Refined Cauchy/Littlewood identities and partition functions of the six-vertex model Michael Wheeler LPTHE (UPMC Paris 6), CNRS (Collaboration with Dan Betea and Paul Zinn-Justin ) 26 June, 2014 . . . . . . Michael Wheeler Refined Cauchy and Littlewood identities
Disclaimer: the word Baxterize does not appear in this talk. . . . . . . Michael Wheeler Refined Cauchy and Littlewood identities
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 ⟨ 0 | Γ + ( x 1 ; t ) . . . Γ + ( x n ; t ) O ( t ; u )Γ − ( y n ; t ) . . . Γ − ( y 1 ; t ) | 0 ⟩ ⟨ 0 | Γ + ( x 1 ; t ) . . . Γ + ( x n ; t )Γ − ( y n ; t ) . . . Γ − ( y 1 ; t ) | 0 ⟩ . . . . . . Michael Wheeler Refined Cauchy and Littlewood identities
Schur polynomials and SSYT The Schur polynomials s λ ( x 1 , . . . , x n ) are the characters of irreducible representations of GL ( n ) . They are given by the Weyl formula: [ ] λ j − j + n det 1 ⩽ i,j ⩽ n x i s λ ( x 1 , . . . , x n ) = ∏ 1 ⩽ i<j ⩽ n ( x i − x j ) A semi-standard Young tableau of shape λ is an assignment of one symbol { 1 , . . . , n } to each box of the Young diagram λ , such that . . The symbols have the ordering 1 < · · · < n . 1 . . The entries in λ increase weakly along each row and strictly down each column. 2 The Schur polynomial s λ ( x 1 , . . . , x n ) is also given by a weighted sum over semi-standard Young tableaux T of shape λ : n ∑ ∏ x #( k ) s λ ( x 1 , . . . , x n ) = k T k =1 . . . . . . Michael Wheeler Refined Cauchy and Littlewood identities
SSYT and sequences of interlacing partitions Two partitions λ and µ interlace , written λ ≻ µ , if λ i ⩾ µ i ⩾ λ i +1 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 (0) ≺ λ (1) ≺ · · · ≺ λ ( n ) ≡ λ } The correspondence works by “peeling away” partition λ ( k ) from T , for all k : 1 1 2 2 4 2 2 3 3 3 4 .4 . . . . . . . . . . . . . . λ (1) ≺ λ (2) ≺ λ (3) ≺ λ (4) T = . . . . . . . . Michael Wheeler Refined Cauchy and Littlewood identities
Plane partitions A plane partition is a set of non-negative integers π ( i, j ) such that for all i, j ⩾ 1 π ( i, j ) ⩾ π ( i + 1 , j ) π ( i, j ) ⩾ π ( i, j + 1) 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 (0) ≺ λ (1) ≺ · · · ≺ λ ( m ) ≡ µ ( n ) ≻ · · · ≻ µ (1) ≻ µ (0) ≡ ∅} . . . . . . Michael Wheeler Refined Cauchy and Littlewood identities
Cauchy identity and plane partitions The Cauchy identity for Schur polynomials, m n ∑ ∏ ∏ 1 s λ ( x 1 , . . . , x m ) s λ ( y 1 , . . . , y n ) = 1 − x i y j λ i =1 j =1 can thus be viewed as a generating series of plane partitions: m n m n ∑ ∏ ∏ ∏ ∏ 1 x | λ ( i ) |−| λ ( i − 1) | y | µ ( j ) |−| µ ( j − 1) | = i j 1 − x i y j π ∈ π m,n i =1 j =1 i =1 j =1 Taking the q -specialization x i = q m − i +1 / 2 and y j = q n − j +1 / 2 , we recover volume-weighted plane partitions: m n m n ∑ ∏ ∏ ∏ ∏ 1 1 q | π | = 1 − q m + n − i − j +1 = 1 − q i + j − 1 π ∈ π m,n i =1 j =1 i =1 j =1 . . . . . . Michael Wheeler Refined Cauchy and Littlewood identities
Symmetric plane partitions Symmetric plane partitions satisfy the condition that π ( i, j ) = π ( j, i ) for all i, j ⩾ 1 . 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. . . . . . . Michael Wheeler Refined Cauchy and Littlewood identities
Littlewood identities and symmetric plane partitions The three (simplest) Littlewood identities for Schur polynomials n ∑ ∏ ∏ 1 1 s λ ( x 1 , . . . , x n ) = 1 − x i x j 1 − x i λ 1 ⩽ i<j ⩽ n i =1 n ∑ ∏ ∏ 1 1 s λ ( x 1 , . . . , x n ) = 1 − x 2 1 − x i x j i i =1 λ even 1 ⩽ i<j ⩽ n ∑ ∏ 1 s λ ( x 1 , . . . , x n ) = 1 − x i x j λ ′ even 1 ⩽ i<j ⩽ n can each be viewed as generating series for symmetric plane partitions, with a (possible) constraint on the partition forming the main diagonal. . . . . . . Michael Wheeler Refined Cauchy and Littlewood identities
Hall–Littlewood polynomials Hall–Littlewood polynomials are t -generalizations of Schur polynomials. They can be defined as a sum over the symmetric group: n ∑ ∏ ∏ x i − tx j 1 x λ i P λ ( x 1 , . . . , x n ; t ) = σ i v λ ( t ) x i − x j σ ∈ S n i =1 1 ⩽ i<j ⩽ n Alternatively, the Hall–Littlewood polynomial P λ ( x 1 , . . . , x n ; t ) is given by a weighted sum over semi-standard Young tableaux T of shape λ : n ( ) ∑ ∏ x #( k ) P λ ( x 1 , . . . , x n ; t ) = ψ λ ( k ) /λ ( k − 1) ( t ) k T k =1 where the function ψ λ/µ ( t ) is given by ( 1 − t m i ( µ ) ) ∏ ψ λ/µ ( t ) = i ⩾ 1 m i ( µ )= m i ( λ )+1 . . . . . . Michael Wheeler Refined Cauchy and Littlewood identities
Path-weighted plane partitions As Vuleti´ c 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 1 − t k . Example of a plane partition with weight (1 − t ) 3 (1 − t 2 ) 4 (1 − t 3 ) 2 shown below: Level-3 Level-2 Level-1 . . . . . . . . . Michael Wheeler Refined Cauchy and Littlewood identities
Hall–Littlewood Cauchy identity and path-weighted plane partitions The Cauchy identity for Hall–Littlewood polynomials, ∞ m i ( λ ) m n ∑ ∏ ∏ ∏ ∏ 1 − tx i y j (1 − t j ) P λ ( x 1 , . . . , x m ; t ) P λ ( y 1 , . . . , y n ; t ) = 1 − x i y j λ i =1 j =1 i =1 j =1 is thus a generating series of (path-weighted) plane partitions: m n m n ∑ ∏ ∏ ∏ ∏ ∏ ( 1 − t i ) p i ( π ) 1 − tx i y j x | λ ( i ) |−| λ ( i − 1) | y | µ ( j ) |−| µ ( j − 1) | = i j 1 − x i y j π ∈ π m,n i ⩾ 1 i =1 j =1 i =1 j =1 Taking the same q -specialization as earlier, we obtain m n ∑ ∏ ∏ ∏ 1 − tq i + j − 1 ( 1 − t i ) p i ( π ) q | π | = 1 − q i + j − 1 π ∈ π m,n i ⩾ 1 i =1 j =1 . . . . . . Michael Wheeler Refined Cauchy and Littlewood identities
Littlewood identities for Hall–Littlewood polynomials The t -analogues of the previously stated Littlewood identities are n ∑ ∏ ∏ 1 − tx i x j 1 P λ ( x 1 , . . . , x n ; t ) = 1 − x i x j 1 − x i λ 1 ⩽ i<j ⩽ n i =1 n ∑ ∏ ∏ 1 − tx i x j 1 P λ ( x 1 , . . . , x n ; t ) = 1 − x 2 1 − x i x j i λ even 1 ⩽ i<j ⩽ n i =1 m i ( λ ) ∞ ∑ ∏ ∏ ∏ 1 − tx i x j (1 − t j − 1 ) P λ ( x 1 , . . . , x n ; t ) = 1 − x i x j λ ′ even i =1 j even 1 ⩽ i<j ⩽ n 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. . . . . . . Michael Wheeler Refined Cauchy and Littlewood identities
t -weighting of symmetric plane partitions 6 6 6 5 6 5 5 5 5 5 4 5 4 5 4 2 4 4 4 4 2 2 2 4 4 4 2 2 2 2 4 4 2 2 1 2 1 2 1 1 1 1 1 1 1 1 0 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∞ m i ( λ ) ∑ ∏ ∏ (1 − t j − 1 ) P λ ( x 1 , . . . , x n ; t ) λ ′ even i =1 j even . . . . . . Michael Wheeler Refined Cauchy and Littlewood identities
Example 1(a): Refined Cauchy identity for Schur polynomials . Theorem . n ∑ ∏ (1 − ut λ i − i + n ) s λ ( x 1 , . . . , x n ) s λ ( y 1 , . . . , y n ) i =1 λ [ 1 − u + ( u − t ) x i y j ] 1 = det (1 − tx i y j )(1 − x i y j ) ∆( x ) n ∆( y ) n 1 ⩽ i,j ⩽ n . . Proof. . Expand the entries of the determinant as formal power series, and use Cauchy–Binet: [ ∞ ] [ 1 − u + ( u − t ) x i y j ] ∑ (1 − ut k ) x k i y k det = det j (1 − tx i y j )(1 − x i y j ) 1 ⩽ i,j ⩽ n 1 ⩽ i,j ⩽ n k =0 n [ ] [ ] ∑ ∏ k j y k i (1 − ut k i ) = det x det i j 1 ⩽ i,j ⩽ n 1 ⩽ i,j ⩽ n i =1 k 1 > ··· >k n ⩾ 0 The proof follows after the change of indices k i = λ i − i + n . . . . . . . . Michael Wheeler Refined Cauchy and Littlewood identities
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