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### Asymptotic of characters of symmetric groups and limit shape of Young diagrams

Valentin Féray

coworkers : Piotr Śniady (Wroclaw), Pierre-Loïc Méliot (Marne-La-Vallée)

Laboratoire Bordelais de Recherche en Informatique CNRS

Séminaire Lotharingien de Combinatoire (64), Lyon, France

Valentin Féray (LaBRI, CNRS) Characters ofSn 2010-03-29 1 / 16

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### Outline of the talk

1 Character values of symmetric groups An exact formula

Asymptotic behaviours

2 Application : limit shape of Young diagrams

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Character values of symmetric groups An exact formula

### Young symmetrizer

Let T be a ﬁlling of λ= (3,2,2).

2 3 6 4 1 7 5 Consider :

row-stabilizer RS(T) =S{2,3,6}×S{1,4}×S{5,7}. column-stabilizer CS(T) =S{2,4,7}×S{1,3,5}.

Valentin Féray (LaBRI, CNRS) Characters ofSn 2010-03-29 3 / 16

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Character values of symmetric groups An exact formula

### Young symmetrizer

Let T be a ﬁlling of λ= (3,2,2).

2 3 6 4 1 7 5

n!sλ

dimλ = X

σ1∈RS(T) σ2∈CS(T)

(−1)σ2ptype(σ2σ1)

Work in progress with P. Śniady : analog for zonal polynomials

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Character values of symmetric groups An exact formula

### An equivalent formulation

Recall : character value χλ(µ) fulﬁllssλ=P

µ χλ(µ)

zµ pµ. n!χλ(π)

dimλ = X

σ2σ1

(−1)σ2Nσ12(λ), where

Deﬁnition

Nσ12(λ) is the number of bijectionsf :{1, . . . ,n} ≃λsuch that for all i,f(i) and f(σ1(i)) (resp.f(σ2(i))) are in the same row(resp.

column).

Valentin Féray (LaBRI, CNRS) Characters ofSn 2010-03-29 4 / 16

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Character values of symmetric groups An exact formula

### Nice behaviour on short permutations

If π ∈Sk

֒→ι Sn, (π(i) =i ∀i >k),

thenNσ12(λ) =0 unlessσ1(i) =σ2(i) =π(i)∀ i >k. In this case the formula becomes :

n!χλ(ι(π))

dimλ = X

σ12Sk σ2σ1

(−1)σ2Nι(σ 1),ι(σ2)(λ)

But Nι(σ 1),ι(σ2)= #{f :{1, . . . ,k}֒→λwith usual conditions}·

(n−k)!

| {z }

choices of the places ofk+1,...,n

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Character values of symmetric groups An exact formula

### Nice behaviour on short permutations

Deﬁnition

Nσ12(λ) is the number of injections f :{1, . . . ,k}֒→λ such that, for all i,f(i) and f(σ1(i)) (resp.f(σ2(i))) are in the same row(resp.

column).

Σπ(λ) := n·(n−1). . .(n−k+1)χλ(ι(π))

dimλ = X

σ12Sk σ2σ1

(−1)σ2Nσ12(λ).

Valentin Féray (LaBRI, CNRS) Characters ofSn 2010-03-29 6 / 16

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Character values of symmetric groups An exact formula

### Forgetting injectivity

Deﬁnition

Nσ12(T) is the number of functions f :{1, . . . ,k}→λsuch that, for all i,f(i) and f(σ1(i)) (resp.f(σ2(i))) are in the same row(resp.

column).

Σπ(λ) := n·(n−1). . .(n−k+1)χλ(ι(π))

dimλ = X

σ12Sk σ2σ1

(−1)σ2Nσ12(λ).

Idea of proof : the total contribution of a non-injective function in rhs is easily seen to be 0.

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Character values of symmetric groups Asymptotic behaviours

### Asymptotics is easy to read on this formula

Model : ﬁx a permutation π0 ∈Sk and a partition λ0⊢k

Consider π=ι(π0) (i.e. we just add ﬁxpoints) and λ=c·λ0=λ multiplied by c (i.e. horizontal lengths are multiplied byc)

7→

Question : asymptotics of χdimλ(π)λ ?

Valentin Féray (LaBRI, CNRS) Characters ofSn 2010-03-29 7 / 16

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Character values of symmetric groups Asymptotic behaviours

### Asymptotics is easy to read on this formula

Model : ﬁx a permutation π0 ∈Sk and a partition λ0⊢k

Consider π=ι(π0) (i.e. we just add ﬁxpoints) and λ=c·λ0=λ multiplied by c (i.e. horizontal lengths are multiplied byc)

7→

Question : asymptotics of χdimλ(π)λ ?

Easy on the N’s :Nσ12(c·λ) =c|C2)|Nσ12(λ)

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Character values of symmetric groups Asymptotic behaviours

### Asymptotics is easy to read on this formula

Model : ﬁx a permutation π0 ∈Sk and a partition λ0⊢k

Consider π=ι(π0) (i.e. we just add ﬁxpoints) and λ=c·λ0=λ multiplied by c (i.e. horizontal lengths are multiplied byc)

7→

Question : asymptotics of χdimλ(π)λ ?

Easy on the N’s :Nσ12(c·λ) =c|C2)|Nσ12(λ) Dominant term of Σπ(λ) :

Nπ,Idk(λ) = Y

µitype(π)

X

j

λµji

Valentin Féray (LaBRI, CNRS) Characters ofSn 2010-03-29 7 / 16

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Character values of symmetric groups Asymptotic behaviours

### Asymptotics is easy to read on this formula

Model : ﬁx a permutation π0 ∈Sk and a partition λ0⊢k

Consider π=ι(π0) (i.e. we just add ﬁxpoints) and λ=c·λ0=λ multiplied by c (i.e. horizontal lengths are multiplied byc)

7→

Question : asymptotics of χdimλ(π)λ ?

Easy on the N’s :Nσ12(c·λ) =c|C2)|Nσ12(λ) Dominant term of Σπ(λ) :

Nπ,Idk(λ) = Y

X λµji

 = Y

pµi(λ)

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Character values of symmetric groups Asymptotic behaviours

### Asymptotics is easy to read on this formula

Model : ﬁx a permutation π0 ∈Sk and a partition λ0⊢k

Consider π=ι(π0) (i.e. we just add ﬁxpoints) and λ=c•λ0=λdilated by c (i.e. horizontal and verticallengths are multiplied byc)

7→

Question : asymptotics of χdimλ(π)λ ?

Valentin Féray (LaBRI, CNRS) Characters ofSn 2010-03-29 7 / 16

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Character values of symmetric groups Asymptotic behaviours

### Asymptotics is easy to read on this formula

Model : ﬁx a permutation π0 ∈Sk and a partition λ0⊢k

Consider π=ι(π0) (i.e. we just add ﬁxpoints) and λ=c•λ0=λdilated by c (i.e. horizontal and verticallengths are multiplied byc)

7→

Question : asymptotics of χdimλ(π)λ ?

Easy on the N’s :Nσ12(c•λ) =c|C1)|+|C(σ2)|Nσ12(λ)

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Character values of symmetric groups Asymptotic behaviours

### Asymptotics is easy to read on this formula

Model : ﬁx a permutation π0 ∈Sk and a partition λ0⊢k

Consider π=ι(π0) (i.e. we just add ﬁxpoints) and λ=c•λ0=λdilated by c (i.e. horizontal and verticallengths are multiplied byc)

7→

Question : asymptotics of χdimλ(π)λ ?

Easy on the N’s :Nσ12(c•λ) =c|C1)|+|C(σ2)|Nσ12(λ) Dominant term of Σπ(λ) :

X

σ1σ2

|C(σ1)|+|C(σ2)|maximal

±Nσ12(λ)

Valentin Féray (LaBRI, CNRS) Characters ofSn 2010-03-29 7 / 16

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Character values of symmetric groups Asymptotic behaviours

### Free cumulants

Σπ(c•λ) = X

σ1σ2

|C(σ1)|+|C(σ2)maximal

±Nσ12(λ)

But,

σ1σ2

|C(σ1)|+|C(σ2)|maximal

≃Q

NCµi ≃Q

Trees(µi).

With generating series, one can prove (Rattan, 2006) rhs =Y

Rµi+1(λ)

Rk : free cumulants deﬁned from the shapeωλ by Biane (1998).

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Character values of symmetric groups Asymptotic behaviours

### Remarks

Works in more general context than sequences c·λ0 and c •λ0 (in fact, works as soon as a sequence of Young diagram has alimit)

These results were already known (Vershik & Kerov 81, Biane 98), but : we provide uniﬁed approach of both cases ;

our bound for error terms are better.

Valentin Féray (LaBRI, CNRS) Characters ofSn 2010-03-29 9 / 16

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Application Limit shape of Young diagrams

### Description of the problem

Consider Plancherel’s probability measure on Young diagrams of size n P(λ) =(dimλ)2

n!

Question : is there a limit shape for (renormalized rotated) Young diagram taken randomly with Plancherel’s measure when n→ ∞?

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Application Limit shape of Young diagrams

### Normalized character values have simple expectations !

Fix π ∈Sn. Let us consider the random variable : Xπ(λ) =χλ(π) = tr ρλ(π)

dimVλ . Let us compute its expectation :

E(Xπ) = 1 n!

X

λn

(dimVλ)·tr ρλ(π)

= 1 n!trL

λ⊢nVλdimVλ(π) = 1

n!trC[Sn](π) Last expression is easy to evaluate :

E(Xπ) =δπ,Idn

Valentin Féray (LaBRI, CNRS) Characters ofSn 2010-03-29 11 / 16

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Application Limit shape of Young diagrams

### Convergence of cumulants

Recall : we proved that Q

iRki+1≈Σk1,...,kr. Thus

E(R2)≈n

E(Ri)≈0 if i >2 Var(Ri)≈0 if i ≥2

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Application Limit shape of Young diagrams

### Convergence of cumulants

Recall : we proved that Q

iRki+1≈Σk1,...,kr. Thus

limE(R2/n) =1 limE

Ri/√ ni

=0 if i >2 lim Var

Ri/√ ni

=0 if i ≥2

Easy to make it formal because Rk ∈Vect(Σπ).

⇒ Random variables Ri/√

ni converge in probability towards the sequence (0,1,0,0, . . .)

General lemma from Kerov :

convergence of cumulants⇒convergence of Young diagrams

Valentin Féray (LaBRI, CNRS) Characters ofSn 2010-03-29 12 / 16

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Application Limit shape of Young diagrams

### Existence of a limiting curve

Theorem (Logan and Shepp 77, Kerov and Vershik 77)

Let us take randomly (with Plancherel measure) a sequence of Young diagram λn of size n. Then, in probability, for the uniform convergence topology on continuous functions, one has :

r45(h1/nn))→δ, where Ωis an explicit function drawn here :

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Application Limit shape of Young diagrams

### Convergence of q-Plancherel measure

Case where expectation of character values are big : there can not be a limit shape after dilatation.

we use the ﬁrst approximation for charactersΣπ(λ)≈Q

pµi(λ).

Valentin Féray (LaBRI, CNRS) Characters ofSn 2010-03-29 14 / 16

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Application Limit shape of Young diagrams

### Convergence of q-Plancherel measure

Case where expectation of character values are big : there can not be a limit shape after dilatation.

we use the ﬁrst approximation for charactersΣπ(λ)≈Q

pµi(λ).

Example : q−Plancherel measure (q<1)

deﬁned using representation of Hecke algebras one can prove

Eqπ) = (1−q)|µ| Q

i1−qµi n(n−1). . .(n− |µ|+1) Thus

Eq(pk)≈ (1−q)k

Q 1−qknk Varq(pk)≈0

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Application Limit shape of Young diagrams

### Convergence of q-Plancherel measure

Case where expectation of character values are big : there can not be a limit shape after dilatation.

we use the ﬁrst approximation for charactersΣπ(λ)≈Q

pµi(λ).

Example : q−Plancherel measure (q<1)

deﬁned using representation of Hecke algebras one can prove

Eqπ) = (1−q)|µ| Q

i1−qµi n(n−1). . .(n− |µ|+1) Thus

limEq(pk/nk) = (1−q)k Q

i1−qk lim Varq(pk/nk) =0

Valentin Féray (LaBRI, CNRS) Characters ofSn 2010-03-29 14 / 16

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Application Limit shape of Young diagrams

### Convergence of q-Plancherel measure

Theorem (F., Méliot, 2010)

Let q<1. In probability, under q-Plancherel measure,

∀k ≥1, pk(λ)

|λ|k −→Mn,q

(1−q)k 1−qk . Moreover,

∀i ≥1, λi

n −→Mn,q (1−q)qi1; We also obtained the second-order asymptotics.

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Application Limit shape of Young diagrams

### End of the talk

Thank you for listening Questions ?

Valentin Féray (LaBRI, CNRS) Characters ofSn 2010-03-29 16 / 16

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