Refined Cauchy/Littlewood identities and partition functions of the six-vertex model

Transcription

1 Refined Cauchy/Littlewood identities and partition functions of the six-vertex model LPTHE (UPMC Paris 6), CNRS (Collaboration with Dan Betea and Paul Zinn-Justin) 6 June, 4

2 Disclaimer: the word Baxterize does not appear in this talk

3 Aim of talk The ASM conjectures were discovered by Mills, Robbins and Rumsey They express the number of ASMs (with additional symmetries) as simple products After Zeilberger s complicated proof of the original conjecture, Kuperberg found a much simpler proof using the six-vertex model Later on, in a real tour de force, Kuperberg computed partition functions of the six-vertex model on a large set of domains All partition functions were expressed in terms of determinants and Pfaffians Given their determinant and Pfaffian form, it is not surprising that they expand nicely in terms of Schur functions What is much more surprising is that they expand nicely in non determinantal symmetric functions as well The results in this talk allow these partition functions to be written, for example, in the form Γ + (x ; t) Γ + (x n ; t)o(t; u)γ (y n ; t) Γ (y ; t) Γ + (x ; t) Γ + (x n ; t)γ (y n ; t) Γ (y ; t)

4 Schur polynomials and SSYT The Schur polynomials s λ (x,, x n) are the characters of irreducible representations of GL(n) They are given by the Weyl formula: [ det i,j n x λ ] j j+n i s λ (x,, x n ) = i<j n (x i x j ) A semi-standard Young tableau of shape λ is an assignment of one symbol {,, n} to each box of the Young diagram λ, such that The symbols have the ordering < < n The entries in λ increase weakly along each row and strictly down each column The Schur polynomial s λ (x,, x n ) is also given by a weighted sum over semi-standard Young tableaux T of shape λ: s λ (x,, x n ) = T n k= x #(k) k

5 SSYT and sequences of interlacing partitions Two partitions λ and µ interlace, written λ µ, if λ i µ i λ i+ across all parts of the partitions It is the same as saying λ/µ is a horizontal strip One can interpret a SSYT as a sequence of interlacing partitions: T = { λ () λ () λ (n) λ} The correspondence works by peeling away partition λ (k) from T, for all k: T = λ () λ λ (3) λ (4)

6 Plane partitions A plane partition is a set of non-negative integers π(i, j) such that for all i, j π(i, j) π(i +, j) π(i, j) π(i, j + ) Plane partitions can be viewed as an increasing then decreasing sequence of interlacing partitions They are equivalent to conjoined SSYT We define the set π m,n = { λ () λ () λ (m) µ (n) µ () µ () }

7 Cauchy identity and plane partitions The Cauchy identity for Schur polynomials, m n s λ (x,, x m )s λ (y,, y n ) = x λ i= j= i y j can thus be viewed as a generating series of plane partitions: m x λ(i) λ (i ) i π π m,n i= n j= y µ(j) µ (j ) j = m n x i= j= i y j Taking the q-specialization x i = q m i+/ and y j = q n j+/, we recover volume-weighted plane partitions: π π m,n q π = m n q m+n i j+ = m i= j= i= n j= q i+j

8 Symmetric plane partitions Symmetric plane partitions satisfy the condition that π(i, j) = π(j, i) for all i, j A symmetric plane partition is determined by an increasing sequence of interlacing partitions (The decreasing part is obtained from the symmetry) They are in one-to-one correspondence with SSYT

9 Littlewood identities and symmetric plane partitions The three (simplest) Littlewood identities for Schur polynomials s λ (x,, x n) = x λ i<j n i x j s λ (x,, x n ) = x λ even i<j n i x j s λ (x,, x n ) = x λ even i<j n i x j n x i= i n x i= i can each be viewed as generating series for symmetric plane partitions, with a (possible) constraint on the partition forming the main diagonal

10 Hall Littlewood polynomials Hall Littlewood polynomials are t-generalizations of Schur polynomials They can be defined as a sum over the symmetric group: P λ (x,, x n; t) = n σ x λ x i i tx j i v λ (t) x σ S n i= i<j n i x j Alternatively, the Hall Littlewood polynomial P λ (x,, x n; t) is given by a weighted sum over semi-standard Young tableaux T of shape λ: P λ (x,, x n ; t) = T n k= ( x #(k) k ψ λ (k) /λ (k )(t) ) where the function ψ λ/µ (t) is given by ψ λ/µ (t) = i m i (µ)=m i (λ)+ ( t m i(µ) )

11 Path-weighted plane partitions As Vuletić discovered, the effect of the t-weighting in tableaux has a nice combinatorial interpretation on plane partitions The refinement is that all paths at level k receive a weight of t k Example of a plane partition with weight ( t) 3 ( t ) 4 ( t 3 ) shown below: Level-3 Level- Level-

12 Hall Littlewood Cauchy identity and path-weighted plane partitions The Cauchy identity for Hall Littlewood polynomials, m i (λ) m n ( t j )P λ (x,, x m ; t)p λ (y,, y n ; t) = λ i= j= is thus a generating series of (path-weighted) plane partitions: ( t i ) p i (π) m x λ(i) λ (i ) i π π m,n i i= n j= Taking the same q-specialization as earlier, we obtain π π m,n i ( t i ) p i (π) m n q π = i= j= y µ(j) µ (j ) j = i= j= tq i+j q i+j m tx i y j x i y j n i= j= tx i y j x i y j

13 Littlewood identities for Hall Littlewood polynomials The t-analogues of the previously stated Littlewood identities are λ even m i (λ) i= j even P λ (x,, x n ; t) = λ λ even P λ (x,, x n; t) = i<j n i<j n ( t j )P λ (x,, x n ; t) = i<j n tx i x j x i x j tx i x j x i x j tx i x j x i x j n x i= i n x i= i These can be regarded as generating series for path-weighted symmetric plane partitions Paths which intersect the main diagonal might not have a t-weight

14 t-weighting of symmetric plane partitions λ even i= m i (λ) j even ( t j )P λ (x,, x n ; t)

15 Example (a): Refined Cauchy identity for Schur polynomials Theorem λ n ( ut λi i+n )s λ (x,, x n )s λ (y,, y n ) i= [ ] u + (u t)xi y j = det (x) n (y) n i,j n ( tx i y j )( x i y j ) Proof Expand the entries of the determinant as formal power series, and use Cauchy Binet: [ ] [ ] u + (u t)xi y j det = det ( ut k )x k i i,j n ( tx i y j )( x i y j ) i,j n yk j k= = k > >k n i= n [ ( ut k i ) det i,j n x k j i The proof follows after the change of indices k i = λ i i + n ] det i,j n [ y k i j ]

16 Example (b): Refined Cauchy identity for Hall Littlewood polynomials Define C n (t; u) = λ m i (λ) ( ut k ) ( t j )P λ (x,, x n ; t)p λ (y,, y n ; t) m (λ) k= i= j= Theorem C n(t; u) = n i,j= ( tx iy j ) (x) n (y) n The specialization u = t is particularly nice: λ i= j= [ ] u + (u t)xi y j det i,j n ( tx i y j )( x i y j ) m i (λ) ( t j )P λ (x,, x n ; t)p λ (y,, y n ; t) = n i,j= ( tx iy j ) (x) n (y) n [ ] ( t) det i,j n ( tx i y j )( x i y j )

17 Example (b): Refined Cauchy identity for Hall Littlewood polynomials Question: What does the refinement do at the level of plane partitions?

18 Example (b): Refined Cauchy identity for Hall Littlewood polynomials Answer: The zero-height entries are treated like the rest

19 Example (c): Refined Cauchy identity for Macdonald polynomials The Cauchy identity for Macdonald polynomials is where b λ (q, t)p λ (x,, x n ; q, t)p λ (y,, y n ; q, t) = λ n i,j= (tx i y j ; q) (x i y j ; q) (x; q) = q k=( k x), b λ (q, t) = q a(s) t l(s)+ q a(s)+ t l(s) s λ Theorem (Kirillov Noumi,Warnaar) λ n ( uq λ i t n i )b λ (q, t)p λ (x,, x n; q, t)p λ (y,, y n; q, t) = i= n i,j= (tx i y j ; q) (x i y j ; q) n i,j= ( x iy j ) i<j n (x i x j )(y i y j ) [ ] u + (u t)xi y j det i,j n ( tx i y j )( x i y j ) Proof Act on the Cauchy identity with a generating series of Macdonald s difference operators The left hand side follows immediately The right hand side follows after acting on the Cauchy kernel, and performing some manipulation

20 Example (a): Refined Littlewood identity for Schur polynomials Theorem λ even n ( ut λi i+n )s λ (x,, x n ) i= [ ] ( u + (u t)xi x j )(x i x j ) = Pf (x i<j n i x j ) i<j n ( tx i x j )( x i x j ) Proof Expand the entries of the Pfaffian and use a Pfaffian analogue of Cauchy Binet: Pf i<j n i<j n δ l,k+ ( ut k )(x l i xk j xk i xl j ) = k > >k n k<l [ ] Pf δ ki,k i<j n j +( ut k i ) det i,j n The Pfaffian in the sum factorizes, to produce the correct (blue) factor and the restriction on the summation [ x k j i ]

21 Example (b): Refined Littlewood identity for Hall Littlewood polynomials Define L n (t; u) = λ even m (λ) ( ut k ) m i (λ) k even i= j even ( t j )P λ (x,, x n ; t) Theorem (DB,MW,PZJ) L n (t; u) = i<j n ( tx i x j ) (x i x j ) Pf i<j n The specialization u = t is again especially nice: [ ] ( u + (u t)xi x j )(x i x j ) ( tx i x j )( x i x j ) λ even m i (λ) i= j even = ( t j )P λ (x,, x n ; t) i<j n ( tx i x j ) (x i x j ) [ Pf i<j n ( t)(x i x j ) ( tx i x j )( x i x j ) ]

22 Example (b): Refined Littlewood identity for Hall Littlewood polynomials At the level of plane partitions, this is (again) a very simple refinement

23 Example (b): Refined Littlewood identity for Hall Littlewood polynomials At the level of plane partitions, this is (again) a very simple refinement

24 Example (c): Refined Littlewood identity for Macdonald polynomials The most fundamental Littlewood identity for Macdonald polynomials is where λ even Conjecture (DB,MW,PZJ) λ even i= i<j n b el λ (q, t)p λ(x,, x n ; q, t) = b el λ (q, t) = s λ l(s) even i<j n q a(s) t l(s)+ q a(s)+ t l(s) n ( uq λ i t n i )b el λ (q, t)p λ(x,, x n ; q, t) = (tx i x j ; q) (x i x j ; q) i<j n ( x i x j ) (x i x j ) Pf i<j n (tx i x j ; q) (x i x j ; q) [ ] ( u + (u t)xi x j )(x i x j ) ( tx i x j )( x i x j )

25 The six-vertex model The vertices of the six-vertex model are x x x y y y a + (x, y) b + (x, y) c + (x, y) x x x y y y a (x, y) b (x, y) c (x, y)

26 The six-vertex model The Boltzmann weights are given by a + (x, y) = tx/y x/y b + (x, y) = t c + (x, y) = ( t) x/y a (x, y) = tx/y x/y b (x, y) = t c (x, y) = ( t)x/y x/y The parameter t from Hall Littlewood is now the crossing parameter of the model The Boltzmann weights obey the Yang Baxter equations (the U q (ŝl ) solution): x y z = x y z

27 Boundary vertices We also require corner vertices which do not depend on a spectral parameter and behave like sources/sinks The corner vertices satisfy a reflection equation: ȳ x ȳ x x y x y = y = y x x ȳ x ȳ x

28 Domain wall boundary conditions The six-vertex model on a lattice with domain wall boundary conditions was first considered by Korepin: x x x 3 x 4 x x 6 ȳ 6 ȳ ȳ 4 ȳ 3 ȳ ȳ This partition function is of fundamental importance in periodic quantum spin chains based on Y(sl ) and U q (ŝl )

29 Domain wall boundary conditions Configurations on this lattice are in one-to-one correspondence with alternating sign matrices: The domain wall partition function was evaluated in determinant form by Izergin: Z ASM (x,, x n ; y,, y n ; t) = n i,j= ( tx [ ] iy j ) i<j n (x i x j )(y i y j ) det ( t) ( tx i y j )( x i y j ) i,j n The DWPF is equal to the right hand side of a refined Cauchy identity: Z ASM (x,, x n; y,, y n; t) = C n(t; t)

30 Half-turn symmetry One can consider those configurations under domain wall boundary conditions which have 8 rotational symmetry The fundamental domain is given by: x 3 x x x x x 3 ȳ 3 ȳ ȳ

31 Half-turn symmetry Configurations on this lattice are in one-to-one correspondence with half-turn symmetric alternating sign matrices: Kuperberg evaluated this partition function as a product of determinants: n i,j= Z HT (x,, x n ; y,, y n ; t) = ( tx iy j ) i<j n (x i x j ) (y i y j ) [ ] [ ] ( t) ( + t)( txi y j ) det det i,j n ( tx i y j )( x i y j ) i,j n ( tx i y j )( x i y j ) In other words, Z HT (x,, x n ; y,, y n ; t) = C n (t; t)c n (t; t)

32 Off-diagonally symmetric boundary conditions Off-diagonally symmetric boundary conditions were introduced by Kuperberg One considers domain wall configurations with reflection symmetry about a diagonal axis, and which have no c vertices on that diagonal The fundamental domain is x 6 x x 4 x 3 x x x x x 3 x 4 x x 6

33 Off-diagonally symmetric boundary conditions Configurations on this lattice are in one-to-one correspondence with off-diagonally symmetric ASMs (OSASMs): Kuperberg evaluated this partition function as a Pfaffian: Z OSASM (x,, x n ; t) = i<j n [ ] ( tx i x j ) (x i x j ) Pf ( t)(x i x j ) ( tx i x j )( x i x j ) i<j n The OSASM partition function is equal to the right hand side of a refined Littlewood identity: Z OSASM (x,, x n ; t) = L n (t; t)

34 Off-diagonally/off-anti-diagonally symmetric boundary conditions Similarly, one can consider domain wall configurations with reflection symmetry in both diagonals, and with no c vertices on those diagonals The fundamental domain is x 4 x 3 x x x x x 3 x x x 4 x 3 x 4

35 Off-diagonally/off-anti-diagonally symmetric boundary conditions Configurations on this lattice are in one-to-one correspondence with off-diagonally/off-anti-diagonally symmetric ASMs (OOSASMs): The partition function can be evaluated as a product of Pfaffians: Z OOSASM (x,, x n ; t) = [ Pf i<j n In other words, ( t)(x i x j ) ( tx i x j )( x i x j ) i<j n ] Pf i<j n ( tx i x j ) (x i x j ) [ ] ( + t)( txi x j )(x i x j ) ( tx i x j )( x i x j ) Z OOSASM (x,, x n ; t) = L n (t; t)l n (t; t)

36 Open questions Expansion of other symmetry classes of ASMs What is the missing operator needed to prove the conjecture? Do these correspondences have a combinatorial meaning? The similarity of the underlying domains on both sides of these correspondences is very curious Can more general objects in the six-vertex/xxz model (form factors/correlation functions) be expanded nicely in terms of Hall Littlewood polynomials? What about more general models, such as the eight-vertex and 8VSOS models?