Stochastic vertex models and generalized Macdonald polynomials - PowerPoint PPT Presentation

Macdonald polynomials Borodin–Petrov polynomials Unification Stochastic vertex models and generalized Macdonald polynomials Michael Wheeler School of Mathematics and Statistics University of Melbourne 26 August, 2016 Michael Wheeler

Macdonald polynomials Borodin–Petrov polynomials Unification Outline 1 Macdonald polynomials 2 Borodin–Petrov polynomials 3 Unification Michael Wheeler

Macdonald polynomials Borodin–Petrov polynomials Unification Charting the territory Symplectic characters BC Hall–Littlewood Koornwinder Schur Q Schur Hall–Littlewood Macdonald ❘ ◗ ❱ P ❙ Borodin–Petrov Grothendieck (Grassmannian) Grothendieck Grothendieck Schubert (general) (conormal) Schubert Conormal (conormal) Michael Wheeler

Macdonald polynomials Borodin–Petrov polynomials Unification Why are Macdonald polynomials interesting? Macdonald polynomials can be considered as grandparents of a host of symmetric functions (monomial, Schur, Hall–Littlewood, Jack, zonal). They are two parameter ( q , t ) generalizations of Schur polynomials. They have led to fascinating problems in pure mathematics, such as the constant term conjecture (proved in full generality by Cherednik) and the positivity conjecture (proved by Haiman). They have appeared in the physics literature, in connection with the 5D AGT correspondence. They were also the inspiration for the Macdonald process of Borodin and Corwin, which generalizes the Schur process introduced by Okounkov and Reshetikhin. In the algebraic Bethe Ansatz for quantum integrable models, symmetric functions arise very naturally: Ψ i 1 ,..., i n ( x 1 , . . . , x n ) = � i 1 , . . . , i n | B ( x 1 ) . . . B ( x n ) | 0 � is symmetric in x 1 , . . . , x n . A key point of this work is to realize Macdonald polynomials via such a fomula. Michael Wheeler

Macdonald polynomials Borodin–Petrov polynomials Unification Macdonald polynomials: first definition Let λ be a partition, λ 1 � · · · � λ n � 0 . The monomial symmetric functions are given by n m λ ( x 1 , . . . , x n ) = ∑ x λ i S λ = S n / S λ ∏ σ ( i ) , n . σ ∈ S λ i = 1 For example: m 2,2,1,1 ( x 1 , x 2 , x 3 , x 4 ) = x 2 1 x 2 2 x 3 x 4 + x 2 1 x 2 x 2 3 x 4 + x 2 1 x 2 x 3 x 2 4 + x 1 x 2 2 x 2 3 x 4 + x 1 x 2 2 x 3 x 2 4 + x 1 x 2 x 2 3 x 2 4 The Macdonald polynomial P λ ( x ; q , t ) is the unique homogeneous, symmetric function which satisfies P λ ( x 1 , . . . , x n ; q , t ) = m λ ( x 1 , . . . , x n ) + ∑ c λ , µ ( q , t ) m µ ( x 1 , . . . , x n ) , µ < λ � P λ , P µ � = 0, λ � = µ , with respect to a certain bilinear form defined on power sums: � p λ , p µ � = δ λ , µ × ( some rational function in q , t ) . Michael Wheeler

Macdonald polynomials Borodin–Petrov polynomials Unification Macdonald polynomials: second definition Introduce a commuting family of difference operators [Macdonald 87], and their generating series: � � n tx i − x j D r n = t r ( r − 1 ) /2 D r n z r . ∑ ∏ i ∈ S ∏ ∏ ∑ T q , x k , D n ( z ) = x i − x j S ⊆ [ 1,..., n ] j �∈ S k ∈ S r = 0 | S | = r The Macdonald polynomials are the unique eigenfunctions of D n ( z ) : n ( 1 + zq λ i t n − i ) P λ ( x 1 , . . . , x n ; q , t ) . ∏ D n ( z ) P λ ( x 1 , . . . , x n ; q , t ) = i = 1 This definition is more appealing for a mathematical physicist, but it gives no clear insight into the structure of P λ ( x 1 , . . . , x n ; q , t ) . Michael Wheeler

Macdonald polynomials Borodin–Petrov polynomials Unification Macdonald polynomials in integrable probability Macdonald processes [Borodin, Corwin 11] are a very general class of stochastic processes which include many others as special cases: Schur processes, totally asymmetric simple exclusion processes, last passage percolation, random directed polymers. They are based on the Macdonald measure on partitions � � ∞ 1 − t n M λ ( ρ 1 ; ρ 2 ) : = P λ ( ρ 1 ) Q λ ( ρ 2 ) 1 ∑ , Π ( ρ 1 ; ρ 2 ) = exp 1 − q n p n ( ρ 1 ) p n ( ρ 2 ) , Π ( ρ 1 ; ρ 2 ) n n = 1 where ρ 1 and ρ 2 are two specializations of the ring of symmetric functions. The action of difference operators on Macdonald polynomials then leads to a natural class of observables for study: � � ∑ λ ∏ n i = 1 ( 1 + zq λ i t n − i ) P λ ( x ) Q λ ( y ) � � e m ( q λ 1 t n − 1 , . . . , q λ n t 0 ) = E Π ( x ; y ) z m � D n ( z ) Π ( x ; y ) � = Π ( x ; y ) z m where we have taken ρ 1 = ( x 1 , . . . , x n ) and ρ 2 = ( y 1 , . . . , y n ) for simplicity. Michael Wheeler

Macdonald polynomials Borodin–Petrov polynomials Unification Non-symmetric Macdonald polynomials The Hecke algebra of type A n − 1 is generated by T 1 , . . . , T n − 1 modulo the relations ( T i − t )( T i + 1 ) = 0, T i T i ± 1 T i = T i ± 1 T i T i ± 1 , T i T j = T j T i , | i − j | > 1. We consider a polynomial representation of the algebra, given by T i = t − tx i − x i + 1 ( 1 − σ i ) , 1 � i � n − 1. x i − x i + 1 Consider the following (commuting) elements of the Hecke algebra: Y i = T i · · · T n − 1 ω T − 1 · · · T − 1 ω g ( x 1 , . . . , x n ) = g ( qx n , x 1 , . . . , x n − 1 ) . i − 1 , 1 Non-symmetric Macdonald polynomials E µ [Cherednik 95], [Opdam 95], [Macdonald 95] are defined as the unique eigenfunctions of these operators: y i ( µ ) = t ρ i ( µ ) q µ i . Y i E µ = y i ( µ ) E µ , Theorem The Macdonald polynomial P λ ( x ; q , t ) is given by P λ ( x 1 , . . . , x n ; q , t ) = ∑ κ σ ( λ ) E σ ( λ ) ( x 1 , . . . , x n ; q , t ) , σ ∈ S λ where the sum is over all distinct permutations of λ . Michael Wheeler

Macdonald polynomials Borodin–Petrov polynomials Unification Another non-symmetric basis In this work we are interested in another basis, f µ ( x 1 , . . . , x n ; q , t ) , defined by f δ 1 ,..., δ n : = E δ 1 ,..., δ n when δ 1 � · · · � δ n f µ 1 ,..., µ i , µ i + 1 ,..., µ n : = T − 1 f µ 1 ,..., µ i + 1 , µ i ,..., µ n when µ i > µ i + 1 . i The transition matrix between f µ and E µ cannot easily be written down, but it is a triangular change of basis. Theorem The Macdonald polynomial P λ ( x ; q , t ) is given by P λ ( x 1 , . . . , x n ; q , t ) = ∑ f σ ( λ ) ( x 1 , . . . , x n ; q , t ) , σ ∈ S λ where the sum is over all distinct permutations of λ . For reasons that will become clear, we call this the ASEP basis, after the asymmetric simple exclusion process. Michael Wheeler

Macdonald polynomials Borodin–Petrov polynomials Unification Matrix product solution of Knizhnik–Zamolodchikov equations These polynomials satisfy the Knizhnik–Zamolodchikov equations [Kasatani, Takeyama 07]:  t f µ 1 ,..., µ i + 1 , µ i ,..., µ n ( x 1 , . . . , x n ) , µ i = µ i + 1  T i f µ 1 ,..., µ i , µ i + 1 ,..., µ n ( x 1 , . . . , x n ) = f µ 1 ,..., µ i + 1 , µ i ,..., µ n ( x 1 , . . . , x n ) , µ i > µ i + 1  f µ n , µ 1 ,..., µ n − 1 ( qx n , x 1 , . . . , x n − 1 ) = q µ n f µ 1 ,..., µ n ( x 1 , . . . , x n ) . We seek a matrix product solution of the above equations: � � Ω µ + ( q , t ) f µ ( x 1 , . . . , x n ) = Tr A µ 1 ( x 1 ) . . . A µ n ( x n ) S This Ansatz works provided that A i ( x ) A i ( y ) = A i ( y ) A i ( x ) tA j ( x ) A i ( y ) − tx − y � � A j ( x ) A i ( y ) − A j ( y ) A i ( x ) = A i ( x ) A j ( y ) x − y S A i ( qx ) = q i A i ( x ) S . Michael Wheeler

Macdonald polynomials Borodin–Petrov polynomials Unification Zamolodchikov–Faddeev algebra The previous relations can be written more succinctly as A 0 ( x )   A 1 ( x )   R ab ( x / y ) A a ( x ) A b ( y ) = A b ( y ) A a ( x ) , A a ( x ) =  .  .   .   A r ( x ) a The R matrix is (a stochastic) higher rank version of the six-vertex model: r � � E ( ii ) E ( ii ) E ( ii ) E ( jj ) + t E ( jj ) E ( ii ) R ab ( x / y ) = ( 1 − tx / y ) ∑ + ( 1 − x / y ) ∑ a a a b b b i = 0 0 � i < j � r � E ( ij ) E ( ji ) + x / y E ( ji ) E ( ij ) � ∑ + ( 1 − t ) . a a b b 0 � i < j � r The operator S satisfies SA ( qx ) = q ∑ r i = 0 iE ( ii ) A ( x ) S . Michael Wheeler

Macdonald polynomials Borodin–Petrov polynomials Unification Matrix product formula Theorem (Cantini–de Gier–Wheeler 2015) Let A i ( x ) be the i th component of A ( x ) , and S be as above. Then 1 − q j − i t ( µ + ) ′ i − ( µ + ) ′ � � � � f µ ( x 1 , . . . , x n ; q , t ) = ∏ A µ 1 ( x 1 ) . . . A µ n ( x n ) S j Tr 1 � i < j � r 1 − q j − i t λ ′ i − λ ′ � � � � ∏ ∑ P λ ( x 1 , . . . , x n ; q , t ) = j Tr A µ 1 ( x 1 ) . . . A µ n ( x n ) S 1 � i < j � r µ ∈ S λ · λ The non-symmetric polynomials f µ can be viewed as generalized ASEP configuration probabilities. Symmetric Macdonald polynomials P λ are the normalizations of these probabilities. Michael Wheeler

Macdonald polynomials Borodin–Petrov polynomials Unification A reminder of the landscape Symplectic characters BC Hall–Littlewood Koornwinder Schur Q Schur Hall–Littlewood Macdonald ❘ ◗ ❱ P ❙ Borodin–Petrov Grothendieck (Grassmannian) Grothendieck Grothendieck Schubert (general) (conormal) Schubert Conormal (conormal) Michael Wheeler