[W3ii] = E[(Tr(W))3] + 3E[Tr(W2)Tr(W)] + 2E[Tr(W3)]

N(N+ 1)(N+ 2) .

3.2 Expectation of products of matrices

Let U be an NNHaar-distributed unitary random matrix and {bi}iN, {Bj}jN be se-

quences ofNNcomplex matrices. In this section, we wish to find a formula for expressionsof the form

E[U B1Ub1U B2U

b2 U Bn1Ubn1U BnU

], (3.5)

wheren N.

Note that (3.5) looks very similar to a general version of the expectation computed inExample1.8. We will use the same approach and first compute the ij th entry of (3.5).

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We haveE[U B1U

b1UB2Ub2 U Bn1U

bn1U BnU]ij

= E [Tr(U B1Ub1UB2U

b2 U Bn1Ubn1U BnU

bn)] , where bn= Ej,i is a basis matrix

=N

i1,...,in,i1,...,in,j1,...,jn,j

1,...,j

n=1

E Ui1j1(B1)j1j1(U)j1i1(b1)i1i2 Uinjn(Bn)jnjn(U)jnin(bn)ini1=

Ni1,...,in,i

1,...,i

n,

j1,...,jn,j1,...,j

n=1

(B1)j1j1(b1)i1i2 (Bn)jnjn(bn)ini1E


=N

i1,...,in,i1,...,i

n,

j1,...,jn,j1,...,j

n=1

(B1)j1j1(b1)i1i2 (Bn)jnjn(bn)ini1

,Sni1i(1) ini(n)j1j(1) jnj(n)Wg

U(1, N)

=

,Sn

WgU(1, N) N

j1,...,jn,j1,...,j

n=1

(B1)j1j1 (Bn)jnjnj1j(1) jnj(n)

N

i1,...,in,i1,...,i

n=1

(b1)i1i2 (bn)ini1i1i(1) ini(n)

=

,Sn

WgU(1, N) N

j1,...,jn=1

(B1)j(1)

j1 (Bn)j

(n)jn

N

i1,...,in=1

(b1)i1i(2) (bn)ini(1)

= ,Sn

WgU(1, N) N

j1,...,jn=1

(B1)j1j1(1) (Bn)jnj1(n)

N

i1,...,in=1

(b1)i1in(1) (bn)ini

n(n)

=

,Sn

WgU(1, N)Tr1(B1, . . . , Bn)Trn(b1, . . . , bn).

Notation 3.6. Let Tr(b , . . . , b n1

, Ej,i) = (Tr(b , . . . , b n1

, 1))ij, where is a permutation in Sn.

Roughly speaking, we do not apply Tr to the cycle of that contains the element n, andthus Tr(b , . . . , b

n1, 1) is a matrix.

For example, suppose S6 such that = (1, 3, 4)(2)(5, 6), then

Tr(b1, . . . , b5, 1) ={Tr(b1b3b4)Tr(b2)} b5b6.

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Theorem 3.7. Using Notation3.6, we have a formula for (3.5):

E[U B1Ub1UB2U

b2 U Bn1Ubn1U BnU

]

=

,SnWgU(1, N)Tr1(B1, . . . , Bn)Trn(b1, . . . , bn1, 1). (3.6)

Proposition 3.8. By the same reasoning as in Proposition3.4, formula (3.6) can also bewritten as a double convolution overSn as,Sn

Tr1(B1, . . . , Bn)WgU(1, N)Trn(b1, . . . , bn1, 1) =

Tr(B) WgU Tr(b) (n),

whereB= (B1, . . . , Bn), b= (b1, . . . , bn1, 1), andTr(b)

() = Tr(b1, . . . , bn1, 1).

Example 3.9. Suppose now that bi =b for all i N and Bj =B for all j N, then (3.5)becomes

E

U BU(bUBU)n1

, (3.7)

wheren 1.

As shown in [5], WgU(, N) is a rational function ofNfor every Sn (this fact shouldbe clear from the definitions given in the previous chapter as well). This and Theorem3.7

together imply that (3.7) can be written in the formn1

k=0akb

k, where ak are coefficients and

b0 =IN.

Note that Example1.8corresponds to the case n = 1, and in this example we will calcu-late (3.7) for the case n= 2 using the table of values in appendix B for unitary Weingartenfunctions.

Since WgU([1, 1], N) = 1

N2 1and WgU([2], N) =

1

N(N2 1), by Theorem3.7, we have

E[UBUbUBU]

= ,S2 WgU(1, N)Tr1(B, B)Tr2(b, 1)=S2

WgU(, N)Tr1(B, B)b +S2

WgU(2, N)Tr1(B, B)Tr(b)IN

=(Tr(B))2

N2 1 b

Tr(B2)

N(N2 1)b

(Tr(B))2Tr(b)

N(N2 1) IN +

Tr(B2)Tr(b)

N2 1 IN

=

Tr(B2)Tr(b)

N2 1

(Tr(B))2Tr(b)

N(N2 1)

IN +

(Tr(B))2

N2 1

Tr(B2)

N(N2 1)

b.

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Therefore E[U BUbUBU] =1

k=0

akbk =a0IN+a1b, where

a0=NTr(B2)Tr(b) (Tr(B))2Tr(b)

N(N2 1) and a1 =

N(Tr(B))2 Tr(B2)

N(N2 1) .

Example 3.10. IfB is a rank-one projection, i.e. all entries ofB are 0 except for one ofthe entries on the main diagonal, which is 1, then

Tr(B) = Tr(B , . . . , B n

) = 1 for all Sn

and thus

E U BU(bUBU)n1

= ,Sn

WgU(1, N)Tr1(B)Trn(b , . . . , b n1

, 1)

=Sn

Sn

WgU(, N)Tr(b , . . . , b n1

, 1)

=n

k=1

(N+ k 1)1

Sn

Tr(b , . . . , b n1

, 1)

(by Proposition2.4)=

n

k=1(N+ k 1)1

n1

k=0 l1++lr=n1k Tr(bl1) Tr(blr) b

k.

To make the above expression simpler, it would require further investigation into thecycle structures of permutations ofSn.

3.3 Matricial cumulants

Definition 3.11. 1. LetXbe a set of random matrices, then expressions of the form

E [Tr(X1, . . . , X n)]

whereXi X and Sn, are called generalized momentsof order nof the set X.

2. LetX and Bbe two sets of random matrices, then the mixed generalized moments ofthe sets X and Bare the generalized moments of the set X B. Note that they canbe computed from expressions of the form

E [Tr(B1X1, . . . , BnXn)] ,

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whereBi B {IN}, Xi X {IN}, and Sn.

Motivated by Propositions 3.4 and 3.8, we will discuss the following questions in thissection:

1. Do we have a convolution formula for moments of the form E [Tr(B1X1, . . . , BnXn)],and thus mixed generalized moments, as well?

2. If so, what are the conditions on the matricial modelsX andB?

It turns out that for two independent n-tuples ofNNrandom matricesX = (X1, . . . , X n)and B = (B1, . . . , Bn), if one of them, say X, has a distribution which is invariant underunitary conjugation, then the moment E [Tr(B1X1, . . . , BnXn)] for any Sn can be writ-ten as a convolution over the symmetric group Sn between the generalized moments ofBwith some matricial cumulant functionCX, as defined by the authors in [3].

Definition 3.12. For n N, let X = (X1, . . . , X n) be any n-tuple of NN randommatrices, the nth UN-cumulant function

CX :Sn C, CX()

is defined by the relation

CX := E[Tr(X)]

N#(1)

,

whereN# is the function in C[Sn] such that N#

() =N#() for Sn and

N#

(1)

is

the inverse ofN#

with respect to the classical convolution operation.

The UN-cumulants ofXare the CX() for single cycles ofSn.

For simplicity, we will refer to UN-cumulant functions and UN-cumulants simply as cu-mulant functions and cumulants from now on. Moreover, whenX1 =X2 = = Xn =X,we will use the notation CX forC(X,...,X).

Example 3.13. 1. When n = 1, the only element in S1 is e = (1).

Therefore N# (e) =N, N#(1) (e) = 1N and thusCX(e) = E[Tre(X)]

N#

(1)(e) = E[tr(X)].

2. Whenn = 2, then S2= {e, (1, 2)} and

N# :e N2, (1, 2)N.

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Therefore

N#(1)

:e 1

N2 1, (1, 2)

1

N(N2 1)and it can be checked that

N#

N#(1)

=

N#(1)

N# =e.

We can calculate the cumulant functions as

C(X1,X2)(e) =S2

E [Tr(X1, X2)]

N#(1)

(1e)

= E [Tre(X1, X2)]

N#(1)

(e) + E

Tr(1,2)(X1, X2)

N#(1)

((1, 2))

=E [Tr(X1)Tr(X2)]

N2 1

E [Tr(X1X2)]

N(N2 1)

=NE [Tr(X1)Tr(X2)] E [Tr(X1X2)]

N(N2 1) ,

C(X1,X2)((1, 2)) =S2

E [Tr(X1, X2)]

N#(1)

(1(1, 2))

= E [Tre(X1, X2)]

N#(1)

((1, 2)) + E

Tr(1,2)(X1, X2)

N#(1)

(e)

=E [Tr(X1)Tr(X2)]

N(N2 1) +

E [Tr(X1X2)]

N2 1

=E [Tr(X1)Tr(X2)] +NE [Tr(X1X2)]

N(N2 1) .

Remark 3.14. Note that we previously defined the functionN# C[Sn] asG whereG() =N#() and, as shown earlier, G is invertible with G1 = WgU whenn N.

Therefore the cumulant functions can be equivalently defined as

CX() =E[Tr(X)] WgU

(),

whereX = (X1, . . . , X n) and Sn.

For consistency, we will use this notation from now on.

Proposition 3.15. Some basic properties of the cumulant functions.

1. For each Sn, (X1, . . . , X n)C(X1,...,Xn)() isn-linear.That is, for eachi where1 i n, ifXi= Yi+Zi where C, then

C(X1,...,Xi,...,Xn)() =C(X1,...,Yi,...,Xn)() +C(X1,...,Zi,...,Xn)().

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2. Since{E[Tr(IN)]} () = E [Tr(IN, . . . , I N)] =N#() =G(),

the generalized moments ofXcan be found from the cumulant function by the inverseformula:

E[Tr(X)] = E[Tr(X)] WgU G= CX E[Tr(IN)].

3. SinceTr(U X1U

, . . . , U X nU

) = Tr(X1, . . . , X n) for any unitary matrixU,

C(UX1U,...,UXnU)() =C(X1,...,Xn)().

4. SinceWgU(, N) depends only on the conjugacy class of,

CX() =Sn

E [Tr(X)]WgU(1, N)

has this property as well.

Therefore the cumulantsCX() of a matrixX for single cycles ofSn are all equal.We will denote by Cn(X) this common value and call it cumulant of order n of thematrixX. In particular, as computed in the previous example,

C1(X) = E[tr(X)], and

C2(X) =E [Tr(X1)Tr(X2)] +NE [Tr(X1X2)]

N(N2 1)

= N

N2 1

E

tr(X2)

E [tr(X)]2

.

Lemma 3.16. LetX = (X1, . . . , X n) be an n-tuple of NN random matrices that areunitarily invariant and let B = (B1, . . . , Bn) be an n-tuple of N N random matricesindependent withX, then

E [Tre(B1X1, . . . , BnXn)] ={E [Tr(B)] CX} (e) ={CB [Tr(X)]} (e).

Proof. For any independent N N Haar-distributed unitary random matrixU, we have

E [Tre(B1X1, . . . , BnXn)]

= E n

i=1 Tr(BiXi)= E

ni=1

Tr(BiU XiU)

=N

i1,...,in,i1,...,i

n,

j1,...,jn,j1,...,j

n=1

E

(B1)i1i1Ui1j1(X1)j1j1Ui1j1 (Bn)ininUinjn(Xn)jnjnUinjn

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=N

i1,...,in,i1,...,i

n,

j1,...,jn,j1,...,j

n=1

E

(B1)i1i1 (Bn)ininE

(X1)j1j1 (Xn)jnjnE


.

Apply Theorem1.5 and rearrange the terms, a similar calculation as the derivation of

formula (3.6) gives usE [Tre(B1X1, . . . , BnXn)]

=

,Sn

WgU(1, N)E [Tr(B)]E [Tr1(X)]

=Sn

E [Tr(B)]Sn

E [Tr(X)]WgU(11, N)

=Sn

E [Tr(B)] CX(1)

={E [Tr(B)] CX} (e).

Note that this is equal toE [Tr(B)] WgU E [Tr(X)]

(e)

={CB [Tr(X)]} (e),

since WgU is a central function on Sn.

Let us now consider the mixed generalized moments with any in Sn. Recall that inNotation3.1,we defined the basis {Ea,b}1a,bN ofMN(C). It has the following additionalproperties.

Proposition 3.17. For all permutations, inSn,

Tr

Ea(1),b1 , . . . , E a(n),bn

= Tr(Ea1,b1, . . . , E an,bn) .

Proof. By definitions, Tr

Ea(1),b1, . . . , E a(n),bn

=

N

i1,...,in=1Ea(1),b1

i1i(1)

Ea(n),bn

ini(n)=

Ni1,...,in=1

a(1),i1b1,i(1)

a(n),inbn,i(n)

=N

i1,...,in=1

a1,i1(1)b1,i(1)

an,i1(n)bn,i(n)

=

Ni1,...,in=1

(Ea1,b1)i1(1)i(1) (Ean,bn)i1(n)i(n)

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=N

i1,...,in=1

(Ea1,b1)i1i(1) (Ean,bn)ini(n)

= Tr(Ea1,b1, . . . , E an,bn) .

Proposition 3.18. For all permutations inSn,

Tr(Ea1,b1X1, . . . , E an,bnXn) = (X1)b1a(1) (Xn)bna(n) .

Proof. Note that for each i, multiplying Eai,bi on the left ofXi, roughly speaking, replacesthe athi row ofXi by its b

thi row and all other rows vanish. Therefore (Eai,biXi)jij(i) is non-

zero only when ji = ai, and it is equal to (Xi)bia(i) .Thus

Tr(Ea1,b1X1, . . . , E an,bnXn)

=

Nj1,...,jn=1

(Ea1,b1X1)j1j(1) (Ean,bnXn)jnj(n)

= (X1)b1a(1) (Xn)bna(n) .

Lemma 3.19. Now for all permutations inSn, we have

E [Tr(Ea1,b1X1, . . . , E an,bnXn)] ={E [Tr (Ea1,b1 , . . . , E an,bn)] CX} ().

Proof. By the previous two propositions,

E [Tr(Ea1,b1X1, . . . , E an,bnXn)]

= E

ni=1

(Xi)bia(i)

= E

ni=1

Tr

Ea(i),biXi

= E

Tre

Ea(1),b1X1, . . . , E a(n),bnXn

= E TrEa(1),b1, . . . , E a(n),bn CX (e)=Sn

E

Tr1

Ea(1),b1, . . . , E a(n),bn

CX()

=Sn

E [Tr1(Ea1,b1 , . . . , E an,bn)] CX()

={E [Tr (Ea1,b1, . . . , E an,bn)] CX} ().

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By n-linearity and using the fact that WgU is a central function on Sn, we have provedthe following theorem.

Theorem 3.20. LetX andBbe two independent sets ofN Nrandom matrices such that

the distribution ofX is unitarily invariant. Then for anynN, letX= (X1, . . . , X n) beann-tuple inX andB= (B1, . . . , Bn) be ann-tuple inB, we have

E [Tr(B1X1, . . . , BnXn)] ={E [Tr(B)] CX} () ={CB [Tr(X)]} ().

This implies the following convolution relation.

Corollary 3.21. With the hypothesis of Theorem3.20,

C(X1B1,...,XnBn) = CX CB.

Proof.

C(X1B1,...,XnBn)= E [Tr(B1X1, . . . , BnXn)] WgU

= E [Tr(B)] CX WgU

=CX CB.

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Chapter 4

Conclusion and Future Work

Throughout this report, Theorem1.5 was used repeatedly. One of the applications was thatit allowed us to derive a formula for expressions of the form ( 3.7):

E U BU(bUBU)n1wheren 1 and U,b, B are as described.

We now briefly discuss why we would be interested in such expressions.

4.1 The Cauchy transform

Let C+ ={z C| Im(z)>0} denote the complex upper half plane and C ={z C| Im(z)< 0}

denote the complex lower half plane. Let be a probability measure on Rand forz R let

G(z) =

R

1

z td(t).

G is the Cauchy transformof the measure and it is analytic on C+ with range contained

in C.

Suppose is compactly supported, denote r:= sup {|t| | t supp()}. Then for|z|> rwe can expand

1

z t =1

z 11 t/z=

1

z

n=0

tn

zn

=n=0

tn

zn+1, t supp().

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The convergence is uniform in t supp(), therefore we can integrate the series term byterm against d(t) and obtain

G(z) =

n=0n

zn+1, |z|> r,

wheren:=R

tnd(t) is the nth moment offor n 0.

Consider now the formal power series

M(z):= 1 +n=1

nzn.

M(z) is the moment-generating seriesof and it is closely related to the Cauchy trans-

form via the relation

G(z) =1

zM

1

z

.

It is advantageous to consider the Cauchy transform because it has nice analytic prop-erties and provides an effective way of recovering the corresponding probability measure concretely via the Stieltjes inversion formula(see Remark 2.20 in [18]for more details).

4.2 The operator-valued Cauchy transform

In this section, we will use some concepts from free probability theory. For a brief introduc-tion of the subject, please see appendixA or [18].

First, note that ifX is a (classical) random variable distributed according to , then the(classical) Cauchy transform can be expressed as

G(z) = E

(z X)1

; z / R,

whereE

denotes expectation with respect to .

Now, ifX is a self-adjoint operator on a Hilbert space, then the operator-valued analyticfunctionRX(z) = (zI X)

1, where Idenotes the identity operator, is called the resolventof the operator X. Following Voiculescu [22], a more general notion of the resolvent ofXcan be defined as follows: for an arbitrary operator b on the same Hilbert space as X,b canbe written as

b= Re(b) +iIm(b),

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where Re(b) = b+b

2 and Im(b) =

b b

2i . It has been noted by Voiculescu in[22] that

RX(b) = (b X)1, Im(b)>0

is an analytic map so that Im (RX(b))< 0. Note that an operator is said to be positiveif its

spectrum consists of non-negative real numbers.

LetA be a von Neumann operator algebra with the normal faithful trace , and letBbea von Neumann subalgebra ofA. A conditional expectation (|B) is a weakly continuouslinear mapA B satisfying the following properties:

1. (1A|B) = 1, and

2. ifb1, b2 B, then (b1ab2|B) =b1(a|B)b2.

In the paper [2], the author showed that if two self-adjoint operatorsa, b A are free (see,e.g. appendixAor [18] for definitions), then the following identity holds for their resolvents:

(Ra+b(z)|a) =Ra((z)).

In other words,

(z1A (a +b))1|a

= ((z)1A a)

1, (4.1)

where(|a) denotes the conditional expectation on the subalgebra generated by a and (z)is an analytic function from C+ to C+.

Let AN and BN be NN Hermitian deterministic matrices, UN be an NN Haar-

distributed unitary random matrix, and put XN=AN+ UNBNUN. The resolvent ofXN isRXN(z) = (zIN XN)

1 and the resolvents ofAN and BNare defined similarly.

It is known that AN and UNBNUN are asymptotically free as N (see, e.g. ap-

pendixA or [18]). Therefore we wish to investigate if (4.1) holds for random matrices in anapproximate sense.

Note that inXN=AN+ UNBNUN, theANpart is deterministic and the UNBNU

N part

is random. Therefore projectingXNonto the algebra generated byANsimply means takingexpectation in the random part and thus

E (zIN (AN+UNBNUN))1= (N(z) AN)1 (4.2)as N , where E denotes expectation with respect to the Haar measure and N(z) MN(C) is an N Ndeterministic matrix.

The right-hand side of (4.2) is equal to RAN(N(z)) which can be considered as a resol-vent ofAN, and note that (4.2) holds only when N .

46


52/59

The inception of this project was motivated by the following question: for a given N,how far is E

(zIN (AN+ UNBNU

N))

1 from being a resolvent ofAN?Observe that ifbN=zINANandz C\R, thenbNis invertible by the spectral theorem

(since AN is Hermitian). Moreover, if|z| is large enough, then b1N= (zIN AN)

1

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Gu_Yinzheng_201308_MSc-Weingarten Function and Random Matrices

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