Matrix product formula for Macdonald polynomials Jan de Gier 19 May 2015, GGI, Firenze Collaborators: Luigi Cantini Michael Wheeler Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 1 / 26
Outline Macdonald polynomials 1 Construction of Matrix Product form 2 General solution and combinatorics 3 Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 2 / 26
Macdonald polynomials I. What are Macdonald polynomials? Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 3 / 26
Macdonald polynomials Symmetric group Let s i ( i = 1 , . . . , n − 1 ) be generators of the symmetric group S n : s i s i + 1 s i = s i + 1 s i s i + 1 s 2 i = 1 ; There exist a natural t -deformation of S n : 2 )( T i + t − 1 1 2 ) = 0 , ( T i − t ( i = 1 , . . . , n − 1 ) , T i T i + 1 T i = T i + 1 T i T i + 1 . This is the Hecke algebra (of type A n − 1 ) and S n is recovered when t → 1. Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 4 / 26
Macdonald polynomials Polynomial action The generators s i act naturally on polynomials: s i f ( . . . , x i , x i + 1 , . . . ) = f ( . . . , x i + 1 , x i , . . . ) i = 1 , . . . n − 1 and the t -deformation also has an action: 2 tx i − x i + 1 = t ± 1 2 − t − 1 T ± 1 ( 1 − s i ) . i x i − x i + 1 Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 5 / 26
Macdonald polynomials Polynomial action The generators s i act naturally on polynomials: s i f ( . . . , x i , x i + 1 , . . . ) = f ( . . . , x i + 1 , x i , . . . ) i = 1 , . . . n − 1 and the t -deformation also has an action: 2 tx i − x i + 1 = t ± 1 2 − t − 1 T ± 1 ( 1 − s i ) . i x i − x i + 1 The shifted operator, T i ( u ) = T i + t − 1 [ u ] = 1 − t u 2 [ u ] , 1 − t . satisfies the Yang–Baxter equation, T i ( u ) T i + 1 ( u + v ) T i ( v ) = T i + 1 ( v ) T i ( u + v ) T i + 1 ( u ) . Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 5 / 26
Macdonald polynomials Nonsymmetric Macdonald polynomials We can extend to the affine Hecke algebra by adding a cyclic shift operator: ( ω f )( x 1 , . . . , x n ) = f ( qx n , x 1 , . . . , x n − 1 ) , ω T i = T i + 1 ω. Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 6 / 26
Macdonald polynomials Nonsymmetric Macdonald polynomials We can extend to the affine Hecke algebra by adding a cyclic shift operator: ( ω f )( x 1 , . . . , x n ) = f ( qx n , x 1 , . . . , x n − 1 ) , ω T i = T i + 1 ω. This algebra has a family of commuting operators (Abelian subalgebra) generated by the Murphy elements: Y i = T i · · · T n − 1 ω T − 1 · · · T − 1 i − 1 . 1 which commute: [ Y i , Y j ] = 0 . Remark: Symmetric functions of { Y i } are central, i.e. commute with all elements in the Hecke algebra. Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 6 / 26
Macdonald polynomials Nonsymmetric Macdonald polynomials Since the Y i commute, they can be diagonalised simultaneously: Definition (Nonsymmetric Macdonald polynomial E λ ) Y i E λ = y i ( λ ) E λ , The index λ = ( λ 1 , . . . , λ n ) is a composition, λ i ∈ N 0 , and y i ( λ ) = t ρ ( λ ) i q λ i Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 7 / 26
Macdonald polynomials Nonsymmetric Macdonald polynomials Since the Y i commute, they can be diagonalised simultaneously: Definition (Nonsymmetric Macdonald polynomial E λ ) Y i E λ = y i ( λ ) E λ , The index λ = ( λ 1 , . . . , λ n ) is a composition, λ i ∈ N 0 , and y i ( λ ) = t ρ ( λ ) i q λ i Example: If λ = ( 3 , 0 , 4 , 4 , 2 ) , then define ρ = ( 2 , 1 , 0 , − 1 , − 2 ) Dominant weight λ + = ( 4 , 4 , 3 , 2 , 0 ) Reorder ρ in the same way as reordering λ + → λ ρ ( λ ) = ( 0 , − 2 , 2 , 1 , − 1 ) . Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 7 / 26
Macdonald polynomials Macdonald polynomials E λ forms a basis in the ring of polynomials with top-degree λ + , E λ ( x 1 , . . . , x n ) = x λ 1 · · · x λ n � c λµ x µ + n 1 µ<λ (summation in dominance ordering) Definition (Symmetric Macdonald polynomials) � P λ + = E λ λ ≤ λ + Macdonald polynomials are ( q , t ) generalisations of Schur polynomials. Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 8 / 26
Construction of Matrix Product form II. Matrix Product Form Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 9 / 26
Construction of Matrix Product form Exchange relations Let δ be the anti-dominant weight ( δ 1 ≤ δ 2 ≤ . . . ≤ δ n ) . Definition (The exchange basis) f δ := E δ f ...,λ i ,λ i + 1 ,... := t − 1 2 T − 1 f ...,λ i + 1 ,λ i ,... λ i > λ i + 1 . i Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 10 / 26
Construction of Matrix Product form Exchange relations Let δ be the anti-dominant weight ( δ 1 ≤ δ 2 ≤ . . . ≤ δ n ) . Definition (The exchange basis) f δ := E δ f ...,λ i ,λ i + 1 ,... := t − 1 2 T − 1 f ...,λ i + 1 ,λ i ,... λ i > λ i + 1 . i Then f solves the exchange equations 1 2 f ...,λ i ,λ i + 1 ,... T i f ...,λ i ,λ i + 1 ,... = t λ i = λ i + 1 , T i f ...,λ i ,λ i + 1 ,... = t − 1 2 f ...,λ i + 1 ,λ i ,... λ i > λ i + 1 , ω f λ n ,λ 1 ,...,λ n − 1 = q λ n f λ 1 ,...,λ n . • Dynamics of the multispecies asymmetric exclusion process • t 1 / 2 -deformed Knizhnik-Zamolodchikov equations Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 10 / 26
Construction of Matrix Product form Matrix product ansatz Assume � � f λ ( x 1 , . . . , x n ) = Tr A λ 1 ( x 1 ) · · · A λ n ( x n ) S , This implies a matrix product for Macdonald polynomials as also � P λ + = f λ λ ≤ λ + ‘Normalisation of ASEP’. Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 11 / 26
Construction of Matrix Product form Matrix product ansatz Assume � � f λ ( x 1 , . . . , x n ) = Tr A λ 1 ( x 1 ) · · · A λ n ( x n ) S , This implies a matrix product for Macdonald polynomials as also � P λ + = f λ λ ≤ λ + ‘Normalisation of ASEP’. The exchange relations imply the following algebra for the ‘matrices’ A : 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 SA i ( qx ) = q i A i ( x ) S , Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 11 / 26
Construction of Matrix Product form Zamolodchikov-Faddeev algebra For λ ⊂ r n the algebra relations can be rephrased by writing A ( r ) ( x ) = ( A 0 ( x ) , . . . , A r ( x )) T , as an ( r + 1 ) -dimensional operator valued column vector. Lemma (ZF algebra) The exchange relations are equivalent to ˇ R ( x , y ) · [ A ( x ) ⊗ A ( y )] = [ A ( y ) ⊗ A ( x )] 2 ( sl r + 1 ) R-matrix of dimension ( r + 1 ) 2 ( r = 1 is the 6-vertex model). ˇ R ( x , y ) is the U 1 t Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 12 / 26
Construction of Matrix Product form Yang-Baxter algebra and Nested Matrix Product Form More familiar is rank r Yang-Baxter algebra: R ( x , y ) · [ L ( x ) ⊗ L ( y )] = [ L ( y ) ⊗ L ( x )] · ˇ ˇ R ( x , y ) Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 13 / 26
Construction of Matrix Product form Yang-Baxter algebra and Nested Matrix Product Form More familiar is rank r Yang-Baxter algebra: R ( x , y ) · [ L ( x ) ⊗ L ( y )] = [ L ( y ) ⊗ L ( x )] · ˇ ˇ R ( x , y ) Assume a solution of the following modified RLL relation � � � � R ( r ) ( x , y ) · ˇ ˜ L ( x ) ⊗ ˜ L ( y ) ⊗ ˜ ˜ · ˇ R ( r − 1 ) ( x , y ) L ( y ) = L ( x ) s ˜ L ij ( qx ) = q i − j ˜ L ij ( x ) s . in terms of an ( r + 1 ) × r operator-valued matrix ˜ L ( x ) = ˜ L ( r ) ( x ) . Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 13 / 26
Construction of Matrix Product form Yang-Baxter algebra and Nested Matrix Product Form More familiar is rank r Yang-Baxter algebra: R ( x , y ) · [ L ( x ) ⊗ L ( y )] = [ L ( y ) ⊗ L ( x )] · ˇ ˇ R ( x , y ) Assume a solution of the following modified RLL relation � � � � R ( r ) ( x , y ) · ˇ L ( x ) ⊗ ˜ ˜ L ( y ) ⊗ ˜ ˜ · ˇ R ( r − 1 ) ( x , y ) L ( y ) = L ( x ) s ˜ L ij ( qx ) = q i − j ˜ L ij ( x ) s . in terms of an ( r + 1 ) × r operator-valued matrix ˜ L ( x ) = ˜ L ( r ) ( x ) . Then A ( r ) ( x ) = ˜ L ( r ) ( x ) · ˜ L ( r − 1 ) ( x ) · · · ˜ L ( 1 ) ( x ) S ( r ) = s ( r ) · s ( r − 1 ) · · · s ( 1 ) Solves the ZF algebra ˇ R ( x , y ) · [ A ( x ) ⊗ A ( y )] = [ A ( y ) ⊗ A ( x )] Jan de Gier Matrix product formula for Macdonald polynomials 19 May 2015 13 / 26
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