Six-vertex model partition functions and symmetric polynomials of - PowerPoint PPT Presentation

Six-vertex model partition functions and symmetric polynomials of type BC Dan Betea LPMA (UPMC Paris 6), CNRS (Collaboration with Michael Wheeler, Paul Zinn-Justin ) Iunius XXVI, MMXIV

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Outline ◮ Symplectic Schur polynomials ◮ Symplectic Cauchy identity and plane partitions ◮ Refined symplectic Cauchy identity ◮ BC Hall–Littlewood polynomials and another refined conjectural Cauchy identity ◮ Six-vertex model with reflecting boundary (UASMs, UUASMs) ◮ Putting it all together ◮ Conclusion

Symplectic Schur polynomials (aka symplectic characters) The symplectic Schur polynomials sp λ ( x 1 , ¯ x 1 , . . . , x n , ¯ x n ) are the irreducible characters x = 1 of Sp (2 n ) . Weyl gives ( ¯ x ) λ j − j + n +1 λ j − j + n +1 � � det x − ¯ x i i 1 � i,j � n sp λ ( x 1 , ¯ x 1 , . . . , x n , ¯ x n ) = � n x i ) � i =1 ( x i − ¯ 1 � i<j � n ( x i − x j )(1 − ¯ x i ¯ x j ) A symplectic tableau of shape λ on the alphabet 1 < ¯ 1 < · · · < n < ¯ n is a SSYT with the extra condition that all entries in row k of λ are at least k . 1 1 ¯ 1 ¯ 2 3 2 ¯ 2 3 ¯ 3 ¯ 3 4 4 n x #( k ) − #(¯ k ) � � sp λ ( x 1 , ¯ x 1 , . . . , x n , ¯ x n ) = k k =1 T

Symplectic tableaux as interlacing sequences of partitions λ (0) ≺ λ (1) ≺ ¯ λ (1) ≺ · · · ≺ λ ( n ) ≺ ¯ λ ( n ) ≡ λ | ℓ (¯ T = {∅ ≡ ¯ λ ( i ) ) � i } ¯ ¯ 1 1 1 2 3 ¯ 2 3 2 ¯ ¯ 4 3 3 4 Example: T = {∅ ≺ (2) ≺ (3) ≺ (3 , 1) ≺ (4 , 2) ≺ (5 , 3) ≺ (5 , 3 , 2) ≺ (5 , 3 , 3 , 1) ≺ (5 , 3 , 3 , 1) } n n x | λ ( i ) |−| ¯ λ ( i − 1) | x | λ ( j ) |−| ¯ λ ( j ) | � � � sp λ ( x 1 , ¯ x 1 , . . . , x n , ¯ x n ) = i j i =1 j =1 T n x 2 | λ ( i ) |−| ¯ λ ( i ) |−| ¯ λ ( i − 1) | � � = i i =1 T

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Plane partitions from SSYT + symplectic tableaux The set of such plane partitions is (Schur left, symplectic Schur right): π m, 2 n = {∅ ≡ λ (0) ≺ λ (1) ≺ · · · ≺ λ ( m ) ≡ ¯ µ ( n ) ≻ µ ( n ) ≻ · · · ≻ ¯ µ (1) ≻ µ (1) ≻ ¯ µ (0) ≡ ∅} (2) ≺ (4 , 2) ≺ (5 , 3 , 2) ≺ (7 , 5 , 3 , 1) ≻ (6 , 5 , 3 , 1) ≻ (5 , 4 , 3) ≻ (4 , 4 , 2) ≻ (4 , 2) ≻ (2 , 1) ≻ (2) ≻ (1)

Symplectic Cauchy identity and associated plane partitions The Cauchy identity for symplectic Schur polynomials, � 1 � i<j � m (1 − x i x j ) � s λ ( x 1 , . . . , x m ) sp λ ( y 1 , ¯ y 1 , . . . , y n , ¯ y n ) = � m � n j =1 (1 − x i y j )(1 − x i ¯ y j ) i =1 λ can now be regarded as a generating series for the plane partitions defined: m n x | λ ( i ) |−| λ ( i − 1) | y 2 | µ ( j ) |−| ¯ µ ( j ) |−| ¯ µ ( j − 1) | � � � = i j π ∈ π m, 2 n i =1 j =1 � 1 � i<j � m (1 − x i x j ) � m � n j =1 (1 − x i y j )(1 − x i ¯ y j ) i =1 What is a “good” q -specialization? We choose x i = q m − i +3 / 2 , y j = q 1 / 2 , giving 1 � i<j � m (1 − q i + j +1 ) � q | π � | q | π o > |−| π e > | = � � m i =1 (1 − q i ) n (1 − q i +1 ) n π ∈ π m, 2 n

Measure on symplectic plane partitions Left is q Volume (rose), right alternates between q Volume (odd slices, in rose) and q − Volume (even positive slices, in coagulated blood). 1 � i<j � m (1 − q i + j +1 ) � � wt ( π ) = � m i =1 (1 − q i ) n (1 − q i +1 ) n π ∈ π m, 2 n

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Refined Cauchy identity for symplectic Schur polynomials Theorem (DB,MW) n � � (1 − t λ i − i + n +1 ) s λ ( x 1 , . . . , x n ) sp λ ( y 1 , ¯ y 1 , . . . , y n , ¯ y n ) = λ i =1 � n i =1 (1 − tx 2 i ) � (1 − t ) � y j ) det � i<j (1 − ¯ (1 − tx i y j )(1 − tx i ¯ y j )(1 − x i y j )(1 − x i ¯ ∆( x ) n ∆( y ) n y i ¯ y j ) 1 � i,j � n Proof. Cauchy-Binet � ∞ n � � (1 − tx 2 � (1 − t k +1 ) x k i ( y k +1 y k +1 i )( y i − ¯ y i ) det {· · · } 1 � i,j � n = det − ¯ ) j j i =1 k =0 1 � i,j � n n � k j � � � � � (1 − t k i +1 ) det y k i +1 y k i +1 = x 1 � i,j � n det − ¯ i j j 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 .

Is there a Hall–Littlewood analogue? Macdonald extended his theory of symmetric functions to other root systems. We will use Hall–Littlewood polynomials of type BC . They have a combinatorial definition (Venkateswaran): K λ ( y 1 , ¯ y 1 , . . . , y n , ¯ y n ; t ) =   n y λ i ( y i − ty j )(1 − t ¯ 1 y i ¯ y j ) � � i � ω   y 2 v λ ( t ) (1 − ¯ i ) ( y i − y j )(1 − ¯ y i ¯ y j ) i =1 ω ∈ W ( BC n ) 1 � i<j � n ◮ Koornwinder (or BC Macdonald) with q = 0 . ◮ t = 0 ⇒ symplectic Schur polynomials. ◮ No known interpretation as a sum over tableaux!

Main conjecture in type BC Conjecture (DB,MW) m i ( λ ) ∞ � � � (1 − t j ) P λ ( x 1 , . . . , x n ; t ) K λ ( y 1 , ¯ y 1 , . . . , y n , ¯ y n ; t ) = λ i =0 j =1 � n i,j =1 (1 − tx i y j )(1 − tx i ¯ y j ) � 1 � i<j � n ( x i − x j )( y i − y j )(1 − tx i x j )(1 − ¯ y i ¯ y j ) � (1 − t ) � × det (1 − tx i y j )(1 − tx i ¯ y j )(1 − x i y j )(1 − x i ¯ y j ) 1 � i,j � n

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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 )

The six-vertex model The Boltzmann weights are given by a + ( x, y ) = 1 − tx/y a − ( x, y ) = 1 − tx/y 1 − x/y 1 − x/y b + ( x, y ) = 1 b − ( x, y ) = t c + ( x, y ) = (1 − t ) c − ( x, y ) = (1 − t ) x/y 1 − x/y 1 − x/y The parameter t from Hall–Littlewood is now the crossing parameter of the model. The Boltzmann weights obey the Yang–Baxter equations: � x � x = � y � y z z � �

Boundary vertices In addition to the bulk vertices, we need U-turn vertices ◭ ◮ • • x x ◮ ◭ 1 / (1 − x 2 ) 1 / (1 − x 2 ) which depend on a single spectral parameter and are spin-conserving.

Boundary vertices satisfy the Sklyanin reflection equation � � � � x ¯ ¯ y x ¯ y ¯ • • = • • x y x y � � � �

Domain wall boundary conditions The six-vertex model on a lattice with domain wall boundary conditions: � � � � � � � x 1 ◮ ◭ � x 2 ◮ ◭ � x 3 ◮ ◭ � x 4 ◮ ◭ � x 5 ◮ ◭ � x 6 ◮ ◭ � � � � � � y 6 ¯ y 5 ¯ y 4 ¯ ¯ y 3 y 2 ¯ y 1 ¯ � � � � � � This partition function (the IK determinant) is of fundamental importance in periodic quantum spin chains and combinatorics.

Reflecting domain wall boundary conditions Interested in the following: � � � � � ¯ x 1 ◮ • � x 1 ◮ � ¯ x 2 ◮ • � x 2 ◮ � ¯ x 3 ◮ • � x 3 ◮ � ¯ x 4 ◮ • � x 4 ◮ � � � � y 4 ¯ y 3 ¯ y 2 ¯ y 1 ¯ � � � � This quantity is important in quantum spin-chains with open boundary conditions.

Reflecting domain wall boundary conditions Configurations on this lattice are in one-to-one correspondence with U-turn ASMs (UASMs): 0 + 0  + − 0  0 0 +   0 + −   0 0 0  0 0 + The partition function is also a determinant (Tsuchiya): Z UASM ( x 1 , . . . , x n ; y 1 , ¯ y 1 , . . . , y n , ¯ y n ; t ) = � n i,j =1 (1 − tx i y j )(1 − tx i ¯ y j ) � 1 � i<j � n ( x i − x j )( y i − y j )(1 − tx i x j )(1 − ¯ y i ¯ y j ) � (1 − t ) � × det (1 − tx i y j )(1 − tx i ¯ y j )(1 − x i y j )(1 − x i ¯ y j ) 1 � i,j � n

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Putting it together Conjecture (DB,MW) Z UASM ( x 1 , . . . , x n ; y 1 , ¯ y 1 , . . . , y n , ¯ y n ; t ) = ∞ m i ( λ ) � � � (1 − t j ) P λ ( x 1 , . . . , x n ; t ) K λ ( y 1 , ¯ y 1 , . . . , y n , ¯ y n ; t ) λ i =0 j =1

We can do more (doubly reflecting domain wall) y 3 y 2 y 1 • • • � ¯ x 1 ◮ • x 1 � x 1 ◮ � ¯ x 2 ◮ • x 2 � x 2 ◮ � ¯ x 3 ◮ • x 3 � x 3 ◮ � � � � � � y 3 ¯ y 3 y 2 ¯ y 2 y 1 ¯ y 1 � � � � � � is a product det 1 × det 2 (Kuperberg) with det 1 already described (with appropriate vertex weights).

The missing determinant A general version of det 2 is: Conjecture (DB,MW,PZJ) m 0( λ ) (1 − ut i − 1 ) b λ ( t ) P λ ( x 1 , . . . , x n ; t ) ˜ K λ ( y ± 1 , . . . , y ± 1 ; 0 , t, ut n − 1 ; t 0 , t 1 , t 2 , t 3 ) = � � 1 n λ i =1 � n n (1 − t 0 x i )(1 − t 1 x i )(1 − t 2 x i )(1 − t 3 x i ) i,j =1 (1 − tx i y j )(1 − tx i ¯ y j ) � (1 − tx 2 i ) � 1 � i<j � n ( x i − x j )( y i − y j )(1 − tx i x j )(1 − ¯ y i ¯ y j ) i =1 y j ) + ( t 2 − u ) x 2 � � 1 − u + ( u − t )( x i y j + x i ¯ i × det := det 2 ( t, t 0 , t 1 , t 2 , t 3 , u ) (1 − x i y j )(1 − tx i y j )(1 − x i ¯ y j )(1 − tx i ¯ y j ) 1 � i,j � n