Difference operators for functions of partitions and its application to hook-content identities (joint with Paul-Olivier Dehaye and Guo-Niu Han) Huan Xiong CNRS – Université de Strasbourg 78th SLC, 28 March 2017 Huan Xiong Difference operators for functions of partitions 28 March 2017 1 / 22
Definitions partition: λ = ( λ 1 , λ 2 , . . . , λ ℓ ) with λ 1 ≥ λ 2 ≥ · · · ≥ λ ℓ > 0 . size: | λ | = � 1 ≤ i ≤ ℓ λ i . Young diagram: boxes arranged in left-justified rows with λ i boxes in the i -th row. hook length: h � := # boxes exactly to the right, exactly above, and � itself. H ( λ ) : the product of all hook lengths in the Young diagram. 2 1 4 3 1 4 5 4 2 9 8 6 3 2 1 Figure: The Young diagram of the partition ( 6 , 3 , 3 , 2 ) and the hook lengths of corresponding boxes. Huan Xiong Difference operators for functions of partitions 28 March 2017 2 / 22
Definitions partition: λ = ( λ 1 , λ 2 , . . . , λ ℓ ) with λ 1 ≥ λ 2 ≥ · · · ≥ λ ℓ > 0 . size: | λ | = � 1 ≤ i ≤ ℓ λ i . Young diagram: boxes arranged in left-justified rows with λ i boxes in the i -th row. hook length: h � := # boxes exactly to the right, exactly above, and � itself. H ( λ ) : the product of all hook lengths in the Young diagram. content: c � := j − i for the box � in the i-th row and j-th column. − 3 − 2 − 2 − 1 0 − 1 0 1 0 1 2 3 4 5 Figure: The contents of the partition ( 6 , 3 , 3 , 2 ) . Huan Xiong Difference operators for functions of partitions 28 March 2017 3 / 22
Definitions partition: λ = ( λ 1 , λ 2 , . . . , λ ℓ ) with λ 1 ≥ λ 2 ≥ · · · ≥ λ ℓ > 0 . size: | λ | = � 1 ≤ i ≤ ℓ λ i . Young diagram: boxes arranged in left-justified rows with λ i boxes in the i -th row. hook length: h � := # boxes exactly to the right, exactly above, and � itself. H ( λ ) : the product of all hook lengths in the Young diagram. content: c � := j − i for the box � in the i-th row and j-th column. standard Young tableau (SYT) of the shape λ : fill in the Young diagram with distinct numbers 1 to | λ | such that the numbers in each row and each column are increasing. f λ : # SYTs of the shape λ . 6 9 3 8 14 2 5 13 7 1 4 10 11 12 Figure: A standard Young tableau of the shape ( 6 , 3 , 3 , 2 ) . Huan Xiong Difference operators for functions of partitions 28 March 2017 4 / 22
RSK algorithm (Robinson-Schensted-Knuth) ⇒ 1 | λ | = n f 2 � λ = 1 . n ! Huan Xiong Difference operators for functions of partitions 28 March 2017 5 / 22
RSK algorithm (Robinson-Schensted-Knuth) ⇒ 1 | λ | = n f 2 � λ = 1 . n ! Theorem (Nekrasov and Okounkov 2003, Westbury 2006, Han 2008) x n � � � � ( 1 − x i ) − 1 − y . f 2 ( y + h 2 = � ) λ n ! 2 n ≥ 0 i ≥ 1 | λ | = n � ∈ λ First proved by Nekrasov and Okounkov in their study of Seiberg-Witten Theory on supersymmetric gauges in particle physics. Huan Xiong Difference operators for functions of partitions 28 March 2017 5 / 22
RSK algorithm (Robinson-Schensted-Knuth) ⇒ 1 | λ | = n f 2 � λ = 1 . n ! Theorem (Nekrasov and Okounkov 2003, Westbury 2006, Han 2008) x n � � � � ( 1 − x i ) − 1 − y . f 2 ( y + h 2 = � ) λ n ! 2 n ≥ 0 i ≥ 1 | λ | = n � ∈ λ First proved by Nekrasov and Okounkov in their study of Seiberg-Witten Theory on supersymmetric gauges in particle physics. Rediscovered independently by Westbury using D’Arcais polynomials and by Han using Macdonald’s identity. Theorem (Han 2008) Let H t ( λ ) be the multiset of the hook lengths of λ which are divisible by t . Then ( 1 − x tk ) t y − tyz � x | λ | � � � � = . 1 − ( yx t ) k � t − z ( 1 − x k ) h 2 � λ ∈P k ≥ 1 h ∈H t ( λ ) Huan Xiong Difference operators for functions of partitions 28 March 2017 5 / 22
RSK algorithm (Robinson-Schensted-Knuth) ⇒ 1 | λ | = n f 2 � λ = 1 . n ! Theorem (Nekrasov and Okounkov 2003, Westbury 2006, Han 2008) x n � � � � ( 1 − x i ) − 1 − y . f 2 ( y + h 2 = � ) λ n ! 2 n ≥ 0 i ≥ 1 | λ | = n � ∈ λ First proved by Nekrasov and Okounkov in their study of Seiberg-Witten Theory on supersymmetric gauges in particle physics. Rediscovered independently by Westbury using D’Arcais polynomials and by Han using Macdonald’s identity. Theorem (Han 2008) Let H t ( λ ) be the multiset of the hook lengths of λ which are divisible by t . Then ( 1 − x tk ) t y − tyz � x | λ | � � � � = . 1 − ( yx t ) k � t − z ( 1 − x k ) h 2 � λ ∈P k ≥ 1 h ∈H t ( λ ) The case z = 0 , y = 1 gives the generating function for the number of partitions. Huan Xiong Difference operators for functions of partitions 28 March 2017 5 / 22
RSK algorithm (Robinson-Schensted-Knuth) ⇒ 1 | λ | = n f 2 � λ = 1 . n ! Theorem (Nekrasov and Okounkov 2003, Westbury 2006, Han 2008) x n � � � � ( 1 − x i ) − 1 − y . f 2 ( y + h 2 = � ) λ n ! 2 n ≥ 0 i ≥ 1 | λ | = n � ∈ λ First proved by Nekrasov and Okounkov in their study of Seiberg-Witten Theory on supersymmetric gauges in particle physics. Rediscovered independently by Westbury using D’Arcais polynomials and by Han using Macdonald’s identity. Theorem (Han 2008) Let H t ( λ ) be the multiset of the hook lengths of λ which are divisible by t . Then ( 1 − x tk ) t y − tyz � x | λ | � � � � = . 1 − ( yx t ) k � t − z ( 1 − x k ) h 2 � λ ∈P k ≥ 1 h ∈H t ( λ ) The case z = 0 , y = 1 gives the generating function for the number of partitions. Another corollary is the Marked hook formula: h 2 = n ( 3 n − 1 ) 1 � f 2 � . λ n ! 2 | λ | = n h ∈H ( λ ) Huan Xiong Difference operators for functions of partitions 28 March 2017 5 / 22
f 2 | λ | ! is called the Plancherel measure of the partition λ . λ 1 � f 2 λ g ( λ ) is called the n-th Plancherel average of the function g ( λ ) . n ! | λ | = n Formulas related to Plancherel measure and Plancherel average appear naturally in the study of Probability Theory, Random Matrix Theory, Mathematical Physics and Combinatorics. Huan Xiong Difference operators for functions of partitions 28 March 2017 6 / 22
f 2 | λ | ! is called the Plancherel measure of the partition λ . λ 1 � f 2 λ g ( λ ) is called the n-th Plancherel average of the function g ( λ ) . n ! | λ | = n Formulas related to Plancherel measure and Plancherel average appear naturally in the study of Probability Theory, Random Matrix Theory, Mathematical Physics and Combinatorics. Problem For which function g ( λ ) , its Plancherel average 1 f 2 � λ g ( λ ) has a nice expression? n ! | λ | = n Huan Xiong Difference operators for functions of partitions 28 March 2017 6 / 22
f 2 | λ | ! is called the Plancherel measure of the partition λ . λ 1 � f 2 λ g ( λ ) is called the n-th Plancherel average of the function g ( λ ) . n ! | λ | = n Formulas related to Plancherel measure and Plancherel average appear naturally in the study of Probability Theory, Random Matrix Theory, Mathematical Physics and Combinatorics. Problem For which function g ( λ ) , its Plancherel average 1 f 2 � λ g ( λ ) has a nice expression? n ! | λ | = n Han 2008 � = 3 n 2 − n 1 f 2 h 2 � � . n ! λ 2 | λ | = n � ∈ λ � = 40 n 3 − 75 n 2 + 41 n 1 � f 2 � h 4 . λ n ! 6 | λ | = n � ∈ λ � = 1050 n 4 − 4060 n 3 + 5586 n 2 − 2552 n 1 � f 2 � h 6 . n ! λ 24 | λ | = n � ∈ λ Huan Xiong Difference operators for functions of partitions 28 March 2017 6 / 22
Conjecture (Han 2008) � ∈ λ h 2 k The Plancherel average of the function g ( λ ) = � � : P ( n ) = 1 � � f 2 h 2 k λ � n ! | λ | = n � ∈ λ is always a polynomial of n for every k ∈ N . Huan Xiong Difference operators for functions of partitions 28 March 2017 7 / 22
Conjecture (Han 2008) � ∈ λ h 2 k The Plancherel average of the function g ( λ ) = � � : P ( n ) = 1 � � f 2 h 2 k λ � n ! | λ | = n � ∈ λ is always a polynomial of n for every k ∈ N . This conjecture was proved and generalized by Stanley. Theorem (Stanley 2010) Let Q 1 and Q 2 be two given symmetric functions. Then the Plancherel average of the function Q 1 ( h 2 � : � ∈ λ ) Q 2 ( c � : � ∈ λ ) : P ( n ) = 1 � f 2 λ Q 1 ( h 2 � : � ∈ λ ) Q 2 ( c � : � ∈ λ ) n ! | λ | = n is a polynomial of n . Olshanski (2010) also proved the content case. Huan Xiong Difference operators for functions of partitions 28 March 2017 7 / 22
An application of Han-Stanley Theorem: Corollary (Okada-Panova 2008) r � r i = 1 ( h 2 � − i 2 ) � 1 � 2 r �� 2 r + 2 � � � ∈ λ � ( n − j ) . n ! = H ( λ ) 2 2 ( r + 1 ) 2 r + 1 r | λ | = n j = 0 Huan Xiong Difference operators for functions of partitions 28 March 2017 8 / 22
An application of Han-Stanley Theorem: Corollary (Okada-Panova 2008) r � r i = 1 ( h 2 � − i 2 ) � 1 � 2 r �� 2 r + 2 � � � ∈ λ � ( n − j ) . n ! = H ( λ ) 2 2 ( r + 1 ) 2 r + 1 r | λ | = n j = 0 Definition Let g ( λ ) be a function defined on partitions. The difference operator D on functions of partitions is defined by � g ( λ + ) − g ( λ ) . Dg ( λ ) := | λ + /λ | = 1 Huan Xiong Difference operators for functions of partitions 28 March 2017 8 / 22
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