Fully Packed Loop Configurations and LittlewoodRichardson - PowerPoint PPT Presentation

Fully Packed Loop Configurations and Littlewood–Richardson coefficients Philippe Nadeau Faculty of Mathematics, University of Vienna FPSAC 22, San Francisco, August 4th 2010.

Rough Outline Fully Packed Loops in a square grid

Rough Outline Fully Packed Loops in a square grid From the square to the triangle

Rough Outline Fully Packed Loops in a square grid From the square to the triangle Fully Packed Loops in a triangle

FPL configurations : Definition Start with the square grid G n with n 2 vertices and 4 n external edges, and pick every other edge on the boundary (starting with the topmost on the left). 14 13 12 1 11 2 10 3 9 4 8 5 6 7

FPL configurations : Definition Start with the square grid G n with n 2 vertices and 4 n external edges, and pick every other edge on the boundary (starting with the topmost on the left). 14 13 12 1 11 A Fully Packed Loop (FPL) 2 10 configuration of size n is a subgraph of G n with exactly 2 3 9 edges incident to each vertex. 4 8 5 6 7

FPL configurations : Enumeration FPL configurations are in simple bijection with numerous objects : alternating sign matrices (ASMs), height matrices, configurations of the six vertex model, Gog triangles,... 14 13 12 1 11 2 10 3 9 4 8 5 6 7

FPL configurations : Enumeration FPL configurations are in simple bijection with numerous objects : alternating sign matrices (ASMs), height matrices, configurations of the six vertex model, Gog triangles,... FPL of size n ASMs of size n

FPL configurations : Enumeration FPL configurations are in simple bijection with numerous objects : alternating sign matrices (ASMs), height matrices, configurations of the six vertex model, Gog triangles,... 14 13 12 1 11 FPL of size n 2 10 3 9 ASMs of size n 4 8 5 6 7

FPL configurations : Enumeration FPL configurations are in simple bijection with numerous objects : alternating sign matrices (ASMs), height matrices, configurations of the six vertex model, Gog triangles,... 14 13 12 1 11 n − 1 (3 i + 1)! � | FPL n | = A n = ( n + i )! 2 10 i =0 3 9 [Zeilberger ’96, Kuperberg ’96] 4 8 5 6 7

FPL configurations : Enumeration FPL configurations are in simple bijection with numerous objects : alternating sign matrices (ASMs), height matrices, configurations of the six vertex model, Gog triangles,... 14 13 12 1 11 n − 1 (3 i + 1)! � | FPL n | = A n = ( n + i )! 2 10 i =0 3 9 [Zeilberger ’96, Kuperberg ’96] 4 8 5 6 7 Why study FPLs rather than ASMs ?

FPL configurations : Refined enumeration Every FPL configuration determines a link pattern on the external edges of the grid G n , where link pattern = set of n noncrossing chords between 2 n labeled points on a disk. 14 13 12 1 11 2 10 3 9 4 8 5 6 7

FPL configurations : Refined enumeration Every FPL configuration determines a link pattern on the external edges of the grid G n , where link pattern = set of n noncrossing chords between 2 n labeled points on a disk. 14 13 12 13 14 12 1 11 1 11 2 10 2 10 3 9 4 7 8 3 5 6 9 4 8 � 2 n � 1 | LP n | = C n := 5 6 7 n + 1 n

FPL configurations : Refined enumeration Main problem : given a link pattern X , how many FPL configurations induce the link pattern X ? We note A X this number. 6 6 6 1 1 1 5 5 5 If X = A X = 2 2 4 2 2 4 4 3 3 3

FPL configurations : Refined enumeration Main problem : given a link pattern X , how many FPL configurations induce the link pattern X ? We note A X this number. 6 6 6 1 1 1 5 5 5 If X = A X = 2 2 4 2 2 4 4 3 3 3 Wieland’s rotation Given a link pattern X , consider the rotated pattern r ( X ) obtained by { i, j } �→ { i + 1 , j + 1 } . Then A X = A r ( X ) . X r ( X )

The Razumov-Stroganov correspondence For i = 1 . . . 2 n let e i act on link patterns by { i, j } , { i + 1 , k } ∈ X → { i, i + 1 } , { j, k } ∈ e i ( X ) . j i j i i + 1 i + 1 e i k k

The Razumov-Stroganov correspondence For i = 1 . . . 2 n let e i act on link patterns by { i, j } , { i + 1 , k } ∈ X → { i, i + 1 } , { j, k } ∈ e i ( X ) . j i j i i + 1 i + 1 e i k k RS correspondence [RS ’01, Cantini and Sportiello ’10] : � ∀ X, 2 nA X = A Y ( i,Y ) ,e i ( Y )= X

The Razumov-Stroganov correspondence For i = 1 . . . 2 n let e i act on link patterns by { i, j } , { i + 1 , k } ∈ X → { i, i + 1 } , { j, k } ∈ e i ( X ) . j i j i i + 1 i + 1 e i k k RS correspondence [RS ’01, Cantini and Sportiello ’10] : � ∀ X, 2 nA X = A Y ( i,Y ) ,e i ( Y )= X These relations completely characterize the A X . Di Francesco and Zinn Justin had previously obtained results on the solutions of these relations → these are now applicable to the quantities A X .

Link patterns X with nice expressions for A X a b a b c i + j + k − 1 X = � � � A X = i + j + k − 2 i =1 j =1 k =1 c [Zinn-Justin, Zuber, Di Francesco]

Link patterns X with nice expressions for A X a b a b c i + j + k − 1 X = � � � A X = i + j + k − 2 i =1 j =1 k =1 c [Zinn-Justin, Zuber, Di Francesco] A X = Complicated determinant formula X = [Caselli and Krattenthaler ’04, Zinn-Justin ’08]

Link patterns X with nice expressions for A X a b a b c i + j + k − 1 X = � � � A X = i + j + k − 2 i =1 j =1 k =1 c [Zinn-Justin, Zuber, Di Francesco] A X = Complicated determinant formula X = [Caselli and Krattenthaler ’04, Zinn-Justin ’08] A X = A n − 1 . X = [Di Francesco and Z-J, Cantini and Sportiello]

Link patterns with nested arches We consider now integers n, m ≥ 0 , and link patterns with m nested arches, and π is a noncrossing matching with n arches. m π X = π ∪ m We will determine an expression for A π ∪ m , based on FPLs in a triangle. ( → The case m = 0 gives the usual numbers A X .)

Link patterns with nested arches We suppose m ≥ 3 n − 1 , and choose k such that 0 ≤ k ≤ m − (3 n − 1) . m π π 4 n − 2 m − 3 n − k + 1 k

Link patterns with nested arches Fixed edges based on a lemma from [de Gier, ’02]. π k 4 n − 2 m − 3 n − k + 1

Link patterns with nested arches Fixed edges based on a lemma from [de Gier, ’02]. Three regions appear R 2 R 1 T π k 4 n − 2 m − 3 n − k + 1

Link patterns with nested arches We can then write, for m ≥ 3 n − 1 and 0 ≤ k ≤ m − (3 n − 1) � |R 1 ( σ, k ) | × t π A π ∪ m = σ,τ × |R 2 ( τ, m − 3 n − k + 1) | σ,τ R 2 R 1 T τ σ π k m − 3 n − k + 1 • σ, τ are words of length 2 n on { 0 , 1 } ; • R 1 ( σ, . ) , R 2 ( τ, . ) are the sets of FPLs in R 1 and R 2 . • t π σ,τ is the number of FPL configurations in the triangle T .

Words and Shapes Let σ = σ 1 . . . σ p be a word in { 0 , 1 } p ; we write | σ | := p . Words ↔ Ferrers shapes in a box. Example : σ = 0101011110 , so | σ | = 10 , | σ | 0 = 4 , | σ | 1 = 6 . σ 0 1

Words and Shapes Let σ = σ 1 . . . σ p be a word in { 0 , 1 } p ; we write | σ | := p . Words ↔ Ferrers shapes in a box. Example : σ = 0101011110 , so | σ | = 10 , | σ | 0 = 4 , | σ | 1 = 6 . σ 0 1 d ( σ ) := the number of boxes in the diagram σ . σ ∗ := (1 − σ p ) · · · (1 − σ 2 )(1 − σ 1 ) In the example, d ( σ ) = 9 and σ ∗ = 1000010101 .

Words and Shapes σ → σ ′ σ ≤ σ ′ At most one more box per column

Words and Shapes σ → σ ′ σ ≤ σ ′ At most one more box per column A semi standard Young tableau of shape σ and entries bounded by N is a filling of the shape σ by integers in { 1 , . . . , N } such that entries are strictly increasing in columns and weakly increasing in rows.

Words and Shapes σ → σ ′ σ ≤ σ ′ At most one more box per column A semi standard Young tableau of shape σ and entries bounded by N is a filling of the shape σ by integers in { 1 , . . . , N } such that entries are strictly increasing in columns and weakly increasing in rows. The number of such tableaux is given by SSY T ( σ, N ) , an 1 H ( σ ) N d ( σ ) . explicit polynomial in N with leading term (Here H ( σ ) is the product of hook lengths of the shape σ .)

Regions R 1 and R 2 Proposition [Caselli,Krattenthaler,Lass,N. ’05] Let σ be a word of length 2 n , and k ∈ N . There is a bijection between FPLs in R 1 ( σ, k ) and semistandard Young tableaux of shape σ and length n + k . R 2 R 1 T τ σ π k m − 3 n − k + 1

Regions R 1 and R 2 Proposition [Caselli,Krattenthaler,Lass,N. ’05] Let σ be a word of length 2 n , and k ∈ N . There is a bijection between FPLs in R 1 ( σ, k ) and semistandard Young tableaux of shape σ and length n + k . So for m ≥ 3 n − 1 (and k = 0 ) we obtain : � |R 1 ( σ, 0) | · t π A π ∪ m = σ,τ · |R 2 ( τ, m − 3 n + 1) | σ,τ � SSY T ( σ, n ) · t π σ,τ · SSY T ( τ ∗ , m − 2 n + 1) = σ,τ