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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 13 October 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 � 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

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

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

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 λ ( σ ) = tr ( π λ ( σ )) , ˆ 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 1013 4 7 6 11

Sum of matrix entries of representations of the symmetric group and its asymptotics Preliminaries Standard Young tableaux 1 2 8 9 12 3 5 1013 4 7 6 11 dim λ := number of SYT of shape 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 1013 4 7 6 11 dim λ := number of SYT of shape λ λ = ( 3 , 2 ) ⇒ dim λ = 5 1 2 3 1 2 4 1 3 4 1 2 5 1 3 5 4 5 3 5 2 5 3 4 2 4

Sum of matrix entries of representations of the symmetric group and its asymptotics Preliminaries Plancherel measure � ( dim λ ) 2 = n ! λ ⊢ n

Sum of matrix entries of representations of the symmetric group and its asymptotics Preliminaries Plancherel measure � ( dim λ ) 2 = n ! λ ⊢ n Plancherel measure To λ ⊢ n we associate the weight dim λ 2 n !

Sum of matrix entries of representations of the symmetric group and its asymptotics Preliminaries Plancherel measure � ( dim λ ) 2 = n ! λ ⊢ n Plancherel measure To λ ⊢ n we associate the weight dim λ 2 n ! 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 � � 1 + 2 θ sin θ + 2 ω x ( θ ) = π cos θ π � � 1 − 2 θ sin θ − 2 ω y ( θ ) = π cos θ π Theorem (Kerov 1999) � wt ( ρ ) χ λ k m k ( ρ ) / 2 H m k ( ρ ) ( ξ k ) n ˆ ρ → 2 k ≥ 2

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

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 First, second, third, . . . biggest Young diagram eigenvalues of a Gaussian random Hermitian matrix Same first order asymptotics

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 First, second, third, . . . biggest Young diagram eigenvalues of a Gaussian random Hermitian matrix Same first order asymptotics Same joint fluctuation (Tracy-Widom law)

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 First, second, third, . . . biggest Young diagram 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

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 1 2 3 T = ⇒ d 3 ( T ) = − 3 4 5

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 1 2 3 T = ⇒ d 3 ( T ) = − 3 4 5 1 3 5 7 1 4 5 7 2 6 2 6 ( 3 , 4 ) = 4 3

Sum of matrix entries of representations of the symmetric group and its asymptotics Young seminormal representation Young seminormal representation  if T = ˜ 1 / d k ( T ) T       � π λ (( k , k + 1 )) T , ˜ if ( k , k + 1 ) T = ˜ 1 1 − T = T d k ( T ) 2       0 else

Sum of matrix entries of representations of the symmetric group and its asymptotics Young seminormal representation 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 0 0 0 1 0 0 0 0 8 / 9 √ √ 1 / 3 0 0 0 0 − 1 / 2 0 0  8 / 9   3 / 4  √     · = 0 0 1 0 0 0 1 / 2 0 0    3 / 4  √     0 0 0 1 0 0 0 0 − 1 / 2    3 / 4  √ 0 0 0 0 − 1 0 0 0 1 / 2 3 / 4 −√ √   − 1 / 3 0 0 2 / 9 2 / 3 √ √ − 1 / 6 0 0  8 / 9 1 / 12  √   = 0 1 / 2 0 0  3 / 4  √   0 0 0 − 1 / 2   3 / 4 −√ 0 0 0 − 1 / 2 3 / 4

Sum of matrix entries of representations of the symmetric group and its asymptotics Partial sums 0 ≤ u ≤ 1 Partial trace π λ ( σ ) i , i � PT λ u ( σ ) := dim λ i ≤ u dim λ

Sum of matrix entries of representations of the symmetric group and its asymptotics Partial sums 0 ≤ u ≤ 1 Partial trace π λ ( σ ) i , i � PT λ u ( σ ) := dim λ i ≤ u dim λ We would like to refine Kerov’s result

Sum of matrix entries of representations of the symmetric group and its asymptotics Partial sums 0 ≤ u ≤ 1 Partial trace π λ ( σ ) i , i � PT λ u ( σ ) := dim λ i ≤ u 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 u dim λ PT π λ ( σ ) = u dim λ

Sum of matrix entries of representations of the symmetric group and its asymptotics Partial sums Decomposition of PT λ =

Sum of matrix entries of representations of the symmetric group and its asymptotics Partial sums Decomposition of PT λ = µ 1 = X

Sum of matrix entries of representations of the symmetric group and its asymptotics Partial sums Decomposition of PT X λ = µ 1 = µ 2 = · · ·

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) χ µ i ( σ ) � PT λ u ( σ ) = dim λ + Rem i < ¯ k

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) χ µ i ( σ ) � PT λ u ( σ ) = dim λ + Rem i < ¯ k π µ ¯ k ( σ ) i , i = dim µ ¯ � dim λ PT µ ¯ k Rem = u ( σ ) k ˜ dim λ i ≤ ˜ u dim µ ¯ k

Sum of matrix entries of representations of the symmetric group and its asymptotics Partial sums Proof π µ 1 ( σ ) 0 πµ 2 ( σ ) π λ ( σ ) = π µ 3 ( σ ) 0 ...

Sum of matrix entries of representations of the symmetric group and its asymptotics Partial sums Proof χ µ j ( σ ) PT λ ( σ ) = � dim λ + Rem i < ¯ k u dim λ π µ 1 ( σ ) 0 πµ 2 ( σ ) π λ ( σ ) = u dim λ π µ 3 ( σ ) 0 ...

Sum of matrix entries of representations of the symmetric group and its asymptotics Partial sums Asymptotics dim µ j � PT λ χ µ j ( σ ) + Rem u ( σ ) = dim λ ˆ j < ¯ k