Character orthogonality for the partition algebra and fixed points of permutations

a

nd

tee

et

Advances in Applied Mathematics 31 (2003) 113–131

www.elsevier.com/locate/yaam

Character orthogonality for the partition algebra afixed points of permutations

John Farinaa,1 and Tom Halversonb,∗,2

a Department of Mathematics, University of California–San Diego, La Jolla, CA 92093, USAb Department of Mathematics and Computer Science, Macalester College, Saint Paul, MN 55105, USA

Received 26 June 2002; accepted 2 August 2002

Abstract

We give a closed formula for the trace of the partition algebraPn(x) acting on an irreduciblerepresentation whose basis is indexed by the set partitions of1, . . . , n. This trace counts the separtitions that are fixed under the action of the symmetric groupSn. We use this trace to determinan analog of the “second orthogonality of characters” formula forPn(x) and to compute the tracof the biregular representation ofPn(x). We use the Schur–Weyl duality betweenPn(r) and thesymmetric groupSr to study fixed points of random permutations inSr . In particular, we computethe joint mixed moments of the random variables Tr(σ j ) for σ ∈ Sr . 2003 Elsevier Inc. All rights reserved.

Keywords:Set partition; Symmetric group; Representation; Character; Joint moments

For x ∈ C∗ andn ∈ Z with n 0, the partition algebraPn(x) is an algebra overCwith a basis indexed by the partitions of1,2, . . . ,2n into subsets. Ifx is not an integerin the range−2n x 2n − 1, thenPn(x) is semisimple. WhenPn(x) is semisimple,its irreducible representations are indexed by the integer partitionsλ k,0 k n. Inthis paper, we study the irreduciblePn(x)-representationM∅

n indexed by the partition∅of 0. The dimension ofM∅

n is thenth Bell numberB(n), and it has a basis indexed by spartitions of1,2, . . . , n.

* Corresponding author.E-mail addresses:[email protected] (J. Farina), [email protected] (T. Halverson).

1 Supported in part by National Science Foundation Grant DMS-980085.2 Supported in part by National ScienceFoundation Grants DMS-0100975 and DMS-980085.

0196-8858/03/$ – see front matter 2003 Elsevier Inc. All rights reserved.doi:10.1016/S0196-8858(02)00555-9

114 J. Farina, T. Halverson / Advances in Applied Mathematics 31 (2003) 113–131

s

r

ds

tric

the

is

artinnics.

We give a closed formula for the characterχ∅n (dµ,n) of M∅

n evaluated on a clasof special elementsdµ,n ∈ Pn(x), where µ k,0 k n. Like conjugacy classrepresentatives in a finite group, it is known that characters ofPn(x) are completelydetermined by their values on thedµ,n. Our formula is

χ∅n (dµ,n) = xn−|µ| ∑

Pµ

∏X∈Pµ

∑d |X

d |X|−1,

where the outer sum is over all set partitionsPµ of µ = µ1, . . . ,µ, the product is oveall partsX of Pµ, and the inner sum is over all positive integersd such thatd divideseachµi ∈ X.

We show that this characterχ∅n has a number of uses and interpretations:

(a) If we let btrn(dµ,n, dν,n) be the simultaneous trace ofdµ,n anddν,n acting onPn(x) byleft and right multiplication, respectively, then we show that

btrn(dµ,n, dν,n) = χ∅2n(dµ∪ν,2n).

As a special case of this formula we get thetrace of the regular representation ofPn(x).(b) We show that ∑

λ

χλn (dµ,n)χ

λn (dν,n) = χ∅

2n(dµ∪ν,2n),

where λ ranges over an index set for the irreducible characters ofPn(x) and χλn

is the irreducible character corresponding toλ. This is an analogue of “the seconorthogonality relation” for characters of a finite group. Ram [Ra, Appendix] provethat a second orthogonality relation exists for any split semisimple algebra.

(c) We show that if|µ| = n, thenχ∅n (dµ,n) counts the number of set partitions of1, . . . , n

that are fixed when the numbers 1, . . . , n are permuted by an element in the symmegroupSn having cycle typeµ.

(d) We apply our result to the study of fixed points of random permutations insymmetric groupSr . Let σ ∈ Sr be uniformly distributed. Then Tr(σ k) is the numberof fixed points ofσk (in the usual permutation representation ofσ ). For a partitionµ = (1a1,2a2, . . .) with |µ| = n we show that

E

n∏

k=1

Tr(σk

)ak

= 1

|Sr |∑σ∈Sr

n∏k=1

Tr(σk

)ak = χ∅n (dµ,n).

This expected value gives joint mixed moments for the random variables Tr(σ 1),

Tr(σ 2), . . . , Tr(σn), and our result forχ∅n (dµ,n) provides a closed formula for th

expected value.

The partition algebra first appeared independently in the work of Jones [Jo] and M[Mar1,Mar2] arising from transfer matrices of lattice models in statistical mecha

J. Farina, T. Halverson / Advances in Applied Mathematics 31 (2003) 113–131 115

ers ofgroup

r–n

s

etric

cond

se the

nality

e setr

the set

raphs).r

A Frobenius formula and Murnaghan–Nakayama rule for the irreducible charactPn(x) were derived by the second author [Ha]. The group algebra of the symmetricC[Sn] and the Brauer algebraBn(x) are subalgebras ofPn(x).

The connection betweenPn(x) and Sr comes from the fact that they are in SchuWeyl duality with each other on tensor space. IfV is the r-dimensional permutatiorepresentation ofSr , then there is an action ofPn(r) on then-fold tensor productV ⊗n thatcommutes with the action ofSr . The algebrasPn(r) andC[Sr ] generate full centralizerof each other in End(V ⊗n), and whenr 2n, Pn(r) ∼= EndSr (V

⊗n). It is from this dualitythat we are able, in Section 5, to derive identities in the character ring of the symmgroupSr .

Much of this paper is inspired by the work of Ram [Ra], who derives a seorthogonality relation for characters of the Brauer algebraBn(x). In Section 5, we followthe work of Diaconis and Shahshahani [DS] and Diaconis and Evans [DE], who usecond orthogonality relations forSn andBn(r) to compute the joint moments of Tr(Mj )

for random matricesM from the unitary groupUr , the orthogonal groupOr , and thesymplectic groupSpr . The connection between these classical groups and the orthogorelations comes via Schur–Weyl duality: the pairsUr andC[Sn], Or andBn(r), andSprand Bn(−r) are centralizers of each other on tensor space. In our case,Sr and Pn(r)

are centralizers of each other, and we use the second orthogonality relation forPn(r) tocompute the joint moments of Tr(σ j ) for σ ∈ Sr .

1. The partition algebra

Let Ωn denote the collection of set partitions (or equivalence relations) on th1, . . . ,2n. The number of these partitions with exactlyk subsets is the Stirling numbeS(2n, k), and the total number of these partitions is

|Ωn| = B(2n) =2n∑

k=1

S(2n, k),

whereB(2n) is the 2nth Bell number (see, for example, [Mac, I.2, Exercise 11]).We identifyπ ∈ Ωn with a partition diagram, which is a simple graph on 2n vertices

arranged in two rows ofn vertices. We label the vertices in the top row with 1, . . . , n

(left-to-right) and label the vertices in the bottom row withn + 1, . . . , 2n (left-to-right). We draw the edges so that the connected components of the graph formpartition π . Two different graphs whose connected components areπ are considered tobe the same partition diagram (so a partition diagram is an equivalence class of gThusπ = 1,2,4,9,10,13, 3, 5,6,7,11,12,14,15, 8,16, is represented by eitheof the following diagrams:

.


on

s

ent the

n

for

nce

In this paper, we do not distinguish betweena set partition and its corresponding partitidiagram. Fora, b ∈ 1, . . . ,2n andπ ∈ Ωn, we write

aπ←→ b if a andb are in the same connected component ofπ. (1)

Thusaπ←→ b if a andb are in the same subset of the set partitionπ .

Let x ∈ C∗ = C \ 0. Define theC-vector space,

Pn(x) = C-spanπ | π ∈ Ωn,

so that the partition diagramsΩn form a basis ofPn(x). We makePn(x) into an associativealgebra by defining a multiplication on the basis elements. Given partition diagramπ1andπ2, define

π1π2 = xγ (π1,π2)π3, (2)

whereγ (π1,π2) andπ3 ∈ Ωn are defined as follows: placeπ1 aboveπ2 and identify thevertices in the bottom row ofπ1 with the corresponding vertices in the top row ofπ2. Callthis new graphG(π1,π2). It contains 3 rows ofn vertices each. Then let

γ (π1,π2) = the number of connected components ofG(π1,π2)

which contain vertices only from the middle row

,

π3 = the partition diagram obtained by considering only

the top and bottom rows ofG(π1,π2), ignoring anyparts entirely in the middle row

.

For example,

= x2 .

This product is associative and independent of the graph that we choose to represpartition. Partition diagram multiplication extends linearly toPn(x) and makes it intoan associative algebra whose identity is idn = 1, n + 1, 2, n + 2, . . . , n,2n. Thedimension ofPn(x) is the Bell numberB(2n), and by conventionP0(x) = C. The partitionalgebraPn(x) is known to be semisimple for allx ∈ C so long asx not an integer in therange−2n x 2n − 1 (see [MS]).

The Brauer algebraBn(x) [Br] is embedded inPn(x) as the span of the partitiodiagrams for which each edge is adjacent to exactly two vertices. The group algebraC[Sn]of the symmetric groupSn is embedded inPn(x) as the span of the partition diagramswhich each edge is adjacent to exactly one vertex in each row.

We use the notations for integer partitions and compositions found in [Mac]. A sequeof nonnegative integersλ = (λ1, . . . , λ) is a composition ofn if |λ| = λ1 + · · · + λ = n.


The compositionλ is a partition ofn, denotedλ n, if |λ| = n andλ1 λ2 · · · λ. Ifλ is a partition, we letmk(λ) be the number of partsλi of λ which are equal tok, and wesometimes use the notationλ = (1m1(λ),2m2(λ), . . .). For example,

µ = (3,1,0,3,4,3,1) is a composition of 15, and

λ = (4,3,3,3,1,1) = (12,33,4

)is a partition of 15.

WhenPn(x) is semisimple, the irreducible representations ofPn(x) are labeled by thepartitions in the set

Λn = λ k | 0 k n (3)

(see [Mar1,Jo]). Forλ ∈ Λn, we letMλn denote the irreduciblePn(x) module indexed byλ

and letχλn denote its character.

Fork 1, define the following elements ofPk(x),

γk = and ek = . (4)

For π ∈ Ωk andτ ∈ Ω we defineπ ⊗ τ to be the diagram inPk+(x) given by drawingτ immediately to the right ofπ . For a compositionµ = (µ1, . . . ,µ) with |µ| = k and0 k n, define

γµ = γµ1 ⊗ · · · ⊗ γµ and dµ,n = γµ ⊗ en−k, (5)

in Ωk andΩn, respectively. We refer to thedµ,n asstandard elementsin Pn(x). Note thatif |µ| = n thenγµ = dµ,n. For example,

γ(3,2,2,1) = d(3,2,2,1),8 = ∈ Ω8,

d(3,2,2,1),10 = ∈ Ω10.

The second author [Ha, Proposition 2.2.1] proves that any character ofPn(x) is completelydetermined by its values on the elements in the setdµ,n | µ ∈ Λn.

2. The representation M∅n

We let Ω∅n ⊆ Ωn denote the subset of partition diagrams that haven + 1, . . . ,2n as

one of its parts. For example,

Ω∅3 =

.


top

[Ha].

s

last

We define the vector space

M∅n = C-span

π

∣∣ π ∈ Ω∅n

,

so that the setΩ∅n is a basis ofM∅

n . If a ∈ Ωn andπ ∈ Ω∅n , thenaπ = xγ (a,π)π ′ with

π ′ ∈ Ω∅n , soM∅

n is a submodule of the (left) regular representation ofPn(x) onPn(x). Wehave

dim(M∅

n

) = ∣∣Ω∅n

∣∣ = B(n),

since the top row of a diagram inΩ∅n partitions the set1, . . . , n into subsets. The bottom

row of a diagram inΩ∅n never changes, so we will identify a such diagram with just its

row. Thus, we identify the diagrams

= .

The representationM∅n is the irreduciblePn(x)-representation indexed by∅ ∈ Λn. This

can be seen from the construction of the irreducible representations in [Mar2,DW], orLet χ∅

n denote the character ofM∅n . If a is an element ofPn(x), we can computeχ∅

n (a)

by

χ∅n (a) =

∑π∈Ω∅

n

aπ |π, (6)

whereaπ |π denotes the coefficient of the basis elementπ whenaπ is expanded in termof the basisΩ∅

n .Characters ofPn(x) are completely determined by their values ondµ,n,µ k, 0

k n (see (5)). The next proposition reduces the problem of computingχ∅n (dµ,n) to that

of counting fixed points ofγµ in Ω∅|µ|.

Proposition 1. Letµ = (µ1, . . . ,µ) be a composition with0 |µ| n.

(a) χ∅n (dµ,n) = xn−|µ|χ∅

|µ|(γµ).

(b) If |µ| = n andπ ∈ Ω∅n , thenγµπ |π = 0 if and only ifγµπ = π .

(c) If |µ| = n, χ∅n (γµ) equals the number of diagrams inΩ∅

n fixed by the permutationγµ.

Proof. (a) If n − |µ| = k > 0, thendµ,n has isolated vertices (no connections) in thek columns. Furthermore, forπ ∈ Ω∅

n , we see thatdµ,nπ will be a scalar multiple of adiagram that also has isolated vertices in the lastk columns. Thusdµ,nπ |π = 0 only if π

has empty vertices in the lastk columns, and in fact

dµ,nπ = xkγµπ∗,


n (see

the

d

whereπ∗ ∈ Ω∅|µ| is the same as the diagram ofπ except with the lastk isolated vertices

deleted. It follows that,χ∅n (dµ,n) = xkχ∅

n−k(γµ).

(b) The action ofγµ on π in the productγµπ is simply to permute the vertices ofπ .Thus,γµπ ∈ Ω∅

n andγµπ |π = 0 if and only ifγµπ = π .(c) This assertion is an immediate consequence of part (b) and of the definitio

Eq. (6)) of the characterχ∅n .

Recall from Section 1, thatγn is then-cycle and thusγn = γ(n) = d(n),n is the standardelement corresponding to the partitionµ = (n).

Lemma 2. If π ∈ Ω∅n , thenγnπ = π if and only if the following condition holds

iπ←→ j if and only if (i + k)

π←→ (j + k), for all k ∈ Z,

wherei + k andj + k are computedmodn.

Proof. The action ofγn onπ is to shift each vertex one position to the left and to shiftfirst vertex to the last position. This action carries connections with it. That is, ifi

π←→ j

then(i − 1)γnπ←→ (j − 1), where we view this subtraction modn. Now, if γnπ = π then

iπ←→ j implies that(i − 1)

π←→ (j − 1). Furthermore ifγnπ = π then for anyk ∈ Z wehaveγ k

n π = π , soiπ←→ j implies that(i − k)

π←→ (j − k). Definition 3. For each divisord of n, denotedd|n, we define a partition diagramyd,n ∈ Ω∅

n

whose connections are given by the rule

ayd,n←→ b if and only if a ≡ b modd.

For example, ifn = 6, then the possibleyd,6 are

y1,6 = , y2,6 = ,

y3,6 = , y6,6 = .

Note thatyd,n hasd connected components each of sizen/d . We refer to the connectecomponents ofyd,n asd-components.

Proposition 4. The set of diagrams inΩ∅n fixed byγn is yd,n | d|n, and

χ∅n (γn) = d(n) = the number of divisors ofn.


2

nce,

t

iedn

en

e

n

Proof. We see that ifd|n, thenyd,n satisfies the condition of Lemma 2, soyd,n is fixedby γn. Now letπ ∈ Ω∅

n such thatγnπ = π , and letd be the minimum distance betweenvertices that are connected by an edge inπ . That is

d =

n, if π has no connections,min

(i − j) modn

∣∣ iπ←→ j, i = j

, otherwise.

Note thati − j is computed modn so that the last vertex and the first vertex are dista1 apart. Choosei and j so thati

π←→ j with (j − i) modn = d . Then by Lemma 2we have(i + k)

π←→ (j + k) for 0 k n. If d does not dividen, then all the verticesin π will be connected and so in factd = 1 (which does dividen). Thus we know thayd,n is a subgraph ofπ . We claim that theyd,n-connections are the only connections inπ .Theyd,n-connections partitionπ into d connected components each of sizen/d . If therewere another connection inπ , thenπ would also have all the other connections implby Lemma 2. In this eventπ would connect two vertices which are closer together thad ,contradicting the minimality ofd . Definition 5. For a compositionµ = (µ1, . . . ,µ) of n and a diagramπ ∈ Ω∅

n , we say thattheµ-blocksof π are the subdiagrams ofπ given by grouping the vertices ofπ into thesubsets

1, . . . ,µ1, µ1 + 1, . . . ,µ1 + µ2, . . . , µ1 + · · · + µ−1 + 1, . . . , n.

Within a µ-block we inherit any connection fromπ , but we ignore connections betwevertices in differentµ blocks. For example, if

π = ,

andµ = (5,4,3), then theµ-blocks ofπ are

.

The next lemma classifies the points inΩ∅n that are fixed byγµ. Example 7 gives som

examples of such fixed points.

Lemma 6. Let µ = (µ1, . . . ,µ) be a composition ofn and letπ ∈ Ω∅n . Thenγµπ = π if

and only if the following conditions hold:

(a) for eachi, theµi -block ofπ is ydi,µi for some divisordi |µi ;(b) if a µi -block of typeydi,µi and aµj -block of typeydj ,µj have connections betwee

them inπ , then(i) di = dj ,


o).

t

e

use

en

(ii) eachdi-component of theµi -block is connected to a uniquedi -component of theµj -block,

(iii) there are no further connections between these two blocks.

Proof. Whenγµ acts onπ the cycleγµi acts on theµi block, so by Proposition 4 theµi

block must be of the formydi,µi for some divisordi of µi . This proves part (a).Now we consider the possible connections between twoµ-blocks. Adi component in

µi can be connected to at most onedj component inµj , otherwise by transitivity those twdj components would be connected to each other (and by definition this cannot happenWhenγµ acts onπ it cyclically permutes thedi components inµi and (simultaneously) icyclically permutes thedj components ofµj . So if α1, . . . , αd1 are thedi components inthe order thatγµ cycles through them andβ1, . . . , βd2 are thedj components ordered thsame way. Then ifα1 is connected toβ1, then for alli, αi is connected toβi . In particular,we must haved1 = d2 for otherwise we get two components in one of theµ-parts connectedto one component in the other. There can be nofurther connections in these blocks becathat would again force two components in one to be connected to one in the other.Example 7. Here we give examples of fixed points forγµ with µ = (9,6,6,4,3). In eachclass theµ-blocks arey3,9, y3,6, y2,6, y2,4, y3,3, respectively, but the connections betwetheµ-blocks vary and are drawn with dashed lines.

π1 = ,

µ1 = 9 µ2 = 6 µ3 = 6 µ4 = 4 µ5 = 3

d1 = 3 d2 = 3 d3 = 2 d4 = 2 d5 = 3

π2 = ,

π3 = ,

π4 = .

Now, we count the number of diagrams that satisfy the conditions of Lemma 6.


s

ese

t

e set

partson.

rtitiony

Proposition 8. For a compositionµ = (µ1, . . . ,µn) of n the number of diagrams inΩ∅n

fixed byγµ is ∑Pµ

∏X∈Pµ

∑d |X

d |X|−1,

where the outer sum is over all set partitionsPµ of µ1, . . . ,µ, the product is over part(subsets) X ∈ Pµ, and the inner sum is over all positive integersd which divide eachelement ofX.

Proof. Let π ∈ Ω∅n be a diagram fixed byγµ. Then by Lemma 6, eachµi -block must

be of the formydi,µi for some divisordi of µi . Furthermore some of theseµ blocks areconnected to others. We say thatµi ∼ µj if there is at least one connection between thtwo blocks. This equivalence relation determines a set partition ofµ1, . . . ,µ. We denotethis set partition byPµ. In this way, each fixed diagramπ determines a set partitionPµ.For example, the underlying set partitionsPi of µ = 9,61,62,4,3, for each fixed poinπi in Example 7, are

P1 = 9, 61, 62, 4, 3, P2 = 9,61, 62,4, 3,P3 = 9,61,3, 62,4, P4 = 9, 61,3, 62, 4.

Now we count the number of fixed diagrams that correspond to this sampartition Pµ. Let X be one of the subsets ofPµ. Then all theµ-blocks of X must beconnected to each other. From Lemma 6(a), we see that for this to happen, each block inXmust be of the formyd,µi for an integerd that is a common divisor of eachµi ∈ X. Fromthe proof of Lemma 6(b), we see that there are exactlyd ways to connect aµi block in Xto aµj block in X. That is, in the notation of the proof of Lemma 6(b), we haved choicesfor β1 and then all the other connections are forced. Now, if there are more than twoin X, then we haved choices for how to connect the next block to these two, and soThus, there ared |X|−1 ways to connect theµ blocks inX.

Now we are ready to complete the proof. Each fixed point determines a set paPµ of µ1, . . . ,µ. This fixed point must have itsµ-blocks connected if and only if theare in the same subsetX of Pµ. Thus eachµi ∈ X must be of the formyd,µi for somecommon divisord of all the elementsµi ∈ X. The connections in two different parts ofPµ

are independent of each other. Therefore,the number of diagrams corresponding toPµ is∏X∈Pµ

∑d |X d |X|−1. Summing over all set partitions gives the result.

Combining Propositions 1 and 8 gives our main result,

Theorem 9. If µ = (µ1, . . . ,µ) is a composition with|µ| = k, 0 k n, then

χ∅n (dµ,n) = xn−kχ∅

k (γµ) = xn−k∑P

∏X∈P

∑d |X

d |X|−1,

µ µ


l

n

e

he

la for

of

where the outer sum is over all set partitionsPµ of µ1, . . . ,µ, the product is over alparts X of Pµ, and the inner sum is over all positive integersd such thatd divides eachµi ∈ X.

Our formula has a simpler statement whenµ is a hook shape, i.e.,µ = (1a, b), or µ hastwo parts:

Corollary 10. Let |µ| = n. Letd(k) denote the number of divisors ofk, and letB(k) denotethekth Bell number(the number of set partitions of1, . . . , k).

(a) If µ = (1a, b) with a, b > 0 then χ∅n (γµ) = (d(b) − 1)B(a) + B(a + 1).

(b) If µ = (n), then χ∅n (γµ) = d(n).

(c) If µ = (1n), then χ∅n (γµ) = χ∅

n (idn) = B(n).

(d) If µ = (a, b), then χ∅n (γµ) = d(a)d(b)+ ∑

t |a,b t, where the sum is over all commodivisorst of a andb.

Proof. For part (a) consider the set partitionsP of µ = b,1,1, . . . ,1. If X ∈ P has theform X = 1, . . . ,1 or X = b,1, . . . ,1, then

∑d |X d |X|−1 = 1, since we can only hav

d = 1. If X = b, then∑

d |X d |X|−1 = ∑d |b d0 = d(b). The number of set partitionsP

with X = b as one of the parts isB(a). This is just the number of set partitions of tremaininga ones. The remaining number of set partitions isB(a + 1) − B(a), so usingTheorem 9,

χ∅n (γµ) = d(b)B(a) + (

B(a + 1) − B(a)) = (

d(b) − 1)B(a) + B(a + 1).

Parts (b), (c), and (d) are proved with the same sort of argument usingµ = n,µ = 1, . . . ,1, andµ = a, b, respectively. Remark 11. The second author [Ha] gives a recursive Murnaghan–Nakayama formuχλ

n (dµ,n), and in particular this gives a recursive formula forχ∅n (dµ,n). However, the closed

formula in Theorem 9 is new.

3. A “second orthogonality relation” for characters of Pn(x)

Fora, b ∈ Pn(x) define the bitrace,

btrn(a, b) =∑

π∈Ωn

aπb|π, (7)

whereaπb|π denotes the coefficient of the basis elementπ when aπb is expanded interms of the basisΩn. If a ∈ Pn(x) let La andRa denote the linear transformations


,

le

of

iblecond

ts

ains

of

re,

Pn(x) induced by the action ofa on Pn(x) by left multiplication and right multiplicationrespectively. Ifa, b ∈ Pn(x), thenLa andRb commute and

btrn(a, b) = Tr(LaRb).

The vector spacePn(x) becomes a module forPn(x) ⊗ Pn(x) by using left multiplicationin the first component and right multiplication in the second component. By doubcentralizer theory (see, for example, [CR, Section 3D]), we have

Pn(x) ∼=⊕λ∈Λn

Mλn ⊗ Mλ

n ,

as aPn(x) ⊗ Pn(x)-module, whereMλn is the irreducible leftPn(x)-module labeled byλ

andMλn is the irreducible rightPn(x)-module labeled byλ. Taking traces on both sides

this identity gives

btrn(a, b) =∑λ∈Λn

χλn (a)χλ

n (b). (8)

This formula is aPn(x)-analog of the second orthogonality relation for the irreduccharacters of a finite group. See [Ra, Appendix] for a discussion of how the seorthogonality relation makes sense forany split semisimple algebra.

Since Pn(x) characters are completely determined by their values on the elemendµ,n | µ ∈ Λn and since the bitrace is a trace in each component, we define

btrn(µ, ν) = btrn(dµ,n, dν,n), for µ,ν ∈ Λn, and

χ∅n (µ) = χ∅

n (dµ,n), for µ ∈ Λn.

Recall thatSn ⊆ Pn(x) is the set of partition diagrams for which each edge contexactly one vertex from each row. Ifπ1,π2 ∈ Ωn, then we say that

π1 ∼ π2 ⇔ π1 = σ−1π2σ for someσ ∈ Sn.

Note thatσ−1πσ is gotten by simultaneously permuting both the top and bottom rowsπ

by σ . Furthermore, ifπ1 ∼ π2 thenχ(π1) = χ(π2) for any characterχ of Pn(x).Forπ ∈ Ωn, define flip(π) ∈ Ωn to be the diagram given by flipping the diagram forπ

over the horizontal axis through its middle, i.e.,

flip−→ .

If σ ∈ Sn thenσ ∼ σ−1, since bothσ andσ−1 have the same cycle type. Furthermoflip(σ ) = σ−1, so flip(σ ) ∼ σ , and this extends to, flip(dµ,n) ∼ dµ,n, for all µ ∈ Λn. If


n

k + = n andµ = (µ1, . . . ,µs) andν = (ν1, . . . , νt ) are compositions with 0 |µ| k

and 0 |ν| , then we can conjugate by the symmetric group to get,

dµ,k ⊗ flip(dν,) ∼ dµ∪ν,n, (9)

whereµ ∪ ν = (µ1, . . . ,µs, ν1, . . . , νt ). For example,d(2,2),5 ⊗ flip(d(3,2),7) ∼ d(2,2,3,2),12as we see,

.

Now we define a bijectionΨ : Ωn → Ω∅2n. Givenπ ∈ Ωn let Ψ (π) ∈ Ω∅

2n be the uniquediagram on a single row of 2n vertices such that:

for all a, b ∈ 1, . . . ,2n, aπ←→ b if and only if a

Ψ (π)←→ b.

For example,

Ψ−→ .

Under this bijection, we have, for allπ ∈ Ωn,

Ψ (dµ,nπdν,n) = (dµ,n ⊗ flip(dν,n)

)Ψ (π). (10)

For example,

Ψ−→ .

Theorem 12. Letµ = (µ1, . . . ,µs) andν = (ν1, . . . , νt ) be compositions with0 |µ| n

and0 |µ| n. Then

btrn(µ, ν) = χ∅2n(µ ∪ ν),

whereµ ∪ ν = (µ1, . . . ,µs, ν1, . . . , νt ) and χ∅2n(µ ∪ ν) is computed by the formula i

Theorem9.

Proof. From (10), we see that under the bijectionΨ we get the identity

dµ,nπdν,n|π = (dµ,n ⊗ flip(dν,n)

)Ψ (π)|Ψ (π),


.y

tric

f

and so from the definition (7) of btrn and the definition (6) ofχ∅2n, we have btrn(µ, ν) =

χ∅2n(dµ,n ⊗ flip(dν,n)). From (9), dµ,n ⊗ flip(dν,n) ∼ dµ∪ν,2n, and so χ∅

2n(dµ,n ⊗flip(dν,n)) = χ∅

2n(dµ∪ν,2n). Comparing Eq. (8) and Theorem 12, we have

Corollary 13 (“Second orthogonality of characters” forPn(x)). For compositionsµ andν

with 0 |µ| n and0 |ν| n, we have∑λ∈Λn

χλn (µ)χλ

n (ν) = χ∅2n(µ ∪ ν),

which can be computed using Theorem9.

The left regular representation ofPn(x) is given by the mapa → La described aboveThe trace Trn(dµ,n) of dµ,n on the regular representation ofPn(x) can be computed bbtr(µ, (1n)). That is, we specializedν,n = d(1n,n) = idn in Theorem 9. Therefore,

Corollary 14. For a compositionµ with 0 |µ| n, the trace ofdµ,n in the regularrepresentation ofPn(x) is given by

Trn(dµ,n) = χ∅2n

(µ ∪ (1n)

)which can be computed using Theorem9.

4. Identities in the character ring of Sr

4.1. Schur–Weyl duality betweenSr andPn(r)

Let V = Cr with basisv1, . . . , vr be the permutation representation of the symme

groupSr so thatσvi = vσ(i), for all σ ∈ Sr . Then-fold tensor product representationV ⊗n

has a basis consisting of simple tensorsvi1 ⊗ · · · ⊗ vin , with 1 ij n, and the action oσ ∈ Sr on a simple tensor is

σ(vi1 ⊗ · · · ⊗ vin) = vσ(i1) ⊗ · · · ⊗ vσ(in).

There is an action ofPn(r) onV ⊗n that commutes withSr . Forπ ∈ Ωn, define

δ(π)i1,...,inin+1,...,i2n

=

1, if ik = i wheneverkπ←→ ,

0, otherwise.

Thenπ acts on simple tensors as follows:

π(vi1 ⊗ · · · ⊗ vin ) =∑

δ(π)i1,...,inin+1,...,i2n

vin+1 ⊗ · · · ⊗ vi2n.

in+1,...,i2n


er

each

ns

ch

Jones [Jo] first showed that this map provides a representation ofPn(r) on V ⊗n.Furthermore, this action commutes with theSr -action and generates the centralizEndSr (V

⊗n). Whenr 2n this representation is faithful, andPn(r) ∼= EndSr (V⊗n).

Let r 2n, and for a partitionλ, defineλ∗ andλ as follows:

if λ = (λ1, . . . , λ) r, then λ∗ = (λ2, . . . , λ) (r − λ1),

if λ = (λ1, . . . , λ) k n, then λ = (r − k,λ1, . . . , λ) r.

Sincer 2n we are guaranteed thatλ is a partition and that 0 |λ∗| n. The sets

Γ nr =

λ r∣∣ |λ∗| n

and Λn = λ k | 0 k n

are in bijection with one another using the following maps, which are inverses ofother,

Γ nr → Λn,

λ → λ∗,and

Λn → Γ nr ,

λ → λ.(11)

Via these bijections, we can use eitherΓ nr or Λn to index the irreducible representatio

and standard elements ofPn(r).We letSλ denote the irreducible representation of the symmetric groupSr indexed by

the partitionλ r. The representationsSλ, λ ∈ Γ nr , are the irreducible summands whi

appear inV ⊗n. Using double centralizer theory, Jones [Jo] proves that, whenr 2n, thedecomposition ofV ⊗n as a bimodule forSr × Pn(r) is

V ⊗n =⊕λ∈Γ n

r

Sλ ⊗ Mλ, (12)

whereMλ is the irreduciblePn(r)-module corresponding toλ.

4.2. Inner products of class functions

Let R(Sr ) denote theC-vector space generated by the class functions onSr . That is,R(Sr ) = f :Sr → C | f (σ) = f (τ) if σ ∼ τ , whereσ ∼ τ denotes thatσ and τ areconjugate inSr . Define an inner product onR(Sr ) by

〈f,g〉 = 1

|Sr |∑σ∈Sr

f (σ )g(σ ), (13)

where denotes complex conjugation. The irreducible charactersψλSn

, λ r, form anorthonormal basis ofR(Sr) with respect to〈 , 〉, so if λ, τ r, then⟨

ψλSr

,ψτSr

⟩ = δλ,τ , (14)

whereδλ,τ is the Kronecker delta.


ceave

for a

Fork 0, define the power-sum symmetric functionpk in the variablesx1, . . . , xr by

pk(x1, . . . , xr ) = xk1 + · · · + xk

r ,

and for a compositionµ = (µ1,µ2, . . . ,µ) define pµ = pµ1pµ2 · · ·pµ. Finally, forσ ∈ Sr , define

pµ(σ) = pµ(ξ1, . . . , ξr ),

where ξ1, . . . , ξr are the eigenvalues ofσ in its permutation representation. Sinthe eigenvalues of permutations inSr are constant over conjugacy classes, we hpµ ∈ R(Sr ). If the eigenvalues ofσ areξ1, . . . , ξr then the eigenvalues ofσk areξk

1 , . . . , ξkr ,

and so

pk(σ ) = Tr(σk

) = (the number of fixed points ofσk

), (15)

where Tr(σ k) is the trace ofσk in the permutation representationV .By taking traces on both sides of (12), the second author [Ha] proves that

compositionµ, with 0 |µ| n, andσ ∈ Sr with r 2n, we have

rn−|µ|pµ(σ) =∑λ∈Γ n

r

χλn (µ)ψλ

Sr(σ ), (16)

or, equivalently,rn−|µ|〈pµ,ψλSr

〉 = χλn (µ).

Proposition 15. If µ andν are compositions with0 |µ| n and0 |ν| n then usingthe inner product inSr with r 2n, we have⟨

rn−|µ|pµ, rn−|ν|pν

⟩ = χ∅n (µ ∪ ν)

which can be explicitly computed by the formula in Theorem9.

Proof. Using equations (14) and (16), we have

⟨xn−|µ|pµ,xn−|ν|pν

⟩ =⟨ ∑λ∈Γ n

r

χλn (µ)ψλ

Sr,

∑τ∈Γ n

r

χτn (ν)ψτ

Sr

⟩=

∑λ∈Γ n

r

∑τ∈Γ n

r

χλn (µ)χτ

n (ν)⟨ψλ

Sr,ψτ

Sr

⟩︸︷︷︸δλ,τ

=∑λ∈Γ n

r

χλn (µ)χλ

n (ν) = χ∅2n(µ ∪ ν),

where the last equality comes from Corollary 13 and the bijection in (11).


g

r

oups,lity

iablesd

4.3. Fixed points of powers of permutations

Let µ n with µ = (1a1,2a2, . . . , a) where againai is the number ofµi equal toi.Then forσ ∈ Sr , we see from (15) that

pµ(σ) =∏

k=1

(Tr

(σk

))ak . (17)

Theorem 16. Let µ = (1a1,2a2, . . . , a) andn = |µ|. If σ is uniformly distributed inSr ,with r n, then we have the following expected value,

E

∏

k=1

(Tr

(σk

))ak

= χ∅

n (µ),

which can be explicitly computed using the closed-formula in Theorem9.

Proof. Let µ = (1a1,2a2, . . . , a) and letn = |µ|. Recall thatψ(r)Sr

is the trivialSr -char-

acter (i.e.,ψ(r)Sr

(σ ) = 1 for all σ ∈ Sr ). Then

E

∏

k=1

(Tr

(σk

))ak

= E

pµ(σ)

(by (17))

= 1

r!∑σ∈Sr

pµ(σ) = 1

r!∑σ∈Sr

pµ(σ)ψ(r)Sr

(σ ) = ⟨pµ,ψ

(r)Sr

⟩= χ∅

n (µ) (by (16)). Let σ be uniformly distributed inSr . Using Corollary 10, we get the followin

presumably well-known specializations of Theorem 16:

ETr(σ )n

= B(n), σ ∈ Sr , r 2n, (18)

ETr

(σa

)Tr

(σb

) = d(a)d(b) +∑t |a,b

t, σ ∈ Sr , r 2(a + b), (19)

ETr

(σa

) = d(a), σ ∈ Sr , r 2a, (20)

whereB(n) is thenth Bell number,d(a) is the number of divisors ofa, andt ranges oveall common divisors ofa andb.

In the study of the distribution of eigenvalues of random matrices in classical grDiaconis and Shahshahani [DS] and Diaconis and Evans [DE] use the second orthogonarelation for the symmetric group to compute the joint moments of the random varTr(Mj ) for M a random matrix in the unitary groupUr , and they use the seconorthogonality relation of Ram [Ra] for the Braueralgebra to determinethe joint moments


the

onre).

ts

stions

.37)

rs,

th.

62.

of Tr(Mj ) for M in the orthogonal groupOr andM in the symplectic groupSpr . Thesecomputations use the Schur–Weyl duality betweenUr andSn, Or , andBn(r), andSpr andBn(−r), respectively. Our results in Theorem 16 are the analogs of these results forduality betweenSr andPn(r).

Suppose thatσ = α1α2 · · ·αt is a decomposition ofσ into disjoint cyclesαi such thatαi

is adi cycle. Thenσk = αk1αk

2 · · ·αkt , andαk

i hasdi fixed points ifdi |k and 0 fixed pointsotherwise. Thus

Tr(σk

) =∑d |k

dcd(σ ), (21)

where the sum is over all divisorsd of k and whereσ hascd(σ ) cycles of lengthd .For i fixed andr large, it is known that theci have an approximate Poisson distributi

with independent parameter 1/i (see for example [DS] or [AT] and the references theThus

ETr

(σk

) −→ E

∑d |k

dxd

asr → ∞, (22)

where thexd are independent Poisson variables with parameter 1/d . Thus, whenµ =(1a1, . . . , nan), our formula forχ∅

|µ|(µ) gives a closed formula for the joint mixed momenof (∑

d1|1d1xd1

)a1

,

(∑d2|2

d2xd2

)a2

, . . . ,

(∑dn|n

dnxdn

)an

, (23)

wherex1, . . . , xn are independent Poisson variables with parameters 1, 12, 1

3, . . . , 1n, respec-

tively.

Acknowledgments

We thank Arun Ram and Persi Diaconis for their generous help in giving suggeand comments on our work.

References

[AT] R. Arritia, S. Tavare, The cycle structure of random permutations, Ann. Probab. 20 (3) (1992) 1567–1591[Br] R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. Math. 38 (19

854–872.[CR] C. Curtis, I. Reiner, Methods of Representation Theory: With Applications to Finite Groups and Orde

Vol. I, Wiley, New York, 1981.[DE] P. Diaconis, S. Evans, Linear functionals of eigenvalues of random matrices, Trans. Amer. Ma

Soc. 353 (7) (2001) 2615–2633.[DS] P. Diaconis, M. Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. A 31 (1994) 49–


niguchi267.-

an

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(6)

[DW] W. Doran, D. Wales, The partition algebra revisited, J. Algebra 231 (1) (2000) 265–330.[Ha] T. Halverson, Characters of the partition algebras, J. Algebra 238 (2001) 502–533.[Jo] V.F.R. Jones, The Potts model and the symmetric group, in: Subfactors: Proceedings of the Ta

Symposium on Operator Algebras, Kyuzeso, 1993, World Scientific, River Edge, NJ, 1994, pp. 259–[Mar1] P. Martin, Temperley–Lieb algebras for nonplanarstatistical mechanics—the partition algebra construc

tion, J. Knot Theory Ramifications 3 (1994) 51–82.[Mar2] P. Martin, The structure of the partition algebras, J. Algebra 183 (2) (1996) 319–358.[MS] P. Martin, H. Saleur, Algebras in higher dimensional statistical mechanics—the exceptional partition (me

field) algebras, Lett. Math. Phys. 30 (1994) 179–185.[Mac] I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford Univ. Press, New Yor

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Character orthogonality for the partition algebra and fixed points of permutations