Steep tilings and sequences of interlaced partitions

Transcription

1 Steep tilings and sequences of interlaced partitions Jérémie Bouttier Joint work with Guillaume Chapuy and Sylvie Corteel arxiv: Institut de Physique Théorique, CEA Saclay Département de mathématiques et applications, ENS Paris Integrability and Representation Theory seminar University of Illinois at Urbana-Champaign 4 September 014 Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

2 Introduction Plane partition Rhombus tiling Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September 014 / 31

3 Introduction Plane partition Rhombus tiling Sequence of interlaced partitions Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September 014 / 31

4 Introduction An (integer) partition λ is a finite non-increasing sequence of integers λ 1 λ λ 3 λ l > 0 (By convention we set λ i = 0 for i l.) We say that λ and µ are (horizontally) interlaced, and denote λ µ, iff λ 1 µ 1 λ µ λ 3 Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

5 Introduction An (integer) partition λ is a finite non-increasing sequence of integers λ 1 λ λ 3 λ l > 0 (By convention we set λ i = 0 for i l.) We say that λ and µ are (horizontally) interlaced, and denote λ µ, iff λ 1 µ 1 λ µ λ 3 Plane partitions correspond to sequences of interlaced partitions: λ ( ) λ ( 1) λ (0) λ (1) λ () with λ (i) = for i large enough. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

6 Introduction Define the size of a partition/plane partition as the sum of its entries. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

7 Introduction Define the size of a partition/plane partition as the sum of its entries. plane partitions q size = k=1 1 (1 q k ) k [McMahon] Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

8 Introduction Define the size of a partition/plane partition as the sum of its entries. plane partitions q size = k=1 1 (1 q k ) k [McMahon] The sequence of interlaced partitions corresponding to a random plane partition drawn with probability proportional to q size (0 q < 1) forms a Schur process [Okounkov-Reshetikin 003]. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

9 Introduction Define the size of a partition/plane partition as the sum of its entries. plane partitions q size = k=1 1 (1 q k ) k [McMahon] The sequence of interlaced partitions corresponding to a random plane partition drawn with probability proportional to q size (0 q < 1) forms a Schur process [Okounkov-Reshetikin 003]. For instance the state λ (0) of the main diagonal is drawn with probability proportional to ( s λ (0)(q 1/, q 3/, q 5/,...)). Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

10 Introduction Define the size of a partition/plane partition as the sum of its entries. plane partitions q size = k=1 1 (1 q k ) k [McMahon] The sequence of interlaced partitions corresponding to a random plane partition drawn with probability proportional to q size (0 q < 1) forms a Schur process [Okounkov-Reshetikin 003]. For instance the state λ (0) of the main diagonal is drawn with probability proportional to ( s λ (0)(q 1/, q 3/, q 5/,...)). How about tilings made of dominos instead of rhombi? Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

11 Motivations In general: statistical mechanics: rhombus/domino tilings = dimer model on honeycomb/square lattice enumerative combinatorics: beautiful enumeration formulas probability theory: determinantal correlations, limit shape phenomena, interesting limiting processes related to random matrices algebraic geometry: Donaldson-Thomas theory For our specific work: understand precisely the connection between domino tilings and interlaced partitions, implicitly hinted at in works of Johansson, Borodin, etc. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

12 Motivations In general: statistical mechanics: rhombus/domino tilings = dimer model on honeycomb/square lattice enumerative combinatorics: beautiful enumeration formulas probability theory: determinantal correlations, limit shape phenomena, interesting limiting processes related to random matrices algebraic geometry: Donaldson-Thomas theory For our specific work: understand precisely the connection between domino tilings and interlaced partitions, implicitly hinted at in works of Johansson, Borodin, etc. Have fun with vertex operators (a recreation after a reading group on the works from the Kyoto school: solitons, infinite dimensional Lie algebras and all that). Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

13 Outline 1 Steep tilings Bijection with sequences of interlaced partitions 3 Enumeration via the vertex operator formalism Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

14 Outline 1 Steep tilings Bijection with sequences of interlaced partitions 3 Enumeration via the vertex operator formalism Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

15 Steep tilings y = x y = x l A domino tiling of the oblique strip x l y x Steepness condition: we eventually find only north or east dominos in the NE direction, south or west in the SW direction. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

16 Steep tilings y = x y = x l A domino tiling of the oblique strip x l y x Steepness condition: we eventually find only north or east dominos in the NE direction, south or west in the SW direction. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

17 Steep tilings y = x y = x l A domino tiling of the oblique strip x l y x Steepness condition: we eventually find only north or east dominos in the NE direction, south or west in the SW direction. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

18 Steep tilings w = ( ) The steepness condition implies that the tiling is eventually periodic in both directions. The two repeated patterns define the asymptotic data w {+, } l of the tiling. For fixed w there is a unique (up to translation) minimal tiling which is periodic from the start. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

19 Examples Domino tilings of the Aztec diamond [Elkies et al.] Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

20 Examples (0, 0) Domino tilings of the Aztec diamond [Elkies et al.] correspond to steep tilings with asymptotic data Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

21 Examples (a) (b) Pyramid partitions [Kenyon, Szendrői, Young] Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

22 Examples (a) (b) (c) Pyramid partitions [Kenyon, Szendrői, Young] Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

23 Examples Pyramid partitions [Kenyon, Szendrői, Young] correspond to steep tilings with asymptotic data Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

24 Outline 1 Steep tilings Bijection with sequences of interlaced partitions 3 Enumeration via the vertex operator formalism Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

25 Particle configurations N E S W To each steep tiling we may associate a particle configuration by filling each square covered by a N or E domino with a white particle, and each square covered by a S or W domino with a black particle. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

26 Particle configurations To each steep tiling we may associate a particle configuration by filling each square covered by a N or E domino with a white particle, and each square covered by a S or W domino with a black particle. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

27 Integer partitions Particles along a diagonal form a Maya diagram which codes an integer partition (here 41). Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

28 Interlacing of particles Between two successive even/odd diagonals, the white particles must be adjacent. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

29 Interlacing of particles Between two successive even/odd diagonals, the white particles must be adjacent. Conversely, between two successive odd/even diagonals, the black particles must be adjacent. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

30 Interlacing of partitions Between two successive even/odd diagonals, a finite number of white particles can be moved one site to the left (+) or to the right ( ) in the Maya diagram (depending on asymptotic data). This corresponds to adding/removing a horizontal strip to the associated partition. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

31 Interlacing of partitions Between two successive even/odd diagonals, a finite number of white particles can be moved one site to the left (+) or to the right ( ) in the Maya diagram (depending on asymptotic data). This corresponds to adding/removing a horizontal strip to the associated partition. Conversely, between two successive odd/even diagonals, a vertical strip is added/removed. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

32 Interlacing of partitions For λ, µ two integer partitions, the following are equivalent: λ/µ is a horizontal strip, λ 1 µ 1 λ µ λ 3, λ i µ i {0, 1} for all i. Notation: λ µ or µ λ. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

33 Interlacing of partitions For λ, µ two integer partitions, the following are equivalent: λ/µ is a horizontal strip, λ 1 µ 1 λ µ λ 3, λ i µ i {0, 1} for all i. Notation: λ µ or µ λ. Similarly, the following are equivalent: λ/µ is a vertical strip, λ 1 µ 1 λ µ λ 3, λ i µ i {0, 1} for all i. Notation: λ µ or µ λ. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

34 The fundamental bijection For a fixed word w {+, } l, there is a one-to-one correspondence between steep tilings of asymptotic data w and sequences of partitions (λ (0), λ (1),..., λ (l) ) satisfying for all k = 1,..., l: λ (k ) λ (k 1) if w k 1 = +, and λ (k ) λ (k 1) if w k 1 =, λ (k 1) λ (k) if w k = +, and λ (k 1) λ (k) if w k =. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

35 The fundamental bijection For a fixed word w {+, } l, there is a one-to-one correspondence between steep tilings of asymptotic data w and sequences of partitions (λ (0), λ (1),..., λ (l) ) satisfying for all k = 1,..., l: λ (k ) λ (k 1) if w k 1 = +, and λ (k ) λ (k 1) if w k 1 =, λ (k 1) λ (k) if w k = +, and λ (k 1) λ (k) if w k =. Examples: Aztec diamond: = λ (0) λ (1) λ () λ (3) λ (4) λ (l) =, Pyramid partitions: = λ (0) λ (1) λ () λ (l) λ (l) =. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

36 The fundamental bijection For a fixed word w {+, } l, there is a one-to-one correspondence between steep tilings of asymptotic data w and sequences of partitions (λ (0), λ (1),..., λ (l) ) satisfying for all k = 1,..., l: λ (k ) λ (k 1) if w k 1 = +, and λ (k ) λ (k 1) if w k 1 =, λ (k 1) λ (k) if w k = +, and λ (k 1) λ (k) if w k =. Examples: Aztec diamond: = λ (0) λ (1) λ () λ (3) λ (4) λ (l) =, Pyramid partitions: = λ (0) λ (1) λ () λ (l) λ (l) =. The size of λ (m) is equal to the number of flips on diagonal m in any shortest sequence of flips between the tiling at hand and the minimal tiling. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

37 The fundamental bijection For a fixed word w {+, } l, there is a one-to-one correspondence between steep tilings of asymptotic data w and sequences of partitions (λ (0), λ (1),..., λ (l) ) satisfying for all k = 1,..., l: λ (k ) λ (k 1) if w k 1 = +, and λ (k ) λ (k 1) if w k 1 =, λ (k 1) λ (k) if w k = +, and λ (k 1) λ (k) if w k =. Examples: Aztec diamond: = λ (0) λ (1) λ () λ (3) λ (4) λ (l) =, Pyramid partitions: = λ (0) λ (1) λ () λ (l) λ (l) =. The size of λ (m) is equal to the number of flips on diagonal m in any shortest sequence of flips between the tiling at hand and the minimal tiling. Under natural statistics we obtain a Schur process [Okounkov-Reshetikhin]. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

38 Flips Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

39 Outline 1 Steep tilings Bijection with sequences of interlaced partitions 3 Enumeration via the vertex operator formalism Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

40 Transfer matrices Enumerating sequences of interlaced partitions is done via transfer matrices, which are here vertex operators : { t µ λ if λ µ λ Γ + (t) µ = µ Γ (t) λ = 0 otherwise { λ Γ +(t) µ = µ Γ t µ λ if λ µ (t) λ = 0 otherwise Example: Aztec diamond: Γ + (z 1 )Γ (z )Γ + (z 3 )Γ (z 4 ) Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

41 Bosonic representation The transfer matrices can be rewritten as Γ ± (t) = exp k 1 t k k α ±k, Γ ±(t) = exp k 1 ( 1) k 1 t k k α ±k where [α n, α m ] = nδ n+m. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September 014 / 31

42 Bosonic representation The transfer matrices can be rewritten as Γ ± (t) = exp k 1 t k k α ±k, Γ ±(t) = exp k 1 ( 1) k 1 t k k α ±k where [α n, α m ] = nδ n+m. This implies that Γ s with the same sign commute, and that we have the following nontrivial commutation relations: λ Γ + (t)γ (u) = 1 1 tu Γ (u)γ + (t) Γ + (t)γ (u) = (1 + tu)γ (u)γ + (t) σ k µ τ σ λ µ k = 0, 1,,... k = 0, 1 k τ Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September 014 / 31

43 Super Schur functions When w consists only of + s, the partition function with fixed boundary conditions is a so-called super Schur function µ Γ + (x 1 )Γ +(y 1 )Γ + (x )Γ +(y ) λ = S λ/µ (x 1, x,... ; y 1, y,...). Super Schur functions may be combinatorially defined in terms of super semistandard tableaux or (reverse) plane overpartitions: Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

44 Pure steep tilings For general asymptotic data and pure ( and ) boundary conditions the partition function is readily evaluated from the commutation relations. Γ + (z 1 ) Γ +(z ) Γ (z 3 ) Γ (z 4 ) Γ Γ + (z 1 ) Γ (z 3 ) Γ (z 4 ) +(z ) (1 + z 1 z 4 )(1 + z z 3 ) (1 z 1 z 3 )(1 z z 4 ) = 1 k 13 k 3 k 14 3 k 4 4 k 13, k 4 = 0, 1,,... k 3, k 14 = 0, 1 Equivalently we have a RSK-type bijection between pure steep tilings and suitable fillings of the Young diagram associated with w. Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

45 Pure steep tilings For a general word w and the q flip specialization, the partition function of pure steep tilings is given by a hook-length type formula: { T w (q) = ϕ i,j (q j i 1 + x if j i odd ), ϕ i,j (x) = 1/(1 x) if j i even 1 i<j l w i =+, w j = q q 6 w = Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

46 Aztec diamonds and pyramids 1 + q 1 + q 1 + q 1 + q q q q q 1 q 1 + q q q q 4 1 q q 5 Aztec diamond w = [Elkies et al., Stanley] T w (q) = (1 + q) 3 (1 + q 3 ) (1 + q 5 ) Pyramid partitions w = Case l [Young]: T w (q) = k 1 (1 + q k 1 ) k 1 (1 q k ) k Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

47 Free boundaries We may also obtain a closed-form formula for the partition function in the case of free boundary conditions v = λ v λ λ thanks to the reflection relations Γ + (t) v = 1 1 tv Γ (tv ) v Γ +(t) v = 1 1 tv Γ (tv ) v σ k λ µ k = 0, 1,,... Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

48 Free boundaries Example: w = u Γ + (y 1 )Γ +(y )Γ + (y 3 )Γ +(y 4 ) v = ( 1 l 1 1 u k v k 1 u k 1 v k y i k=1 i=1 ) ϕ i,j (u k v k y i y j ) 1 i<j l u Γ (i) + (y i ) Γ (j) + (y j ) Γ (i) (v y i ) Γ (j) (v y j ) v Γ (i) + (u v y i ) Γ (j) + (u v y j ) Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

49 Periodic boundary conditions When identifying the left and right boundaries we obtain a cylindric steep tiling. The corresponding sequence of interlaced partitions form a periodic Schur process [Borodin]. The partition function may still be written as an infinite product. Tr [ Γ + (z 1 ) Γ +(z ) Γ (z 3 ) Γ (z 4 ) q H ] = Tr [ Γ + (qz 1 )Γ +(qz ) Γ (z 3 ) Γ (z 4 ) q H ] Example: w = + + [ Tr Γ + (z 1 )Γ +(z )Γ (z 3 )Γ (z 4 )q H] = (1 + z 1 z 4 )(1 + z z 3 ) (1 z 1 z 3 )(1 z z 4 ) k=1 (1 + q k 1 x 1 x 4 )(1 + q k 1 x x 3 ) (1 q k )(1 q k 1 x 1 x 3 )(1 q k 1 x x 4 ) Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

50 Further work Correlation functions [joint with C. Boutillier and S. Ramassamy]: straightforward to compute for particles in the pure case, thanks to their free fermionic nature less trivially we deduce an explicit expression for the inverse Kasteleyn matrix, which yields domino correlations more involved in the periodic case [Borodin], how about free boundary case? Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

51 Further work Correlation functions [joint with C. Boutillier and S. Ramassamy]: straightforward to compute for particles in the pure case, thanks to their free fermionic nature less trivially we deduce an explicit expression for the inverse Kasteleyn matrix, which yields domino correlations more involved in the periodic case [Borodin], how about free boundary case? Random generation and limit shapes [joint with D. Betea and M. Vuletić] Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

52 Further work More general setting [BBCCR]: Rail Yard Graphs (interpolate between lozenge and domino tilings) connection with octahedron recurrence/cluster algebras? deformations? (e.g. Schur McDonald) all covered y = 0 none covered } } R+. L+. R. R+. L. R... l 1 r+1 } none covered y = 0 } all covered Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31

53 Further work More general setting [BBCCR]: Rail Yard Graphs (interpolate between lozenge and domino tilings) connection with octahedron recurrence/cluster algebras? deformations? (e.g. Schur McDonald) all covered y = 0 none covered } } R+. L+. R. R+. L. R... l 1 r+1 } none covered y = 0 } all covered Thanks for your attention! Jérémie Bouttier (IPhT/DMA) Steep tilings and interlaced partitions 4 September / 31