Hook formulas for skew shapes Greta Panova (University of - PowerPoint PPT Presentation

Skew HLF Bijections Lattice paths Asymptotics of skew SYTs Multivariate: tilings and beyond Hook formulas for skew shapes Greta Panova (University of Pennsylvania) joint with Alejandro Morales (UCLA), Igor Pak (UCLA) 78th S´ eminaire Lotharingien de combinatoire March 2017 1

Skew HLF Bijections Lattice paths Asymptotics of skew SYTs Multivariate: tilings and beyond Standard Young Tableaux Irreducible representations of S n : Specht modules S λ , for all λ ⊢ n . Basis for S λ : Standard Young Tableaux of shape λ : λ = (2 , 2 , 1): : 1 2 1 2 1 3 1 3 1 4 3 4 3 5 2 4 2 5 2 5 5 4 5 4 3 Hook-length formula [Frame-Robinson-Thrall]: | λ | ! 5! dim S λ = # { SYTs of shape λ } = f λ = = � u ∈ λ h u 4 ∗ 3 ∗ 2 ∗ 1 ∗ 1               Hook length of box u = ( i , j ) ∈ λ : h u = λ i − j + λ ′ j − i + 1 = # ∈ u             2

Skew HLF Bijections Lattice paths Asymptotics of skew SYTs Multivariate: tilings and beyond Counting skew SYTs 2 4 Outer shape λ , inner shape µ , e.g. for λ = (5 , 4 , 4 , 2) , µ = (3 , 2 , 1) 1 5 3 6 8 7 9 Jacobi-Trudi [Feit 1953]: � ℓ ( λ ) � 1 f λ/µ = | λ/µ | ! · det . ( λ i − µ j − i + j )! i , j =1 3

Skew HLF Bijections Lattice paths Asymptotics of skew SYTs Multivariate: tilings and beyond Counting skew SYTs 2 4 Outer shape λ , inner shape µ , e.g. for λ = (5 , 4 , 4 , 2) , µ = (3 , 2 , 1) 1 5 3 6 8 7 9 Jacobi-Trudi [Feit 1953]: � ℓ ( λ ) � 1 f λ/µ = | λ/µ | ! · det . ( λ i − µ j − i + j )! i , j =1 Littlewood-Richardson : f λ/µ = � c λ µ,ν f ν ν 3

Skew HLF Bijections Lattice paths Asymptotics of skew SYTs Multivariate: tilings and beyond Counting skew SYTs 2 4 Outer shape λ , inner shape µ , e.g. for λ = (5 , 4 , 4 , 2) , µ = (3 , 2 , 1) 1 5 3 6 8 7 9 Jacobi-Trudi [Feit 1953]: � ℓ ( λ ) � 1 f λ/µ = | λ/µ | ! · det . ( λ i − µ j − i + j )! i , j =1 Littlewood-Richardson : f λ/µ = � c λ µ,ν f ν ν f δ n +2 /δ n = E 2 n +1 : No product formula, e.g. λ/µ = δ n +2 /δ n : x 2 x 3 x 4 1 + E 1 x + E 2 2! + E 3 3! + E 4 4! + . . . = sec( x ) + tan( x ) . Euler numbers: 2 , 5 , 16 , 61 .... 3

Skew HLF Bijections Lattice paths Asymptotics of skew SYTs Multivariate: tilings and beyond Hook-Length formula for skew shapes Theorem (Naruse, SLC, September 2014) 1 f λ/µ = | λ/µ | ! � � h ( u ) , D ∈E ( λ/µ ) u ∈ [ λ ] \ D where E ( λ/µ ) is the set of excited diagrams of λ/µ . Excited diagrams: E ( λ/µ ) = { D ⊂ λ : obtained from µ via } q 3 q 5 q 5 q 7 q 9 � 1 1 1 1 1 � f (4321 / 21) = 7! 1 4 · 3 3 + 1 3 · 3 3 · 5 + 1 3 · 3 3 · 5 + 1 2 · 3 3 · 5 2 + = 61 1 2 · 3 2 · 5 2 · 7 4

Skew HLF Bijections Lattice paths Asymptotics of skew SYTs Multivariate: tilings and beyond Hook-Length formula for skew shapes q 3 q 5 q 5 q 7 q 9 q 3 q 5 q | T | = s λ/µ (1 , q , q 2 , . . . ) = � (1 − q ) 4 (1 − q 3 ) 3 + 2 × (1 − q ) 3 (1 − q 3 ) 3 (1 − q 5 ) T ∈ SSYT (4321 / 21) q 7 q 9 + (1 − q ) 2 (1 − q 3 ) 3 (1 − q 5 ) 2 + (1 − q ) 2 (1 − q 3 ) 2 (1 − q 5 ) 2 (1 − q 7 ) Theorem (Morales-Pak-P) q λ ′ � j − i � q | T | = � � � . 1 − q h ( i , j ) T ∈ SSYT ( λ/µ ) D ∈E ( λ/µ ) ( i , j ) ∈ [ λ ] \ D Theorem (Morales-Pak-P) � � q h ( u ) q | π | = � � � . 1 − q h ( u ) u ∈ S π ∈ RPP ( λ/µ ) S ∈ PD ( λ/µ ) where PD ( λ/µ ) is the set of pleasant diagrams. Other recent proof by [M. Konvalinka] 4

Skew HLF Bijections Lattice paths Asymptotics of skew SYTs Multivariate: tilings and beyond Algebraic proof for SSYTs: 6 1 3 [Ikeda-Naruse, Kreiman]: Let w � v be Grassman- y 6 − y 3 5 6 y 6 − y 1 nian permutations whose unique descent is at po- 6 sition d with corresponding partitions µ ⊆ λ ⊆ 4 5 y 5 − y 1 y 5 − y 3 5 d × ( n − d ). Then y 4 − y 1 y 4 − y 3 2 3 4 4 � � � � � [ X w ] v = y v ( d + j ) − y v ( d − i +1) . 1 2 3 y 2 − y 1 � 2 D ∈E ( λ/µ ) ( i , j ) ∈ D 1 v = 245613, w = 361245 5

Skew HLF Bijections Lattice paths Asymptotics of skew SYTs Multivariate: tilings and beyond Algebraic proof for SSYTs: 6 1 3 [Ikeda-Naruse, Kreiman]: Let w � v be Grassman- y 6 − y 3 5 6 y 6 − y 1 nian permutations whose unique descent is at po- 6 sition d with corresponding partitions µ ⊆ λ ⊆ 4 5 y 5 − y 1 y 5 − y 3 5 d × ( n − d ). Then y 4 − y 1 y 4 − y 3 2 3 4 4 � � � � � [ X w ] v = y v ( d + j ) − y v ( d − i +1) . 1 2 3 y 2 − y 1 � 2 D ∈E ( λ/µ ) ( i , j ) ∈ D 1 v = 245613, w = 361245 Factorial Schur functions: � d det � ( x j − a 1 ) · · · ( x j − a µ i + d − i ) s ( d ) i , j =1 µ ( x | a ) := , � 1 ≤ i < j ≤ d ( x i − x j ) [Knutson-Tao, Lakshmibai–Raghavan–Sankaran] Schubert class at a point: v = ( − 1) ℓ ( w ) s ( d ) � � y v (1) , . . . , y v ( d ) | y 1 , . . . , y n − 1 � [ X w ] . µ � 5

Skew HLF Bijections Lattice paths Asymptotics of skew SYTs Multivariate: tilings and beyond Algebraic proof for SSYTs: 6 1 3 [Ikeda-Naruse, Kreiman]: Let w � v be Grassman- y 6 − y 3 5 6 y 6 − y 1 nian permutations whose unique descent is at po- 6 sition d with corresponding partitions µ ⊆ λ ⊆ 4 5 y 5 − y 1 y 5 − y 3 5 d × ( n − d ). Then y 4 − y 1 y 4 − y 3 2 3 4 4 � � � � � [ X w ] v = y v ( d + j ) − y v ( d − i +1) . 1 2 3 y 2 − y 1 � 2 D ∈E ( λ/µ ) ( i , j ) ∈ D 1 v = 245613, w = 361245 Factorial Schur functions: � d det � ( x j − a 1 ) · · · ( x j − a µ i + d − i ) s ( d ) i , j =1 µ ( x | a ) := , � 1 ≤ i < j ≤ d ( x i − x j ) [Knutson-Tao, Lakshmibai–Raghavan–Sankaran] Schubert class at a point: v = ( − 1) ℓ ( w ) s ( d ) � � y v (1) , . . . , y v ( d ) | y 1 , . . . , y n − 1 � [ X w ] . µ � Evaluation at y = 1 , q , q 2 , ... , v ( d + 1 − i ) = λ i + d + 1 − i , x i → y v ( i ) = q λ i + d +1 − i → Jacobi-Trudi det[ � µ j + d − j ( q λ i + d +1 − i − q r )] d s ( d ) r =1 i , j =1 µ ( q v (1) , . . . | 1 , q , . . . ) = = .... i < j ( q λ + d +1 − i − q λ j + d +1 − j ) � ... [ simplifications ] ... = det[ h λ i − i − µ j + j (1 , q , . . . )] = s λ/µ (1 , q , . . . ) 5

Skew HLF Bijections Lattice paths Asymptotics of skew SYTs Multivariate: tilings and beyond Combinatorial proofs: Hillman-Grassl algorithm/map Φ: Reverse Plane Partitions of shape λ to Arrays of shape λ : 0 1 2 → 0 1 2 → 0 0 1 → 0 0 1 → 0 0 1 , 0 0 0 RRP P = 1 1 3 1 1 3 0 0 3 0 0 2 0 0 1 0 0 0 2 1 0 0 0 0 0 0 0 → 1 0 0 → 1 0 0 → 1 0 0 → 1 0 1 =: Array A = Φ( P ) 0 0 0 0 0 0 0 0 1 0 0 2 0 0 2 1 1 1 1 1 Weight ( P ) = 0 + 1 + 2 + 1 + 1 + 3 + 2 = 10 = � i , j A i , j hook ( i , j ) = 1 ∗ 5 + 1 ∗ 2 + 2 ∗ 1 + 1 ∗ 1 = weight ( A ) 6

Skew HLF Bijections Lattice paths Asymptotics of skew SYTs Multivariate: tilings and beyond Combinatorial proofs: Hillman-Grassl algorithm/map Φ: Reverse Plane Partitions of shape λ to Arrays of shape λ : 0 1 2 → 0 1 2 → 0 0 1 → 0 0 1 → 0 0 1 , 0 0 0 RRP P = 1 1 3 1 1 3 0 0 3 0 0 2 0 0 1 0 0 0 2 1 0 0 0 0 0 0 0 → 1 0 0 → 1 0 0 → 1 0 0 → 1 0 1 =: Array A = Φ( P ) 0 0 0 0 0 0 0 0 1 0 0 2 0 0 2 1 1 1 1 1 Weight ( P ) = 0 + 1 + 2 + 1 + 1 + 3 + 2 = 10 = � i , j A i , j hook ( i , j ) = 1 ∗ 5 + 1 ∗ 2 + 2 ∗ 1 + 1 ∗ 1 = weight ( A ) 0 0 0 0 0 0 0 0 0 0 0 Φ Φ Φ 1 1 0 1 1 1 0 1 1 1 0 0 2 2 0 0 1 1 2 2 1 0 2 2 1 0 1 1 2 1 1 3 1 1 Theorem (Morales-Pak-P) The restricted Hillman-Grassl map is a bijection to the SSYTs of shape λ/µ to the excited arrays (diagrams in E ( λ/µ ) with nonzero entries on the broken diagonals) . d 1 µ d 1 ( µ ) d 1 ( D ) λ λ λ A ∅ A µ A D 6

Skew HLF Bijections Lattice paths Asymptotics of skew SYTs Multivariate: tilings and beyond Combinatorial proofs: 0 0 0 0 0 0 0 0 0 0 0 Φ Φ Φ 1 1 0 1 1 1 0 1 1 1 0 0 2 2 0 0 1 1 2 2 1 0 2 2 1 0 1 1 2 1 1 3 1 1 Theorem (Morales-Pak-P) The restricted Hillman-Grassl map is a bijection to the SSYTs of shape λ/µ to the excited arrays (diagrams in E ( λ/µ ) with nonzero entries on the broken diagonals) . d 1 µ d 1 ( µ ) d 1 ( D ) λ λ λ A ∅ A µ A D Proof sketch: Issue: enforce 0s on µ and strict increase down columns on λ/µ . Show Φ − 1 ( A ) is column strict in λ/µ + support in λ/µ via properties of RSK (each diagonal of P is shape of RSK tableau on the corresponding subrectangle of A ) Thus, Φ − 1 is injective: restricted arrays → SSYTs of shape λ/µ . Bijective: use the algebraic identity. 6

Skew HLF Bijections Lattice paths Asymptotics of skew SYTs Multivariate: tilings and beyond Hillman-Grassl on skew RPPs Without the restriction of strictly increasing columns, we have skew reverse plane partitions and a wider class of arrays/diagrams, called pleasant diagrams : PD ( λ/µ ). – supersets of E ( λ/µ ), identified by the “high peaks”. D ∗ = ̺ 1 ( S ) λ/µ S D = ̺ 2 ( S ) 7