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

### 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

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 =

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

### 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

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

### 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

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

### 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

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

### 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

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

### 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

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

### 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

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

### 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

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

### 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

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

### 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

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

### 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

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"*

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

### 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} )

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

### 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

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

### 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

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

### 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

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

### 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

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

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

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

### 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

### 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

###

###

###

###

###

###

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

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

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

### Visually

### π ^{λ} (σ) =

### u dim λ

### u dim λ

### PT

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

### 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)

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

### 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)

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

### 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)

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

### 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)

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

### 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)

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

### Proof

### π ^{λ} (σ) =

### π

^{µ}

^{1}

### (σ)

πµ2(σ)

### π

^{µ}

^{3}

### (σ)

**0**

**0**

### . ..

### u dim λ

### u dimλ

### PT ^{λ} (σ) = P

### i< k ¯ χ

^{µ}

^{j}

### (σ)

### dim λ + Rem

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

### Proof

### π ^{λ} (σ) =

### π

^{µ}

^{1}

### (σ)

πµ2(σ)

### π

^{µ}

^{3}

### (σ)

**0**

**0**

### . ..

### u dim λ

### u dimλ

### PT ^{λ} (σ) = P

### i< k ¯ χ

^{µ}

^{j}

### (σ)

### dim λ + Rem

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

### 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 )

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

### 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|−m}

^{2}

^{1}

^{(ρ)}

### Y

### k≥2

### k ^{m}

^{k}

^{(ρ)/2} H _{m}

k

### (ρ) (ξ _{k} )

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

### Asymptotics

### PT _{u} ^{λ} (σ) = X

### j < k ¯

### dim µ _{j} dim λ χ ˆ ^{µ}

^{j}

### (σ)

### | {z }

### + Rem

### A · n ^{−}

^{|ρ|−m}

^{2}

^{1}

^{(ρ)}

### B

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

### Theorem (Kerov 1993)

### X

### j< k ¯

### dim µ _{j}

### dim λ → A (deterministic)

### Theorem (Kerov 1999)

### n

^{|ρ|−m}

^{2}

^{1}

^{(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

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

### Theorem (Kerov 1993)

### X

### j< k ¯

### dim µ _{j}

### dim λ → A (deterministic)

### Theorem (Kerov 1999)

### n

^{|ρ|−m}

^{2}

^{1}

^{(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

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

### Theorem (Kerov 1993)

### X

### j< k ¯

### dim µ _{j}

### dim λ → A (deterministic)

### Theorem (Kerov 1999)

### n

^{|ρ|−m}

^{2}

^{1}

^{(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

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

### First, a definition

### Contents

### c(2) := col(2) − row (2)

### 0 1 2 3 4

### -1 0 1 2

### -2 -1

### -3

### -4

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

### First, a definition

### Contents

### c(2) := col(2) − row (2)

### 0 1 2 3 4

### -1 0 1 2

### -2 -1

### -3

### -4

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

### 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** ^{. ..}

###

###

###

### 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** ^{. ..}

###

###

###

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)

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)

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)

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)

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)

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)

### 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

### 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

### 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

### 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}

### !

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|−m}

^{2}

^{1}

^{(ρ)}

### µ % λ = X

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|−m}

^{2}

^{1}

^{(ρ)}

### µ % λ = X

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|−m}

^{2}

^{1}

^{(ρ)}

### µ % λ = X

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|−m}

^{2}

^{1}

^{(ρ)}

### µ % λ = X

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|−m}

^{2}

^{1}

^{(ρ)}

### µ % λ = X

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

### 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

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

### 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

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

### 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...

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

### Partial sum

### PS _{u} ^{λ} (σ) := X

### i,j≤u dim λ

### π ^{λ} (σ) _{i,j}

### dim λ

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

### Visually

### π ^{λ} (σ) =

### u dim λ

### u dim λ

### PS

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

### 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)]

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

### 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)]

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

### 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)]

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