Gaussian fluctuations of representations of wreath products

0.18 MB·English

by  Piotr Sniady

GAUSSIAN FLUCTUATIONS OF REPRESENTATIONS
OF WREATH PRODUCTS
7
0
0 PIOTRS´NIADY
2

n ABSTRACT. We studytheasymptoticsofthereduciblerepresentations
a ofthewreathproductsG S =Gq⋊S forlargeq,whereGisafixedfi-
J ≀ q q
nitegroupandSqisthesymmetricgroupinqelements;inparticularfor
3
G=Z/2Zwerecoverthehyperoctahedralgroups.Wedecomposesuch

] areduciblerepresentationofG Sq asasumofirreduciblecomponents
T (or,equivalently,asacollection≀oftuplesofYoungdiagrams)andweask
R whatisthecharacterofarandomlychosencomponent(or,whatarethe
. shapesofYoungdiagramsinarandomlychosentuple).Ourmainresult
h
t is thatfor a largeclass of representationsthe fluctuationsof characters
a
(andfluctuationsoftheshapeoftheYoungdiagrams)areasymptotically
m
Gaussian. Theconsideredclassconsistsoftherepresentationsforwhich
[
the characters asymptotically almost factorize and it includes, among

2 others,theleftregularrepresentationthereforeweprovetheanalogueof
v Kerov’scentrallimittheoremforwreathproducts.
9
8
0
1
0
6
0 1. INTRODUCTION
/
h 1.1. Formulation of the problem: asymptotics of representations. One
t
a of the classical problems in the theory of asymptotic combinatorics con-
m
cerns the limit behavior of representations of large groups. The above for-
:
v mulation of the problem is very general and vague, so let us be more spe-
Xi cific. An example of a question which fits into the above category is the
following one: sequences (G ) and (ρ ) are given, where for each q N
r q q
a ∈
we know that G is a finite group and ρ is a finite-dimensional represen-
q q
tationofG ;wedecomposeρ asasumofirreduciblecomponentsandwe
q q
ask about asymptotic properties of a random summand. Of course, if the
irreducible representations of the groups G do not have a nice common
q
structure then it is not clear which asymptoticproperties of a random sum-
mand could be studied and for this reason we should restrict our attention
to ‘nice’ groups G for which the irreducible representations can be nicely
q
describedin auniformway.
In this article we are concerned with the case when G = G S =
q q
Gq ⋊ S is the wreath product of a fixed finite group G by a sym≀metric
q
group S . However, before we come to this topic in Section 1.3 we will
q
1
2 PIOTRS´NIADY
have a closer look (in Section 1.2 below) on a more developed case of the
symmetricgroups.
1.2. Asymptoticsofrepresentations ofthesymmetricgroups. Thesim-
plestexampleofsuch‘nice’groupsisthesequenceofthesymmetricgroups
S —in this case the irreducible representations of S are enumerated by
q q
the Young diagrams with q boxes. Our problem therefore asks about the
statistical properties of some random Young diagrams as their size tends
to infinity. This problem was studied in much detail: Logan and Shepp
[LS77] and Vershik and Kerov [VK77] studied the situation when ρ is
q
the left-regular representation (which corresponds to, so called, Plancherel
measure on the irreducible representations of S ) and they proved that the
q
corresponding Young diagrams concentrate (after some simple geometric
rescaling) around some limit shape. Biane [Bia98, Bia01] generalized this
resultandheprovedtheconcentrationoftheshapesforaverylargeclassof
representations of S ; he also found a connection between the asymptotic
q
theoryoftherepresentationsofthesymmetricgroupsandVoiculescu’sfree
probabilitytheory[VDN92].
Theaboveresultscan beviewedasan analogueofthelawoflargenum-
bers, therefore it was very tempting to check if some kind of central limit
theorem could be true. The first result of this kind was found by Kerov
[Ker93b] (see also Ivanov and Olshanski [IO02]) who proved that for the
left-regular representations the fluctuations of a random Young diagram
around the limit shape are asymptotically Gaussian. Hora [Hor02, Hor03]
found the non-commutative aspect of this result. In a previous paper
[S´ni06b] we proved that an analogue of the Kerov’s central limit theorem
holds true for a large class of representations ρ with, so called, approxi-
q
matefactorizationof characters.
1.3. The main result: asymptotics of representations of the wreath
products. It seems that our understanding of the asymptotic properties of
thesymmetricgroupsis reaching alevelofmaturityand for thisreason the
timehascometohavealookonsomeotherclassicalseries offinitegroups
and in this article we will concentrate on wreath products G S , where G
q
≀
isafixed finitegroup.
Since G S = Gq⋊S is equal to a semidirectproduct of a symmetric
q q
≀
group S therefore the structures of the groups G S and S and of their
q q q
≀
irreducible representations are very closely related to each other; for this
reason our strategy in this article is to provide a setup in which the known
results and methods concerning S [S´ni06b] could be directly applied to
q
G S . Wewilldoit inSection 2.
q
≀
GAUSSIANFLUCTUATIONSOFREPRESENTATIONSOFWREATHPRODUCTS 3
Section 3 contains the main result, namely that the methods which we
usedforthestudyofthesymmetricgroups[S´ni06b]can beextendedtothe
wreath products. To be more explicit: we will introduce a class of repre-
sentationswithapproximatefactorizationof characters;thisclass willturn
outtobebigenoughtocontainalotofnaturalexamplesandwillbeclosed
undersomenaturaloperationsonrepresentationssuchasinduction,restric-
tion, outer and tensor product. We will prove that for the representations
with approximate factorization of characters an analogue of the Kerov’s
theorem [Ker93b] holds true, in otherwords thefluctuations of a randomly
chosenirreduciblecomponentare asymptoticallyGaussian.
2. PRELIMINARIES
2.1. Partial permutations and disjoint products. Ivanov and Kerov
[IK99] defined a partial permutationas a pairα = (π,A),where A (called
support of α) is any subset of {1,...,q} and π : {1,...,q} {1,...,q} is
a bijection which is equal to identity outside of A. The natural product of
partialpermutationsis givenby
→
(π ,A )(π ,A ) = (π π ,A A ).
1 1 2 2 1 2 1 2
∪
PartialpermutationsformasemigroupPS ;inthisarticleweareinterested
q
also in the corresponding semigroup algebra C(PS ) which should be re-
q
garded as an analogue of the permutation group algebra C(S ) equipped
q
withsomeadditionalstructure.
The algebra C(PS ) can be also equipped with a different product ,
q
•
calleddisjointproduct,givenongenerators by
(π π ,A A ) ifA A = ,
(π ,A ) (π ,A ) = 1 2 1∪ 2 1∩ 2 ∅
1 1 2 2
• 0 otherwise.
(cid:14)
2.2. Conjugacy classes for symmetric groups. Let integer numbers
k ,...,k 1 be given. We define the normalized conjugacy class
1 m
indicator to ≥be a central element in the group algebra C(S ) given by
q
[KO94, Bia03, S´ni06a,S´ni06b]
(2.1) Σ = (a ,a ,...,a ) (a ,a ,...,a ),
k1,...,km 1,1 1,2 1,k1 ··· m,1 m,2 m,km
a
X
wherethesumrunsoverallone–to–onefunctions
a : {r,s} : 1 r m,1 s k {1,...,q}
r
≤ ≤ ≤ ≤
and (a1,1,a1,2,(cid:8)...,a1,k1)···(am,1,am,2,...,a(cid:9)m,km) denotes the product
ofdisjointcycles. Ofcourse,ifq < k + +k →thentheabovesumruns
1 m
···
overtheemptyset and Σ = 0.
k1,...,km
4 PIOTRS´NIADY
In otherwords, if k k we considera Young diagram with the
1 m
≥ ··· ≥
rows of the lengths k ,...,k and all ways of filling it with the elements
1 m
oftheset {1,...,q}in such away that noelement appears morethanonce.
Each such a filling can be interpreted as a permutation when we treat the
rows of the Young tableau as disjoint cycles. It follows that Σ is a
k1,...,km
linear combinationof permutations which in the cycle decompositionhave
cycles of length k ,...,k (and, additionally, q − (k + + k ) fix-
1 m 1 m
···
points). Each such a permutation appears in Σ with some positive
k1,...,km
integermultiplicitydependingonthesymmetryofthetuplek ,...,k .
1 m
It is natural to regard Σ as an element of C(PS ), where we treat
k1,...,km q
each summandon theright-hand-sideof(2.1)as apartial permutationwith
thesupportequal totheset{a }.
r,s
2.3. Algebra ofconjugacy classes. Thesequenceofthealgebras C(PS )
q
forms an inverse system with the maps C(PS ) C(PS ) equal to re-
q+1 q
strictions. We consider a vector space spanned by the family of elements
lim Σ lim C(PS ). It is easy to check that both the natural
k1,...,km ∈ q →
product and the disjoint product of two such elements is again in this form
ther←efore theaboveve←ctorspace is an algebra bothwhen as theproduct we
takethenaturalproduct(wewilldenotethisalgebraby A)and whenas the
productwetakethedisjointproduct (wewilldenotethisalgebraby A•).
2.4. Wreath product and its representations. For a finite group G its
wreath product with the symmetric group G S is a semidirect product
q
Gq⋊S , whereS acts on Gqbypermutingth≀efactors.
q q
We denoteby G^ the set of the (equivalenceclasses of) irreducible repre-
sentationsofthefinitegroupG. Foranirreduciblerepresentationζ G^ we
denoteby p C(G) thecorrespondingminimalcentral projection∈.
ζ
∈
Let a Young diagram λ with |λ| boxes be given and let ρ : S
λ |λ|
End(W ) be the corresponding irreducible representation of the symmet-
λ
ric group. For an irreducible representation ζ : G End(V ) we consider
ζ →
the tensor power representation ζ⊗|λ| of the group G|λ| acting on the vector
space
→
(2.2) (V )⊗|λ|.
ζ
Pleasenoticethat(2.2)isalsoarepresentationofS ,wherethesymmetric
|λ|
groupsactsbypermutingthefactors;inthisway(2.2)isarepresentationof
thesemidirectproductG|λ|⋊S = G S .
|λ| q
≀
Also, W is a representation of G|λ|⋊ S , where S acts on this space
λ |λ| |λ|
byρ andG|λ|acts triviallyas identity.
λ
Thus,thetensorproduct
(2.3) W (V )⊗|λ|
λ ζ
⊗
GAUSSIANFLUCTUATIONSOFREPRESENTATIONSOFWREATHPRODUCTS 5
is also a representation of G S . It will be the basic building block in the
q
≀
constructionofallirreduciblerepresentationsofG S . Thefollowingwell
q
≀
known result gives a complete classification of the irreducible representa-
tionsofthewreath products.
Proposition 2.1. Let Λ : G^ Y be a function on the set of irreducible
representations of G valued in the set of the Young diagrams Y such that
|Λ(ζ)| = q.
ζ∈G^ →
The representation
P
(2.4) ρ = W V⊗|Λ(ζ)| Gq⋊Sq
Λ (cid:16)Oζ∈G^ Λ(ζ)⊗ ζ (cid:17)x ζ∈G^(G|Λ(ζ)|⋊S|Λ(ζ)|)
Q
is irreducible; furthermore every irreducible representation of Gq⋊ S is
q
ofthisform.
Example 2.2. We consider the case when G = Z/2Z = {0,1}. Then
\
therearetwoirreduciblerepresentationsofZ/2Z,namelyZ/2Z = {ζ ,ζ },
1 2
where ζ is the trivial representation and ζ is the“alternating” representa-
1 2
tion
ζ (0) = 1, ζ (1) = 1,
1 1
ζ (0) = 1, ζ (1) = −1.
2 2
Thespaces V , V onwhich theserepresentationsact areone-dimensional.
1 2
In the case when ζ = ζ the representation (2.3) of (Z/2Z) S takes a
i |λ|
≀
simplerform
W (V )⊗|λ| =∼ W ;
λ i λ
⊗
onwhichthegroupS actsbyρ andtheelements(z ,...,z ) (Z/2Z)|λ|
|λ| λ 1 λ
∈
acteithertriviallyasidentity(ifζ = ζ )orbymultiplyingby(−1)z1+···+z|λ|
1
(ifζ = ζ ).
2
\
The function Λ : Z/2Z Y can be identified with a pair of Young
diagramsλ ,λ ,whereλ = Λ(ζ ). Therepresentation(2.4)canbewritten
1 2 i i
moreexplicitlyas
→
(Z/2Z)q⋊Sq
ρ = W W .
λ1,λ2 (cid:16) λ1 ⊗ λ2(cid:17)x((Z/2Z)|λ1|⋊S|λ1|)×((Z/2Z)|λ2|⋊S|λ2|)
2.5. Normalizedtrace. Foramatrixx Mn(C)wedefineitsnormalized
∈
trace
1
trx = Trx,
n
whereTrx denotestheusualtrace ofx. In thisway thenormalizedtraceof
theidentitymatrixfulfillstr1 = 1.
6 PIOTRS´NIADY
2.6. The mainhomomorphism. ThefollowingLemmaisofgreat impor-
tance for this article: it provides a homomorphismfrom the tensor product
A⊗|G^| to C(G S ); in this way every representation of the wreath product
q
≀
G S definesarepresentationofthetensorproductA⊗|G^|andthequestions
q
≀
concerning the asymptotic behavior of the wreath products are reduced to
thecorrespondingquestionsconcerning thesymmetricgroups.
Foranirreduciblerepresentationζ G^,thecorrespondingminimalcen-
tralprojectionp C(G)and apartia∈l permutation(π,A) PS weset
ζ q
∈ ∈
(2.5) φ (π,A) = (r r )π C(Gq⋊S ),
ζ 1 q q
×···× ∈
where
p ifm A,
r = ζ ∈
m
1 otherwise.
(cid:14)
Lemma 2.3.
(1) The map φ extends to a homomorphism of algebras φ :
ζ ζ
C(PS ) C(Gq ⋊ S ), where we consider PS equipped with
q q q
thenaturalproduct;
(2) Let us fix for a moment some numbering of the irreducible rep-
→
resentations G^ = {ζ ,...,ζ }. The map φ : C(PS ) ⊗|G^|
1 |G^| q
C(Gq⋊S ) givenon simpletensorsby
q
(cid:0) (cid:1)
→
φ (π ,A ) (π ,A ) = φ (π ,A )
1 1 ⊗···⊗ |G^| |G^| ζj j j
(cid:0) (cid:1) 1≤Yj≤|G^|
isa homomorphismofalgebras.
(3) Let ρ be an irreducible representation of G S , as presented in
Λ q
≀
Proposition2.1. Foranya ,...,a A
1 |G^| ∈
(2.6) trρ φ(a a ) = trρ (a ),
Λ 1⊗···⊗ |G^| Λ(ζj) j
(cid:0) (cid:1) 1≤Yj≤|G^|
whereρ istheirreduciblerepresentationof S correspond-
Λ(ζj) |Λ(ζj)|
ingtotheYoungdiagramΛ(ζ ).
j
Proof. If a permutation π leaves a set A invariant then the elements
r r and π C(Gq ⋊ S ) contributing to (2.5) commute. Let
1 q q
× ··· × ∈
(π(1),A(1)),(π(2),A(2)) PS and (π(3),A(3)) := (π(1),A(1))(π(2),A(2));
q
∈
GAUSSIANFLUCTUATIONSOFREPRESENTATIONSOFWREATHPRODUCTS 7
itfollowsthat
φ (π(1),A(1))φ (π(2),A(2)) =
ζ ζ
π(1)(r(1) r(1))(r(2) r(2))π(2) =
1 ×···× q 1 ×···× q
π(1)(r(3) r(3))π(2) = π(1)π(2)(r(3) r(3)) =
1 ×···× q 1 ×···× q
φ (π(3),A(3))
ζ
whichfinishestheproofofpoint(1).
Inordertoprovepoint(2)itisenoughtoprovethatifζ(1) = ζ(2)thenthe
6
elements φ (π(1),A(1)) and φ (π(2),A(2)) commute. This holds true
ζ(1) ζ(2)
because when the sets A(1),A(2) are disjoint then permutations π(1),π(2)
commute;when thesets A(1),A(2)arenotdisjointthen
φ (π(1),A(1))φ (π(2),A(2)) = 0 = φ (π(2),A(2))φ (π(1),A(1)).
ζ(1) ζ(2) ζ(2) ζ(1)
Since a ,...,a A therefore φ(a a ) C(Gq⋊ S ) is
1 |G^| ∈ 1 ⊗ ···⊗ |G^| ∈ q
central and Frobenius reciprocity can be applied to calculate the characters
of the induced representation (2.4). This shows that the left-hand side of
(2.6)is equalto thecharacteroftherepresentation
W V⊗|Λ(ζj)|
Λ(ζj)⊗ ζj
1≤Oj≤|G^|
ofthegroup
G := G|Λ(ζj)|⋊S
Λ |Λ(ζj)|
1≤Yj≤|G^|(cid:0) (cid:1)
evaluated on the element φ(a a ) . Notice that the element
1⊗ ···⊗ |G^| GΛ
oftheform

φ (π ,A ) (π ,yA )
1 1 ⊗···⊗ |G^| |G^| GΛ
representson thisspa(cid:0)ceas (cid:1)
y
Λ(ζ ) (π ) 1⊗|Λ(ζj)| ifA is asubsetoftheset ofthe
1≤j≤|G^| j j ⊗ j
elementspermutedby subgroup
N (cid:2)(cid:0) (cid:1) (cid:3)
 S|Λ(ζj)|ofGΛforeach j;
0 otherwise.


Itfollowsthattheelementφ(a1⊗···⊗a|G^|) GΛ representsas
Λ(ζj) (aj) 1y⊗|Λ(ζj)|
⊗
1≤Oj≤|G^|(cid:2)(cid:0) (cid:1) (cid:3)
whichfinishestheproofofpoint(3). (cid:3)
8 PIOTRS´NIADY
Wedefine thenatural producton A⊗|G^|by
(2.7) (a a )(b b ) = a b a b ;
1⊗···⊗ |G^| 1⊗···⊗ |G^| 1 1⊗···⊗ |G^| |G^|
andthedisjointproduct by
•
(2.8) (a a ) (b b ) = (a b ) (a b ).
1⊗···⊗ |G^| • 1⊗···⊗ |G^| 1• 1 ⊗···⊗ |G^|• |G^|
ThusthemapφdescribedintheaboveLemmadefinesahomomorphism
ofalgebrasφ : A⊗|G^| C(Gq⋊S ),whereastheproductinA⊗|G^|wetake
q
thenatural product.
Weconsiderthemapφ˜ : A A⊗|G^|givenby
→ ζi
φ˜ (x) = 1 1 x 1 1;
ζi ⊗···⊗ ⊗ ⊗ ⊗···⊗
→
i−1times |G^|−itimes
in this way φ = φ φ˜ |. Fr{ozm th}e follow|ing{ozn w}e will not use any
ζi ◦ ζi
explicitnumberingoftherepresentationsinG^.
2.7. Elements of the group algebra as random variables. Let us fix
somefinite-dimensionalrepresentationρ ofthewreathproductG S . We
q q
cantreatanycommutingfamilyofelementsofthegroupalgebraC≀(G S )
q
≀
as afamilyofrandom variablesequippedwiththemean valuegivenby the
normalizedcharacter:
(2.9) EX := χ (X) = trρ (X).
ρq q
Itshouldbestressedthatinthegeneral casewetreatelementsofC(G S )
q
≀
as random variables only on a purely formal level; in particular we do not
treat themas functionson someKolmogorovprobabilityspace.
Lemma 2.3 allows us to consider a representation ρ φ of A⊗|G^|; for
q
◦
simplicitywewilldenotethisrepresentationbythesamesymbolρ . Inthis
q
way as random variables we may take the elements of the algebra A⊗|G^|.
This algebra can be equipped either with the natural product (2.7) and we
denotetheresultingcumulants(callednaturalcumulants)byk(X ,...,X )
1 n
or with the disjoint product (2.8) and we denote the resulting cumulants
•
(called disjointcumulants)by k•(X ,...,X ).
1 n
2.8. CanonicalprobabilitymeasureonYoungdiagramsassociatedtoa
representation. There is a special case when it is possible to give a truly
probabilisticinterpretationtothemeanvalue(2.9): itiswhenforthefamily
of random variables we take the center of C(G S ) (respectively, the al-
q
≀
gebra A⊗|G^| equipped with the natural product). The center of C(G S ) is
q
≀
isomorphic(viaFouriertransform)tothealgebraoffunctionsonirreducible
representationsofG S andtheexpectedvalue(2.9)correspondsunderthis
q
≀
isomorphism to the probability measure on irreducible representations (or,
equivalently,onfunctionsΛ : G^ YasdescribedinProposition2.1)such
→
GAUSSIANFLUCTUATIONSOFREPRESENTATIONSOFWREATHPRODUCTS 9
FIGURE 2.1. Youngdiagramassociatedwithapartition8 =
4+3+1.
that the probability of a given irreducible representation is proportional to
thetotaldimensionofall irreduciblecomponentsofthistypeinρ .
q
2.9. GeneralizedYoungdiagrams. LetλbeaYoungdiagram. Weassign
to it a piecewise affine function ωλ : R R with slopes 1, such that
±
ωλ(x) = |x| for large |x| as it can be seen on the example from Figure 2.2.
BycomparingFigure2.1andFigure2.2onecaneasilyseethatthegraphof
→
ωλcanbeobtainedfromthegraphicalrepresentationoftheYoungdiagram
by an appropriate mirror image, rotation and scaling by the factor √2. We
call ωλthegeneralized Young diagram associatedwith theYoung diagram
λ[Ker93a, Ker98,Ker99].
The class of generalized Young diagrams consists of all functions ω :
R R which are Lipschitz with constant 1 and such that ω(x) = |x| for
large|x|andofcoursenoteverygeneralizedYoungdiagramcanbeobtained
bytheaboveconstructionfrom someYoungdiagramλ.
→
The setup of generalized Young diagrams is very useful in the study of
theasymptoticproperties since it allows us to define easily various notions
ofconvergence oftheYoungdiagram shapes.
2.10. Functionals of the shape of Young diagrams. The main result of
this article is that the fluctuations of the shape of some random Young dia-
gramsconverge(after somerescaling)toa Gaussiandistribution. Since the
spaceof(generalized)Youngdiagramsisinfinite-dimensionalthereforewe
need to be very cautious when dealing with such statements. In fact, we
will consider a family of functionals on Young diagrams and we show that
the joint distributionof each finite set of these functionals converges to the
Gaussiandistribution.
The functionals mentioned above are given as follows: for a Young di-
agram λ and the corresponding generalized Young diagram ω we denote
σ(x) = ω(x)−|x| [Bia98,IO02]and considerthefamilyofmaps
2
(2.10) p˜ (λ) = xnσ′′(x)dx.
n
R
Z
10 PIOTRS´NIADY
3 2 1 0 1 2 3
− − −
FIGURE 2.2. Generalized Youngdiagramassociatedwitha
partition8 = 4+3+1.
Sinceσ′′ makessenseasadistributionandσiscompactlysupportedhence
thecollection p˜ (λ) determinestheYoungdiagramλ uniquely.
n n
2.11. Transiti(cid:0)on me(cid:1)asure of a Young diagram. To any generalized
Young diagram ω we can assign the unique probability measure µω on
R,called transitionmeasureofω, theCauchy transformofwhich
1
(2.11) G (z) = dµω(x)
µω
z−x
R
Z
isgivenby
1 1 1
(2.12) logG (z) = − log(z−x)ω′′(x)dx = − ω′(x)dx
µω
2 2 z−x
R R
Z Z
for every z / R. For a Young diagram λ we will write µλ as a short hand
∈
of µωλ. This definition may look artificial but it turns out [Ker93a, OV96,
Bia98, Oko00] that it is equivalent to natural representation-theoretic defi-
nitions which arise by studying the representation ρ together with the in-
q
clusionS S .
q q+1
⊂
Forp > 0andaYoungdiagramλweconsidertherescaled(generalized)
Youngdiagramωpλgivenbyωpλ : x pωλ x . Informallyspeaking,the
7 p
symbolpλcorresponds to the shapeof theYoung diagram λ geometrically
(cid:0) (cid:1)
scaledbyfactorp(inparticular,ifλhasqboxesthenpλhasp2qboxes). It
→
is easy to see that (2.12) implies that the corresponding transition measure
µpλis adilationofµλ:
(2.13) µpλ = D µλ.
p