Sum of matrix entries of representations of the symmetric group and its asymptotics

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Sum of matrix entries of representations of the symmetric group and its asymptotics

Dario De Stavola

7 September 2015

Advisor: Valentin Féray

Affiliation: University of Zürich

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Sum of matrix entries of representations of the symmetric group and its asymptotics Preliminaries

Partitions

A partition λ ` n is a non increasing sequence of positive integers λ = (λ 1 , . . . , λ l )

such that P λ _i = n Example

λ = (3, 2) ` 5

λ =

(3)

Representations

A representation of S n is a morphism π : S n → GL(V ) where V is finite dimensional C vector space

Irreducible representations of S _n ←→ partitions λ ` n

π ^λ , dim λ := dim V ^λ

χ ^λ (σ) = tr(π ^λ (σ)), χ ˆ ^λ (σ) = tr(π ^λ (σ))

dim λ

(4)

Representations

A representation of S n is a morphism π : S n → GL(V ) where V is finite dimensional C vector space

Irreducible representations of S _n ←→ partitions λ ` n

π ^λ , dim λ := dim V ^λ

χ ^λ (σ) = tr(π ^λ (σ)), χ ˆ ^λ (σ) = tr(π ^λ (σ))

dim λ

(5)

Representations

A representation of S n is a morphism π : S n → GL(V ) where V is finite dimensional C vector space

Irreducible representations of S _n ←→ partitions λ ` n

π ^λ , dim λ := dim V ^λ

χ ^λ (σ) = tr(π ^λ (σ)), χ ˆ ^λ (σ) = tr(π ^λ (σ))

dim λ

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Representations

A representation of S n is a morphism π : S n → GL(V ) where V is finite dimensional C vector space

Irreducible representations of S _n ←→ partitions λ ` n

π ^λ , dim λ := dim V ^λ

χ ^λ (σ) = tr(π ^λ (σ)), χ ˆ ^λ (σ) = tr(π ^λ (σ))

dim λ

(7)

Standard Young tableaux

1 2 8 9 12 3 5 10 13 4 7 6 11

dim λ := number of SYT of shape λ

λ = (3, 2) ⇒ dim λ = 5 1 2 3

4 5

1 2 4 3 5

1 3 4 2 5

1 2 5 3 4

1 3 5

2 4

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Standard Young tableaux

1 2 8 9 12 3 5 10 13 4 7 6 11 dim λ := number of SYT of shape λ

λ = (3, 2) ⇒ dim λ = 5 1 2 3

4 5

1 2 4 3 5

1 3 4 2 5

1 2 5 3 4

1 3 5

2 4

(9)

Standard Young tableaux

1 2 8 9 12 3 5 10 13 4 7 6 11 dim λ := number of SYT of shape λ

λ = (3, 2) ⇒ dim λ = 5 1 2 3

4 5

1 2 4 3 5

1 3 4 2 5

1 2 5 3 4

1 3 5

2 4

(10)

Plancherel measure

X

λ`n

(dim λ) ² = n!

Plancherel measure

To λ ` n we associate the weight ^dim _n! ^λ

²

Probability on the set Y n of partitions of n

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Plancherel measure

X

λ`n

(dim λ) ² = n!

Plancherel measure

To λ ` n we associate the weight ^dim _n! ^λ

²

Probability on the set Y n of partitions of n

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Plancherel measure

X

λ`n

(dim λ) ² = n!

Plancherel measure

To λ ` n we associate the weight ^dim _n! ^λ

²

Probability on the set Y n of partitions of n

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Sum of matrix entries of representations of the symmetric group and its asymptotics Motivations

Limit shape

λ distributed with the Plancherel measure and renormalized, then

*Image from D. Romik "The Surprising Mathematics of Longest

Increasing Subsequences"*

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ω _x (θ) =

1 + 2θ π

sin θ+ 2 π cos θ

ω _y (θ) =

1 − 2θ π

sin θ− 2 π cos θ

Theorem (Kerov 1999)

n

|ρ|−m1(ρ)

2

χ ˆ ^λ _ρ → Y

k≥2

k ^m

^k

^(ρ)/2 H _m

_k

_(ρ) (ξ _k )

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Relations with random matrices

Rows λ ₁ , λ ₂ , λ ₃ , . . . of a random Young diagram

First, second, third, . . . biggest eigenvalues of a Gaussian random Hermitian matrix

Same first order asymptotics Same joint fluctuation (Tracy-Widom law)

Similar tools: moment method, link with free probability theory

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Relations with random matrices

Rows λ ₁ , λ ₂ , λ ₃ , . . . of a random Young diagram

First, second, third, . . . biggest eigenvalues of a Gaussian random Hermitian matrix

Same first order asymptotics

Same joint fluctuation (Tracy-Widom law)

Similar tools: moment method, link with free probability theory

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Relations with random matrices

Rows λ ₁ , λ ₂ , λ ₃ , . . . of a random Young diagram

First, second, third, . . . biggest eigenvalues of a Gaussian random Hermitian matrix

Same first order asymptotics Same joint fluctuation (Tracy-Widom law)

Similar tools: moment method, link with free probability theory

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Relations with random matrices

Rows λ ₁ , λ ₂ , λ ₃ , . . . of a random Young diagram

First, second, third, . . . biggest eigenvalues of a Gaussian random Hermitian matrix

Same first order asymptotics Same joint fluctuation (Tracy-Widom law)

Similar tools: moment method, link with free probability theory

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Sum of matrix entries of representations of the symmetric group and its asymptotics Young seminormal representation

Signed distance

d k (T ) = length of northeast path from k to k + 1 or − length of southwest path from k to k + 1

T = 1 2 3

4 5 ⇒ d 3 (T ) = −3

(3, 4)

1 3 5 7 2 6 4

=

1 4 5 7

2 6

3

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Signed distance

d k (T ) = length of northeast path from k to k + 1 or − length of southwest path from k to k + 1

T = 1 2 3

4 5 ⇒ d 3 (T ) = −3

(3, 4)

1 3 5 7 2 6 4

=

1 4 5 7

2 6

3

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Signed distance

d k (T ) = length of northeast path from k to k + 1 or − length of southwest path from k to k + 1

T = 1 2 3

4 5 ⇒ d 3 (T ) = −3

(3, 4)

1 3 5 7 2 6 4

=

1 4 5 7

2 6

3

(22)

Young seminormal representation

π ^λ ((k, k + 1)) _T, _T _˜ =



 

 

 

 

1/d _k (T ) if T = ˜ T

q 1 − _d ¹

k

(T)

²

if (k, k + 1)T = ˜ T

0 else

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Example

λ = (3, 2)

π ^λ ((2, 4, 3)) = π ^λ ((3, 4)(2, 3)) = π ^λ ((3, 4))π ^λ ((2, 3))

=







−1/3 √

8/9 0 0 0

√

8/9 1/3 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 −1







· 





1 0 0 0 0

0 −1/2 √

3/4 0 0

0 √

3/4 1/2 0 0

0 0 0 −1/2 √

3/4

0 0 0 √

3/4 1/2







=







−1/3 − √

2/9 √

2/3 0 0

√

8/9 −1/6 √

1/12 0 0

0 √

3/4 1/2 0 0

0 0 0 −1/2 √

3/4

0 0 0 − √

3/4 −1/2







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Sum of matrix entries of representations of the symmetric group and its asymptotics Partial sums

0 ≤ u ≤ 1

Partial trace

PT _u ^λ (σ) := X

i≤u dim λ

π ^λ (σ) _i,i dim λ

We would like to refine Kerov’s result

The partial trace has been studied in random matrix theory,

e.g. for orthogonal random matrices

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0 ≤ u ≤ 1

Partial trace

PT _u ^λ (σ) := X

i≤u dim λ

π ^λ (σ) _i,i dim λ

We would like to refine Kerov’s result

The partial trace has been studied in random matrix theory,

e.g. for orthogonal random matrices

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Visually

π ^λ (σ) =

u dim λ

PT

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Decomposition of PT

λ =

µ 1 = µ 2 = · · ·

Proposition (DS)

PT _u ^λ (σ) = X

i< k ¯

χ ^µ

ⁱ

(σ)

dim λ + Rem

Rem = X

i ≤˜ u dimµ

k¯

π ^µ

^k^¯

(σ) _i,i

dim λ = dim µ ¯ k

dim λ PT _u _˜ ^µ

^¯^k

(σ)

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Decomposition of PT

λ = X

µ 1 =

µ 2 = · · ·

Proposition (DS)

PT _u ^λ (σ) = X

i< k ¯

χ ^µ

ⁱ

(σ)

dim λ + Rem

Rem = X

i ≤˜ u dimµ

k¯

π ^µ

^k^¯

(σ) _i,i

dim λ = dim µ ¯ k

dim λ PT _u _˜ ^µ

^¯^k

(σ)

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Decomposition of PT

λ = X µ 1 = µ 2 = · · ·

Proposition (DS)

PT _u ^λ (σ) = X

i< k ¯

χ ^µ

ⁱ

(σ)

dim λ + Rem

Rem = X

i ≤˜ u dimµ

k¯

π ^µ

^k^¯

(σ) _i,i

dim λ = dim µ ¯ k

dim λ PT _u _˜ ^µ

^¯^k

(σ)

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Decomposition of PT

λ = µ 1 = µ 2 = · · ·

Proposition (DS)

PT _u ^λ (σ) = X

i< k ¯

χ ^µ

ⁱ

(σ)

dim λ + Rem

Rem = X

i≤ u ˜ dimµ

k¯

π ^µ

^k^¯

(σ) _i,i

dim λ = dim µ k ¯

dim λ PT _u _˜ ^µ

^¯^k

(σ)

(31)

Decomposition of PT

λ = µ 1 = µ 2 = · · ·

Proposition (DS)

PT _u ^λ (σ) = X

i< k ¯

χ ^µ

ⁱ

(σ)

dim λ + Rem

Rem = X

i≤ u ˜ dim µ

k¯

π ^µ

^k^¯

(σ) _i,i

dim λ = dim µ k ¯

dim λ PT _u _˜ ^µ

^¯^k

(σ)

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Proof

π ^λ (σ) =

π

^µ¹

(σ)

πµ2(σ)

π

^µ³

(σ)

0

. ..

u dim λ

u dimλ

PT ^λ (σ) = P

i< k ¯ χ

^µ^j

(σ)

dim λ + Rem

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Proof

π ^λ (σ) =

π

^µ¹

(σ)

πµ2(σ)

π

^µ³

(σ)

0

. ..

u dim λ

u dimλ

PT ^λ (σ) = P

i< k ¯ χ

^µ^j

(σ)

dim λ + Rem

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Asymptotics

PT _u ^λ (σ) = X

j< k ¯

dim µ j

dim λ χ ˆ ^µ

^j

(σ) + Rem

F sc (c )n ⁻

|ρ|−m1(ρ)

2

Y

k≥2

k ^m

^k

^(ρ)/2 H _m

_k

_(ρ) (ξ k )

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Asymptotics

PT _u ^λ (σ) = X

j < k ¯

dim µ _j dim λ χ ˆ ^µ

^j

(σ)

| {z }

+ Rem

F _sc (c )n ⁻

^|ρ|−m²¹^(ρ)

Y

k≥2

k ^m

^k

^(ρ)/2 H _m

k

(ρ) (ξ _k )

(36)

Asymptotics

PT _u ^λ (σ) = X

j < k ¯

dim µ _j dim λ χ ˆ ^µ

^j

(σ)

| {z }

+ Rem

A · n ⁻

^|ρ|−m²¹^(ρ)

B

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Theorem (Kerov 1993)

X

j< k ¯

dim µ _j

dim λ → A (deterministic)

Theorem (Kerov 1999)

n

^|ρ|−m²¹^(ρ)

χ ˆ ^λ (σ) → B (random)

Theorem (DS)

n

|ρ|−m1(ρ) 2

X

j< k ¯

dim µ j

dim λ χ ˆ ^µ

^j

(σ) → AB

The two objects are asymptotically independent

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Theorem (Kerov 1993)

X

j< k ¯

dim µ _j

dim λ → A (deterministic)

Theorem (Kerov 1999)

n

^|ρ|−m²¹^(ρ)

χ ˆ ^λ (σ) → B (random)

Theorem (DS)

n

|ρ|−m1(ρ) 2

X

j< k ¯

dim µ j

dim λ χ ˆ ^µ

^j

(σ) → AB

The two objects are asymptotically independent

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Theorem (Kerov 1993)

X

j< k ¯

dim µ _j

dim λ → A (deterministic)

Theorem (Kerov 1999)

n

^|ρ|−m²¹^(ρ)

χ ˆ ^λ (σ) → B (random)

Theorem (DS)

n

|ρ|−m1(ρ) 2

X

j< k ¯

dim µ j

dim λ χ ˆ ^µ

^j

(σ) → AB

The two objects are asymptotically independent

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Sum of matrix entries of representations of the symmetric group and its asymptotics Jucys-Murphy elements

First, a definition

c(2) := col(2) − row (2)

0 1 2 3 4

-1 0 1 2

-2 -1

-3

-4

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First, a definition

c(2) := col(2) − row (2)

0 1 2 3 4

-1 0 1 2

-2 -1

-3

-4

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Jucys-Murphy elements

J k := (1, k) + (2, k) + . . . + (k − 1, k) ∈ Z (C[S n ])

π ^λ (J _k ) =







c _T

₁

(

^k

) 0

c _T

₂

(

^k

)

0 ^{. ..}







(43)

Jucys-Murphy elements

J k := (1, k) + (2, k) + . . . + (k − 1, k) ∈ Z (C[S n ])

π ^λ (J k ) =







c _T

₁

(

^k

) 0

c _T

₂

(

^k

)

0 ^{. ..}







(44)

n 2

χ

^λ

(τ ) =

χ

^λ

(J

₂

+ . . . + J

_n

)

=

n

P

i=2

χ

^λ

(J

_i

) =

n

P

i=2 dimλ

P

k=1

c

_T_k

(

ⁱ

) = dim λ P

2∈λ

c( 2 )

n 2

ˆ

χ ^λ (transposition) = X

2 ∈λ

c (2)

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n 2

χ

^λ

(τ ) =

χ

^λ

(J

₂

+ . . . + J

_n

) =

n

P

i=2

χ

^λ

(J

_i

)

=

n

P

i=2 dimλ

P

k=1

c

_T_k

(

ⁱ

) = dim λ P

2∈λ

c( 2 )

n 2

ˆ

χ ^λ (transposition) = X

2 ∈λ

c (2)

(46)

n 2

χ

^λ

(τ ) =

χ

^λ

(J

₂

+ . . . + J

_n

) =

n

P

i=2

χ

^λ

(J

_i

) =

n

P

i=2 dimλ

P

k=1

c

_T_k

(

ⁱ

)

= dim λ P

2∈λ

c( 2 )

n 2

ˆ

χ ^λ (transposition) = X

2 ∈λ

c (2)

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n 2

χ

^λ

(τ ) =

χ

^λ

(J

₂

+ . . . + J

_n

) =

n

P

i=2

χ

^λ

(J

_i

) =

n

P

i=2 dimλ

P

k=1

c

_T_k

(

ⁱ

) = dim λ P

2∈λ

c ( 2 )

n 2

ˆ

χ ^λ (transposition) = X

2 ∈λ

c (2)

(48)

n 2

χ

^λ

(τ ) = χ

^λ

(J

₂

+ . . . + J

_n

) =

n

P

i=2

χ

^λ

(J

_i

) =

n

P

i=2 dimλ

P

k=1

c

_T_k

(

ⁱ

) = dim λ P

2∈λ

c ( 2 )

n 2

ˆ

χ ^λ (transposition) = X

2 ∈λ

c (2)

(49)

n 2

χ

^λ

(τ ) = χ

^λ

(J

₂

+ . . . + J

_n

) =

n

P

i=2

χ

^λ

(J

_i

) =

n

P

i=2 dimλ

P

k=1

c

_T_k

(

ⁱ

) = dim λ P

2∈λ

c ( 2 )

n 2

ˆ

χ ^λ (transposition) = X

2 ∈λ

c (2)

(50)

Considering χ ^λ (J 2 + . . . + J n ) we get

n 2

ˆ

χ ^λ (transposition) = X

2 ∈λ

c (2)

Considering χ ^λ _l

Q

i=1

(J ₂ ^ν

ⁱ

+ . . . + J _n ^ν

ⁱ

)

we get

(51)

Considering χ ^λ (J 2 + . . . + J n ) we get

n 2

ˆ

χ ^λ (transposition) = X

2 ∈λ

c (2)

Considering χ ^λ _l

Q

i=1

(J ₂ ^ν

ⁱ

+ . . . + J _n ^ν

ⁱ

)

we get

(52)

Considering χ ^λ (J 2 + . . . + J n ) we get

n 2

ˆ

χ ^λ (transposition) = X

2 ∈λ

c (2)

Considering χ ^λ _l

Q

i=1

(J ₂ ^ν

ⁱ

+ . . . + J _n ^ν

ⁱ

)

we get

c _ρ n ^↓(|ρ|−m

¹

^(ρ)) χ ˆ ^λ _ρ =

l

Y

i=1

X

2 ∈λ

c ( 2 ) ^ν

ⁱ

!

− X

˜ ρ<ρ

c _ρ _˜ n ^↓(|˜ ^ρ|−m

¹

^{( ˜} ^ρ)) χ ˆ ^λ _ρ _˜

where ρ i = ν i + 1

(53)

Considering χ ^λ (J 2 + . . . + J n ) we get

n 2

ˆ

χ ^λ (transposition) = X

2 ∈λ

c (2)

Considering χ ^λ _l

Q

i=1

(J ₂ ^ν

ⁱ

+ . . . + J _n ^ν

ⁱ

)

we get

ˆ χ ^λ (σ)n

|ρ|−m1(ρ)

2

∼

l

Y

i=1

X

2 ∈λ

c (2) ^ν

ⁱ

!

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Sum of matrix entries of representations of the symmetric group and its asymptotics Proof

ˆ χ ^µ (σ)n

|ρ|−m1(ρ)

2

∼ Y ^l

i =1

X

2 ∈µ

c( 2 ) ^ν

ⁱ

!

=

l

Y

i=1

X

2 ∈λ

c( 2 ) ^ν

ⁱ

− c (

^X

) ^ν

ⁱ

!

o

l

Y

i=1

X

2 ∈λ

c( 2 ) ^ν

ⁱ

!

o ˆ

χ ^λ (σ)n

^|ρ|−m²¹^(ρ)

µ % λ = X

(55)

ˆ χ ^µ (σ)n

|ρ|−m1(ρ)

2

∼ Y ^l

i =1

X

2 ∈µ

c( 2 ) ^ν

ⁱ

!

=

l

Y

i=1

X

2 ∈λ

c( 2 ) ^ν

ⁱ

− c (

^X

) ^ν

ⁱ

!

o

l

Y

i=1

X

2 ∈λ

c( 2 ) ^ν

ⁱ

!

o ˆ

χ ^λ (σ)n

^|ρ|−m²¹^(ρ)

µ % λ = X

(56)

ˆ χ ^µ (σ)n

|ρ|−m1(ρ)

2

∼ Y ^l

i =1

X

2 ∈µ

c( 2 ) ^ν

ⁱ

!

=

l

Y

i=1

X

2 ∈λ

c( 2 ) ^ν

ⁱ

− c (

^X

) ^ν

ⁱ

!

o

l

Y

i=1

X

2 ∈λ

c( 2 ) ^ν

ⁱ

!

o ˆ

χ ^λ (σ)n

^|ρ|−m²¹^(ρ)

µ % λ = X

(57)

ˆ χ ^µ (σ)n

|ρ|−m1(ρ)

2

∼ Y ^l

i =1

X

2 ∈µ

c( 2 ) ^ν

ⁱ

!

=

l

Y

i=1

X

2 ∈λ

c( 2 ) ^ν

ⁱ

− c (

^X

) ^ν

ⁱ

!

o

l

Y

i =1

X

2 ∈λ

c( 2 ) ^ν

ⁱ

!

o ˆ

χ ^λ (σ)n

^|ρ|−m²¹^(ρ)

µ % λ = X

(58)

ˆ χ ^µ (σ)n

|ρ|−m1(ρ)

2

∼ Y ^l

i =1

X

2 ∈µ

c( 2 ) ^ν

ⁱ

!

=

l

Y

i=1

X

2 ∈λ

c( 2 ) ^ν

ⁱ

− c (

^X

) ^ν

ⁱ

!

o

l

Y

i =1

X

2 ∈λ

c( 2 ) ^ν

ⁱ

!

o ˆ

χ ^λ (σ)n

^|ρ|−m²¹^(ρ)

µ % λ = X

(59)

n

|ρ|−m1(ρ)

2

X

j< k ¯

dim µ _j dim λ χ ˆ ^µ

^j

(σ)

o

n

|ρ|−m1(ρ) 2



 X

j< k ¯

dim µ j

dim λ



 χ ˆ ^λ (σ)

↓

A · B

(60)

n

|ρ|−m1(ρ)

2

X

j< k ¯

dim µ _j dim λ χ ˆ ^µ

^j

(σ)

o

n

|ρ|−m1(ρ) 2



 X

j< k ¯

dim µ j

dim λ



 χ ˆ ^λ (σ)

↓

A · B

(61)

Telescopic sum

PT _u ^λ (σ) = X

j < k ¯

1

dim µ ⁽¹⁾ _j dim λ χ ˆ ^µ

(1)

j

(σ) + ^P

j<k¯2 dimµ(2)

j dimλ

χ ˆ

^µ

(2)

j

(σ) + . . .

Unfortunately, I cannot prove convergence...

(62)

Partial sum

PS _u ^λ (σ) := X

i,j≤u dim λ

π ^λ (σ) _i,j

dim λ

(63)

Visually

π ^λ (σ) =

u dim λ

PS

(64)

Decomposition of PS

σ ∈ S _r

PS _u ^λ (σ) = X

j < k ¯

dim µ _j

dim λ PS ₁ ^µ

^j

(σ) + Rem

= X

τ∈S

r

E ^r _PL [ ˆ χ ^· (τ )PS ₁ ^· (σ)] PT _u ^λ (τ )

And we have convergence PS _u ^λ (σ) → uE ^r _PL [PS ₁ ^· (σ)]

(65)

Decomposition of PS

σ ∈ S _r

PS _u ^λ (σ) = X

j< k ¯

dim µ _j

dim λ PS ₁ ^µ

^j

(σ)

| {z }

+ Rem

= X

τ∈S

r

E ^r _PL [ ˆ χ ^· (τ )PS ₁ ^· (σ)] PT _u ^λ (τ )

And we have convergence PS _u ^λ (σ) → uE ^r _PL [PS ₁ ^· (σ)]

(66)

Decomposition of PS

σ ∈ S _r

PS _u ^λ (σ) = X

j< k ¯

dim µ _j

dim λ PS ₁ ^µ

^j

(σ)

| {z }

+ Rem

= X

τ∈S

r

E ^r _PL [ ˆ χ ^· (τ )PS ₁ ^· (σ)] PT _u ^λ (τ )

And we have convergence PS _u ^λ (σ) → uE ^r _PL [PS ₁ ^· (σ)]

(67)

T HAN K YOU

Sum of matrix entries of representations of the symmetric group and its asymptotics