Matrix product formula for Macdonald polynomials
aa r X i v : . [ m a t h - ph ] J u l Matrix product formula for Macdonald polynomials Luigi Cantini, Jan de Gier and Michael Wheeler Laboratoire de Physique Th´eorique et Mod´elisation (CNRS UMR 8089), Universit´ede Cergy-Pontoise, F-95302 Cergy-Pontoise, France, , Department of Mathematics and Statistics, The University of Melbourne, 3010VIC, AustraliaE-mail: [email protected], [email protected], [email protected] Abstract.
We derive a matrix product formula for symmetric Macdonaldpolynomials. Our results are obtained by constructing polynomial solutionsof deformed Knizhnik–Zamolodchikov equations, which arise by consideringrepresentations of the Zamolodchikov–Faddeev and Yang–Baxter algebras in termsof t -deformed bosonic operators. These solutions are generalised probabilities forparticle configurations of the multi-species asymmetric exclusion process, and forma basis of the ring of polynomials in n variables whose elements are indexed bycompositions. For weakly increasing compositions (anti-dominant weights), these basiselements coincide with non-symmetric Macdonald polynomials. Our formulas implya natural combinatorial interpretation in terms of solvable lattice models. They alsoimply that normalisations of stationary states of multi-species exclusion processes areobtained as Macdonald polynomials at q = 1. atrix product formula for Macdonald polynomials
1. Introduction
Symmetric Macdonald polynomials [1, 2] are families of multivariable orthogonalpolynomials indexed by partitions whose coefficients depend rationally on twoparameters q and t . In the case q = t they degenerate to the more familiar Schurfunctions which encode characters of irreducible representations of the symmetric group.One way to define Macdonald polynomials is as joint eigenfunctions of a family ofcommuting operators in the double affine Hecke algebra [3, 4]. They can also be definedcombinatorially as generating functions [5, 6, 7], or via symmetrisation of non-symmetricMacdonald polynomials that are computed from Yang-Baxter graphs [8, 9].Another family of commuting operators can be constructed in the Hecke algebra,corresponding to commuting transfer matrices in the theory of Yang–Baxter solvablelattice models [10]. Those operators are solutions of the Yang–Baxter algebra generatedby an R -matrix arising from a quantum group. Eigenfunctions of transfer matricesare generally complicated objects that can be constructed using the so-called Betheansatz. In cases where the eigenvalue is simple it is sometimes possible to constructexplicit eigenfunctions of transfer matrices, and a well developed technique to constructsuch eigenfunctions is the matrix product algebra [11] for the asymmetric exclusionprocess. In an inhomogeneous setting the matrix product algebra is known as theZamolodchikov–Faddeev (ZF) algebra [12, 13]. The latter algebra is related to the t -deformed Knizhnik–Zamolodchikov equation [14, 15].Opdam [16] and Cherednik [3, 4] generalised Macdonalds construction to a non-symmetric setting and defined families of non-symmetric Macdonald polynomials whichare indexed by tuples of integers called compositions. There exists a basis in thering spanned by non-symmetric Macdonald polynomials whose defining equations areexchange relations which are equivalent to the t -deformed Zamolodchikov–Faddeevalgebra. Elements of this basis specialise to probabilities of particle configurations ofthe multi-species asymmetric exclusion process. Solutions to the ZF algebra can beobtained from solutions of the Yang–Baxter algebra, but not all solutions of the lattergive non-trivial solutions of the former. Polynomial solutions of the deformed ZF andKZ algebras have received recent attention in the context of the Razumov–Stroganov–Cantini–Sportiello theorem and alternating-sign matrices [17, 18, 19, 20, 22, 21, 23].In this paper we put several ingredients together to show that a solution ofthe Yang–Baxter algebra arising for the multi-species asymmetric exclusion processand based on t -deformed bosons generates non-trivial polynomial solutions of theZamolodchikov–Faddeev algebra. We show that those solutions lead to matrix productformulas for certain basis functions in the ring of polynomials spanned by the non-symmetric Macdonald polynomials, and by symmetrisation this implies a matrix productformula for symmetric Macdonald polynomials. We furthermore provide a combinatorialinterpretation of our formulas in terms of two-dimensional lattice configurations of bosontrajectories.For q = 1 the homogeneous limit of our formulas are steady state probabilities of atrix product formula for Macdonald polynomials In the following we shall make extensive use of bosonic operators called t -deformedoscillators. These operators appear naturally in the context of solvable lattice modelsand in matrix product formulas for the asymmetric simple exclusion process (ASEP)[24, 34]. They are defined by ak = tka, a † k = t − ka † ,aa † = 1 − k, a † a = 1 − t − k. (1)A consequence of these relations is the following equation which we shall use frequently, aa † − ta † a = 1 − t. (2)A faithful representation of this algebra on the Fock space Span {| m i} ∞ m =0 is given by a | m i = (1 − t m ) | m − i , a † | m i = (1 − t m +1 ) | m + 1 i ,k = diag { , t, t , . . . } . (3) The second algebraic ingredient we need is an action of the Hecke algebra on polynomials.We start by denoting by S n the symmetric group on n elements, i.e. S n is the Weylgroup of type A n − . The symmetric group has a natural action on compositions λ ∈ N n given by s i ( . . . , λ i , λ i +1 , . . . ) = ( . . . , λ i +1 , λ i , . . . ) . (4)The symmetric group also has a natural action on polynomials, given by s i f ( . . . , x i , x i +1 , . . . ) = f ( . . . , x i +1 , x i , . . . ) . (5)A t -deformation of (5) may be defined using divided differences leading to theDemazure operator T ± i = t ± / − t − / tx i − x i +1 x i − x i +1 (1 − s i ) , (6)which for t → T i satisfies the relations of the Heckealgebra of type A n − ,( T i − t / )( T i + t − / ) = 0 , ( i = 1 , . . . , n − T i T i ± T i = T i ± T i T i ± , (7) T i T j = T j T i | i − j | ≥ . atrix product formula for Macdonald polynomials T i ( u ) = T i + t − / [ u ] , [ u ] = 1 − t u − t . (8)which 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 ) . (9)We furthermore define the following shift operator compatible with the affine Heckealgebra of type A n − , ( ωf )( x , . . . , x n ) = f ( qx n , x , . . . , x n − ) , (10) ωT i = T i +1 ω. (11)The affine Hecke algebra formed by the operators (6) and ω has an Abelian subalgebragenerated by the Murphy elements, which are defined as Y i = T i · · · T n − ωT − · · · T − i − . (12)These operators mutually commute and central elements in the Hecke algebra canbe constructed by taking symmetric combinations of the Murphy elements. Jointeigenfunction of the operators Y i are non-symmetric Macdonald polynomials.For later use it is convenient to develop some notation. Representations of S n are indexed by partitions, which are those compositions for which λ ≥ λ ≥ . . . ≥ λ n .Compositions are naturally ordered under the dominance order ≥ on N n , which is definedas λ ≥ µ if k X i =1 ( λ i − µ i ) ≥ k = 1 , . . . , n. (13)Let λ be a composition and let λ + be the dominant weight of λ in the dominanceorder on compositions, i.e. λ + is a permutation of λ such that λ ≥ λ ≥ . . . ≥ λ n .Let w + be the smallest word such that λ = w + · λ + . The permutation w − is obtainedby labeling each entry from λ with a number from 1 to n , from the biggest entry tothe smallest and from the left to the right. For instance λ = (3 , , , , ⇒ w − =(3 , , , ,
4) and so w + = (3 , , , ,
2) and λ + = (4 , , , , δ = (0 , , , , ρ ( λ ) := w + · ρ, (14)where ρ = ( n − , n − , . . . , − ( n − n = 5 and therefore ρ = (2 , , , − , −
2) and ρ ( λ ) = (0 , − , , , − atrix product formula for Macdonald polynomials We are interested in finding explicit matrix product formulas for Macdonaldpolynomials. We first derive matrix product formulas for polynomials which arelinear combinations of non-symmetric Macdonald polynomials. These polynonmials aredefined by a system of exchange equations closely related to the q -deformed Knizhnik–Zamolodchkov ( q KZ) equation. We note here that in our notation the parameter that isusual called q is replaced by t in order to make connection to the literature on Macdonaldtheory.Following Kasatani and Takeyama [20], polynomial solutions to the (reduced) q KZequations, sometimes called exchange relations, can be obtained from eigenfunctions ofthe Y i operators. Let δ be the anti-dominant weight and let E δ be the non-symmetricMacdonald polynomial solving the eigenvalue equation Y i E δ = y i ( δ ) E δ , (15)where y i ( δ ) = t ρ ( δ ) i q δ i . We now define another set of polynomials, which are linearcombinations of the Macdonald polynomials E λ , and for which we will be able to findcompact explicit expression. Define f δ := E δ ,f ...,λ i ,λ i +1 ,... := t − / T − i f ...,λ i +1 ,λ i ,... λ i > λ i +1 . (16)Note that our notation differs slightly from [20] as (18) contains a factor t − / on theright hand side, and consequently some other details below are different.Then f solves the q KZ equations T i f ...,λ i ,λ i +1 ,... = t / f ...,λ i ,λ i +1 ,... λ i = λ i +1 , (17) T i f ...,λ i ,λ i +1 ,... = t − / f ...,λ i +1 ,λ i ,... λ i > λ i +1 , (18) ωf λ n ,λ ,...,λ n − = q λ n f λ ,...,λ n . (19)Writing q = t u we define the elements of the spectral vector h λ i of a composition λ as, h λ i i = ρ i ( λ ) + uλ i . y i ( λ ) = t h λ i i . (20)The non-symmetric Macdonald polynomials are obtained from E δ by the action ofBaxterised operators: E s i λ = T i ( h λ i i +1 − h λ i i ) E λ , λ i < λ i +1 . (21)These polynomials satisfy Y i E λ = y i ( λ ) E λ . (22)Below we will derive a matrix product expression for the functions f λ , and whilewe will not give an explicit matrix product expression for the polynomials E λ , these atrix product formula for Macdonald polynomials f λ using (21). In fact, the two families ofpolynomials are related via a triangular change of basis: E λ = X µ ≤ λ c λµ ( q, t ) f µ , f λ = X µ ≤ λ d λµ ( q, t ) E µ (23)for suitable rational coefficients c λµ ( q, t ) and d λµ ( q, t ).
2. Matrix Product, Yang–Baxter and Zamolodchikov–Faddeev algebras
The aim of this section is to obtain a matrix product expression for E δ ( x , . . . , x n ),the non-symmetric Macdonald polynomial indexed by the anti-dominant weight δ . Ourapproach is to write an Ansatz for the polynomials f λ , which generalize E δ , and to showthat this Ansatz obeys the q KZ equations (17)–(19). The Ansatz is as follows:Ω λ + f λ ( x , . . . , x n ) = Tr h A λ ( x ) · · · A λ n ( x n ) S i , (24)where Ω λ + is a normalisation factor to be determined later and A ( x ) , A ( x ) , . . . , A r ( x )and S are matrices satisfying the exchange relations A i ( x ) A i ( y ) = A i ( y ) A i ( x ) , (25) tA j ( x ) A i ( y ) − tx − yx − y (cid:16) A j ( x ) A i ( y ) − A j ( y ) A i ( x ) (cid:17) = A i ( x ) A j ( y ) , (26) SA i ( qx ) = q i A i ( x ) S, (27)for all 0 ≤ i < j ≤ r . It is straightforward to demonstrate that the Ansatz (24) is afaithful solution to the q KZ relations (17)–(19). The validity of (17) is ensured by (25),since f λ ( x , . . . , x n ) is symmetric in x i , x i +1 when λ i = λ i +1 . Accordingly, the action of(1 − s i ) gives zero in this case, and we find that T i f λ ,...,λ i ,λ i +1 ,...,λ n ( x , . . . , x n ) = t / f λ ,...,λ i +1 ,λ i ,...,λ n ( x , . . . , x n ) (28)by inspection. In a similar vein, when λ i > λ i +1 , direct application of (26) allows us toconclude that T i f λ ,...,λ i ,λ i +1 ,...,λ n ( x , . . . , x n ) = t − / f λ ,...,λ i +1 ,λ i ,...,λ n ( x , . . . , x n ) . (29)Finally, to prove (19), we observe thatΩ λ + f λ n ,λ ,...,λ n − ( qx n , x , . . . , x n − ) = Tr (cid:0) A λ n ( qx n ) A λ ( x ) · · · A λ n − ( x n − ) S (cid:1) , = Tr (cid:0) A λ ( x ) · · · A λ n − ( x n − ) SA λ n ( qx n ) (cid:1) = q λ n Ω λ + f λ ,...,λ n ( x , . . . , x n ) , (30)where we have used the cyclicity of the trace and the exchange relation (27) to reachthe final equality. atrix product formula for Macdonald polynomials Solutions to the relations (25)–(26) can be recovered from the Yang–Baxter algebracorresponding to the quantum group U t / ( A (1) r ), or rather a twisted version of it [35].For models based on U t / ( A (1) r ), the R -matrix can be expressed in the formˇ R ( r ) ( x, y ) = r +1 X i =1 E ( ii ) ⊗ E ( ii ) + x − ytx − y X ≤ i 3. Low rank examples Before presenting the general construction we will first display some explicit examples.For convenience we define the following functions b + = t ( x − y ) tx − y , b − = t − b + = x − ytx − y , (41) c + = 1 − b + = y ( t − tx − y , c − = 1 − b − = x ( t − tx − y . Then for r = 1, we can trivially solve (36) for ˜ L (1) ( x ): c − b + b − c + 00 0 0 1 · " x ! ⊗ y ! = " y ! ⊗ x ! . Hence we see that ˜ L (1) ( x ) = A (1) ( x ) = x ! , (42)is a rank 1 solution to (33). The corresponding solution to the Yang–Baxter algebra(32) is equal to L (1) ( x ) = axa † x ! , (43)where the operators a , a † and k satisfy the t -oscillator relations (1). We note thattrivialising the t -oscillator by sending a † , a k 0, we reduce the rank, andthus obtain the solution ˜ L (1) ( x ): axa † x ! x x ! . (44) atrix product formula for Macdonald polynomials The rank 2 case is less trivial, giving rise to operator valued solutions for A (2) ( x ). Inthe case r = 2, solving equation (36) for ˜ L (2) ( x ), we find that c − b + c − b + b − c + c − b + 00 0 b − c + b − c + 00 0 0 0 0 0 0 0 1 · axk xa † x ⊗ ayk ya † y = ayk ya † y ⊗ axk xa † x · c − b + b − c + 00 0 0 1 , (45)Using (39) we thus construct a solution of the ZF algebra in the following way: A (2) ( x ) = ˜ L (2) ( x ) · ˜ L (1) ( x ) = axk xa † x x ! = xakxxa † + x . (46)The associated rank 2 solution to the Yang–Baxter algebra is L (2) ( x ) = a a xa † k xk xa † xa a † x , (47)where { a , a † , k } and { a , a † , k } are two commuting copies of the t -oscillator algebra(47). The map a † , a k L (2) ( x ) by one L (2) ( x ) a xk xk xa † xa † x ⇒ ˜ L (2) ( x ) = a xk xa † x , (48)where the indices of t -oscillators are redundant in the final matrix, since we no longerneed to distinguish between the two copies of the algebra.We note that low-rank examples of L -matrices based on deformed oscillators havebeen treated in earlier literature, for example, [24, 37, 38, 39, 40]. atrix product formula for Macdonald polynomials We look at an explicit example for rank 2 taking δ = (0 , , , , , ρ ( δ ) = ( − , − , , − , , q KZ equation satisfies the following equations: Y E δ = t − / E δ Y E δ = qt − E δ Y E δ = t − / E δ Y E δ = q t / E δ Y E δ = qtE δ Y E δ = q t / E δ . (49)Using the notation q = t u , [ m ] = 1 − t m − t , (50)it can be verified that the following polynomial solves (49), E δ ( x , . . . , x ; q, t ) = x x x x + t [3 + u ] ( x + x ) x x x x ( x + x )+ t [2][3 + u ][4 + u ] x x x x x x . (51)We now verify the matrix product formΩ δ + E δ ( x , . . . , x ; q = t u , t ) = Tr (cid:2) A ( x ) A ( x ) A ( x ) A ( x ) A ( x ) A ( x ) S (cid:3) , (52)for this explicit solution. From (46) we see that A ( x ) = 1 + xa,A ( x ) = xk, (53) A ( x ) = xa † + x , and using (27) we note that S should satisfy qSa − aS = 0 , Sa † − qa † S = 0 . (54)Taking the explicit representation (3) for the t -oscillators, S has the form S = k u = diag { , t u , t u , . . . } = diag { , q, q , . . . } . (55)Up to a normalisation, the nonsymmetric Macdonald polynomial E δ is now representedin matrix product form byTr (cid:2) (1 + x a ) (1 + x a ) x kx kx (cid:0) a † + x (cid:1) x (cid:0) a † + x (cid:1) S (cid:3) = x x x x Tr (cid:2)(cid:0) x x k + ( x + x )( x + x ) ak a † + x x a k ( a † ) (cid:1) S (cid:3) , (56)where other terms involving unequal powers of a and a † have zero trace. Normalisingwith Ω δ + = Ω = Tr( k S ) we finally get E δ ( x , . . . , x ; q = t u , t ) = x x x x + x x x x ( x + x )( x + x ) t Tr aa † k S Tr k S + x x x x x x t Tr a ( a † ) k S Tr k S , (57)which can be shown to equal (51). We give the details of calculating the traces inSection 8. atrix product formula for Macdonald polynomials 4. Matrix product for general rank The ZF algebra for the case x = y = 1 is known as the matrix product algebra forthe multi-species asymmetric exclusion process. A general rank solution for this case,first for t = ∞ and then in terms of t -oscillators, was recently obtained by severalauthors in a sequence of works [26, 27, 28, 29, 30]. A generalisation of these resultsthat includes a spectral parameter which can be chosen inhomogeneously was foundearlier in [31] and independently in the case of super-algebras in [32]. In Section 4.1we shall illuminate the very natural combinatorial structure of this solution. We notethat a different inhomogeneous generalisation of the multi-secpies ASEP, with species-dependent hopping parameters, was studied in [41, 43, 42, 44]. Theorem 1. Consider a matrix L ( r ) ( x ) whose entries are given by L ( r ) ij ( x ) = x Q rm = i +1 k m , i = jxa j a † i Q rm = i +1 k m , i > j , i < j (58) for all ≤ i, j ≤ r , and L ( r )0 j = a j , ≤ j ≤ r, L ( r ) i ( x ) = xa † i r Y m = i +1 k m , ≤ i ≤ r, L ( r )00 = 1 , (59) where { a i , a † i , k i } , ≤ i ≤ r are r commuting copies of the t -oscillator algebra (47).Then this L matrix satisfies the intertwining equation (32) . An example of of L for r = 3 is given in (70). A proof of Theorem 1 can be obtainedby a long brute force check of the intertwining equations (32), by distinguishing manydifferent cases. Here we present a more compact and elegant proof. First we start withan easy general property of the solutions of the intertwining equations (32). Lemma 1. Let L ( x ) be a ( r + 1) × ( r + 1) matrix of operators depending on a spectralparameter x , and { v , . . . , v r } and { u , . . . , u r } two constant vectors of commutingoperators that also commute with the entries of L ( x ) . Define the matrix L ( x ) = diag { v , . . . , v r } · L ( x ) · diag { u , . . . , u r } . (i) If L ( x ) is a solution of the intertwining equations (32), then L ( x ) solves (32).(ii) Conversely, if L ( x ) is a solution of the intertwining equations (32) and the operators v i , u j are non identically zero, then L ( x ) solves (32). Next we want to describe how to get a solution of the intertwining equations (32) ofrank r from a solution of rank r − 1. For 0 ≤ m ≤ r − atrix product formula for Macdonald polynomials I m,κ : Mat( r ) → Mat( r + 1) I m,κ ( M ) ij = M ij ≤ i ≤ m − , ≤ j ≤ r − ,κδ j,r i = m, ≤ j ≤ r,M i − ,j m + 1 ≤ i ≤ r, ≤ j ≤ r − , ≤ i ≤ m j = r. (60)Here is an example for r = 3, g = g g g g g g g g g −→ I ,κ ( g ) = g g g 00 0 0 κg g g g g g (61) Lemma 2. Let L ( x ) be a solution of the intertwining equations (32) of rank r − suchthat the first m rows are independent of x , while the last r − m are linear in x . Thematrix I m,κ ( L ( x )) is a solution of the intertwining equations (32) of rank r if and onlyif the operator κ satisfies the following commutations κL ij ( x ) = L ij ( x ) κ ∀ j, ≤ i ≤ m − κL ij ( x ) = tL ij ( x ) κ ∀ j, m ≤ i ≤ r − Proof. The proof of this lemma is a straightforward check of (32) .We have already seen that the matrix L ( a, a † , x ) = axa † x ! (63)satisfies the intertwining equations (32) of rank 1, and we set L (1) ( x ) = L ( a , a † , x ) . (64)Now we are going to construct L ( r ) ( x ) starting from L ( r − ( x ) in a recursive way. Thefirst ingredient we need is the auxiliary matrix L (+ ,r ) ( x ) = I ,P (cid:0) diag { , Q , . . . , Q } · L ( r − ( x ) (cid:1) (65)where P , Q commute with the entries of L ( r − ( x ) and satisfy P Q = tQ P . (66)Assuming that L ( r − ( x ) satisfies the intertwining equations and applying Lemma 1 andLemma 2 it follows that L (+ ,r ) ( x ) satisfies the intertwining equations (32).The second ingredient we need is the matrix ¯ L (+ ,r ) i,j ( x ) defined by¯ L (+ ,r ) ( x ) = I r − ,P k r · · · I ,P k r (cid:0) L ( a r , a † r , x ) · diag { , Q } (cid:1) (67) atrix product formula for Macdonald polynomials P , Q commute with a r , a † r , k r and t -commute among themselves P Q = tQ P . By applying Lemma 1 and repeatedly Lemma 2 we have that ¯ L (+ ,r ) ( x ) also satisfies theintertwining equations (32).Then we put the two ingredients together and define L ( r +1) ( x ) := ¯ L (+ ,r ) ( x ) · L (+ ,r ) ( x ) , (68)which obviously satisfies the intertwining equations (32) if L ( r ) ( x ) does. Comparing L ( r +1) ( x ) and L ( r +1) ( x ), we see that L ( r +1) ( x ) = diag { , Q P , Q P , . . . , Q P , } · L ( r +1) ( x ) · diag { , . . . , , Q P } , (69)therefore Theorem 1 follows from the second point of Lemma 1 and induction on r . L -matrix While the L -matrix as given by equations (58) and (59) solves the intertwining equation(32), the precise form of the entries is quite mysterious without any further explanation.The aim of this section is to show that all entries can be deduced from some governingcombinatorial rules. To that end, we denote the entries of the L -matrix graphically bytiles. The left and right edges of the tiles are either unoccupied (corresponding withindex 0), or occupied by a boson of colour i (corresponding with index i ). For example,for rank 3, the entries of the L -matrix are encoded as follows: L (3) ( x ) = = a a a xk k a † xk k xk a † xk a † a xk xa † xa † a xa † a x (70)where the indexing conventions are = 1, = 2, = 3, with an empty edge representingthe index 0.This information alone is sufficient to label the entries of the L -matrixunambiguously. But for the purpose of motivating the form of the entries it is useful toconsider, further to this, lattice paths which are generated by the bosons. The horizontaledges of tiles can be occupied by arbitrary numbers of bosons from each of the families,and we adopt the convention that the bosons are ordered from darkest to lightest,reading from left to right. In passing from the bottom horizontal edge to the top one,the occupation number of any species of boson can go up/down by 1 (representing theaction of a creation/annihilation operator), or remain the same (representing the actionof some diagonal operator). We keep track of these transitions between states by simply“connecting the dots”. For example, atrix product formula for Macdonald polynomials a † a (creation of a type 2 boson, annihilation of atype 1, with time flow up the page). These graphical conventions suffice to explain allcreation/annhilation operators appearing in the L -matrix (70).In order to specify the zero entries of (70), as well as the inclusion of the k i operators,we need two rules for the lattice paths thus constructed: It is forbidden to create a boson of type i and simultaneously annihilate one of type j , when i < j . We obtain a factor of t every time a type j line horizontally crosses a type i line,where i > j .Taking into account these rules, we can now reproduce the exact form of all entriesof the L -matrix. The zero entries correspond with tiles forbidden by rule . Rule explains which entries are dressed by k i operators. For example, (71) corresponds to L (3)2 , = k a † a , with k producing a factor of t m , where m is the number of type 3 bosonspresent. We easily deduce this from rule , since we will have exactly m horizontalcrossings of type 3 lines by the left-turning type 2 line. A further example:corresponds with L (3)1 , = k k a † , since the left-turning type 1 boson horizontally crossesall type 2 and 3 lines present, and we must keep track of these crossings by the inclusionof k k . To obtain a solution to the rank-reduced intertwining equation (36) we choose therepresentation of the first t -oscillator algebra to be trivial: a = a † = 1 , k = 0 . With this choice of representation, it is easy to see that the first two columns of the L -matrix (58), (59) become equal, leading to an immediate solution of (36) (by simplyomitting one of the redundant columns from the L -matrix). For the example (70), after atrix product formula for Macdonald polynomials t -oscillator algebra, we obtain˜ L (3) ( x ) = = a a xk k xk a † xk xa † xa † a x . (72)Up to some elementary transformations, this is exactly the rank 3 solution obtained in[29] (see equation (47) therein), although here we write our operators with subscripts todistinguish commuting copies of the t -oscillator algebra, rather than the tensor productnotation employed in [29]. Having obtained solutions of (36) for all r , we construct solutions of the ZF algebravia the prescription (39). This is conceptually straightforward, although it introducesa slight notational complexity: we must now distinguish not only between differentfamilies of t -bosons (which we have done so far by using subscripts), but also betweenoperators which act at different lattice sites (which we now do by placing superscriptson our operators). For example the rank 3 solution of the ZF algebra is given by A (3) ( x ) = a a xk k xk a † xk xa † xa † a x (3) · a xk xa † x (2) · x ! (1) = A ( x ) A ( x ) A ( x ) A ( x ) . (73)where the superscript placed on a matrix indicates that all operators within that matrixacquire that superscript. In terms of the graphical conventions that we have introduced,the components A i ( x ) of the rank r solution A ( r ) ( x ) are given by rows of tiles of length r . The left boundary of the row is occupied by a particle of colour i , for 1 ≤ i ≤ r , orunoccupied in the case i = 0. The right boundary is always unoccupied. For example,formulating (73) in terms of tiles, we obtain A (3) ( x ) = (3) · (2) · (1) = A ( x ) A ( x ) A ( x ) A ( x ) . (74)From this it is easy to extract individual components, for example: A ( x ) = (3) (2) (1) + (3) (2) (1) + (3) (2) (1) xk (3)3 a † x k (3)3 k (2)2 x k (3)3 a † a (2)2 atrix product formula for Macdonald polynomials i corresponded with the t -oscillator algebra { a i , a † i , k i } . However (with the conventionsthat we have adopted) in passing to the full solution of the ZF algebra, which is valuedon V ( r ) ⊗ · · · ⊗ V (1) , the notions of colour and family no longer coincide across all V ( j ) .For example in (74), a particle of colour corresponds with the algebra { a , a † , k } when in column 3, with { a , a † , k } when in column 2, and with the trivialized algebra { a , a † , k } when in column 1. The general rule is that a particle of colour i in column r − j corresponds with the t -oscillator family i − j , and this shift should be kept in mindin the subsequent sections. Having constructed solutions of (36) for all values of r , we now seek an explicit operator s which satisfies (38). This is very easily achieved by considering the form of the entriesof ˜ L ( x ). We let s be factorized over the non-trivial copies of the t -oscillator algebra,with the following commutation relations: s = s r · · · s , a i s i = q i − s i a i , a † i s i = q − i s i a † i , k i s i = s i k i , (75)where as usual all operators carry the superscript ( r ), which we have suppressed forvisual clarity. Proposition 1. Equation (38) is satisfied with s as defined by (75) . We obtain a valid solution of (75) by letting s i = k ( i − ui = diag { , q ( i − , q i − , . . . } i (76)which is the representation that we use in all subsequent sections. 5. Transition formulas The aim of this section is to write a recursion relation for the function f λ ( x , . . . , x n )for any composition λ , using the fact that both A ( r ) ( x ) and S ( r ) are recursively defined.Indeed, we can rewrite equations (39) and (40) as A ( r ) ( x ) = ˜ L ( r ) ( x ) A ( r − ( x ) (77) S ( r ) = s ( r ) S ( r − (78)where the action in the vector space V ( r ) is explicitly factorized out. This suggests thedefinition of a transition matrix: T λ,µ ( x , . . . , x n ) := Tr h ˜ L λ ,µ ( x ) · · · ˜ L λ n ,µ n ( x n ) s i , atrix product formula for Macdonald polynomials r ) which we have suppressed, and λ, µ arecompositions whose parts lie in [0 , , . . . , r ] and [0 , , . . . , r − λ ∗ be thecomposition obtained by subtracting 1 from all non-zero parts of λ : λ ∗ i = max( λ i − , . Theorem 2. Let λ be a composition whose parts satisfy ≤ λ i ≤ r , for all ≤ i ≤ n .The following recursion relation holds: f λ ( x , . . . , x n ) = r − Y i =1 − q i Y j ≤ i t m j ( λ ) ! X µ ∈ S n · ( λ ∗ ) + T λ,µ ( x , . . . , x n ) f µ ( x , . . . , x n ) , (79) where m i ( λ ) is the number of parts of size i in λ and the sum is taken over allcompositions µ that are a permutation of λ ∗ .Proof. Rewrite (24) asΩ λ + f λ ( x , . . . , x n ) = Tr [ A λ ( x ) · · · A λ n ( x n ) S ] , for some proportionality factor Ω λ + = Ω ( r ) λ + that remains to be determined (and inparticular needs to be shown different from zero). We remark that the factor Ω ( r ) λ + isthe same for all partitions λ with the common re-ordering λ + . Combining (80) withequations (77) and (78), we obtainΩ ( r ) λ + f λ ( x , . . . , x n ) = X µ Tr h ˜ L ( r ) λ ,µ ( x ) · · · ˜ L ( r ) λ n ,µ n ( x n ) s ( r ) i Tr (cid:2) A ( r − µ ( x ) · · · A ( r − µ n ( x n ) S ( r − (cid:3) , (80)which can be rewritten asΩ ( r ) λ + f λ ( x , . . . , x n ) = X µ T λ,µ ( x , . . . , x n )Ω ( r − µ + f µ ( x , . . . , x n ) . (81)At this stage the sum in (81) is taken over all compositions µ , with parts in[0 , , . . . , r − Proposition 2. For any two compositions λ and µ , if µ + = ( λ ∗ ) + then T λ,µ ( x , . . . , x n ) = 0 .Proof. Given a composition λ , let m i ( λ ) be the number of its parts of size i . We wishto show that if m i ( λ ) = m i − ( µ ) for some i ≥ 2, then T λ,µ ( x , . . . , x n ) = 0.For each i ≥ 2, let i ( λ, µ ) be the number of pairs ( λ k , µ k ) = ( i, i − L ( x ), we see that the number of a † i operators appearing in T λ,µ ( x , . . . , x n ) is equal to m i ( λ ) − i ( λ, µ ). Conversely, the number of a i operatorsappearing in this trace is given by m i − ( µ ) − i ( λ, µ ). In order for the trace to benon-vanishing, these two numbers must be equal for all i ≥ atrix product formula for Macdonald polynomials ( r ) λ + f λ ( x , . . . , x n ) = Ω ( r − λ ∗ ) + X µ ∈ S n · ( λ ∗ ) + T λ,µ ( x , . . . , x n ) f µ ( x , . . . , x n ) (82)To complete the proof of (79), it remains only to demonstrate the following result. Proposition 3. The proportionality factors appearing in (82) satisfy Ω ( r ) λ + = Ω ( r − λ ∗ ) + r − Y i =1 − q i t λ ′ − λ ′ i +1 , (83) where λ ′ is the conjugate of λ + .Proof. Without losing generality, we ease notation by assuming that λ is a partition. Wecompare coefficients of the monomial x λ := x λ · · · x λ n n on both sides of (82). Since thematrix elements T λ,µ ( x , . . . , x n ) are at most degree 1 in each x i , the only contributionto x λ in the right hand side comes from µ = λ ∗ . Therefore we can writeΩ ( r ) λ x λ = Ω ( r − λ ∗ T λ,λ ∗ ( x , . . . , x n ) x λ ∗ = Ω ( r − λ ∗ T λ,λ ∗ (1 , . . . , x λ , (84)or more simply, Ω ( r ) λ = Ω ( r − λ ∗ T λ,λ ∗ (1 , . . . , T λ,λ ∗ (1 , . . . , 1) = r − Y i =1 Tr (cid:2) k m ( λ ) · · · k m i ( λ ) k iu (cid:3) = r − Y i =1 − t m ( λ ) · · · t m i ( λ ) q i . Thanks to this proposition, (82) implies (79) provided that Ω ( r − λ ∗ is non-vanishing.This can be deduced inductively on r , using (83) with Ω (0) = 1. Indeed, we find thatΩ ( r ) λ = r Y j =1 r − j Y i =1 − q i Q j ≤ k ≤ i ≤ n . In this case the summation on the right hand side is trivial, and werecover f λ ( x , . . . , x n ) = n Y i =1 x i ! f λ ∗ ( x , . . . , x n ) . atrix product formula for Macdonald polynomials 6. Combinatorial interpretation of matrix product formula In view of the graphical representation of the elements of the ˜ L -matrices, it is possibleto interpret the matrix product formula (24) entirely in terms of lattice paths. Thisleads to a combinatorial rule for f λ ,...,λ n ( x , . . . , x n ), for any composition λ .We consider a lattice formed by n rows of tiles, with associated spectral parameters x , . . . , x n , where the index of the spectral parameters increases as we go from the toprow to the bottom. Each row is r tiles wide, where r is the size of the largest partin λ . For each λ i > 0, there is a boson of colour i incident on the left boundaryof the i th row. However, the rows i for which λ i = 0 have no particle at the leftboundary. The right boundary of the lattice is completely unoccupied by particles. Thelattice thus constructed reproduces all terms in the matrix product (24), since each rowcorresponds with one of the components A i ( x ) of A ( r ) ( x ). For example, for r = 3 and λ = (0 , , , , , A ( x ) A ( x ) A ( x ) A ( x ) A ( x ) A ( x ) S ) = (3) (2) (1) x x x x x x Some care needs to be taken in regard to the boundary conditions on the top andbottom edges of the lattice. Since we are taking a trace in (24), the correct way toview the lattice is with the top and bottom edges identified, so that lattice paths arepermitted to “wrap around” the cylinder thus formed. This ensures that in each columnof the lattice, there is conservation between the number of particles entering/leaving thatcolumn (in other words, all creation/annihilation operators come in pairs, ensuring anon-vanishing trace). The sole exception to this rule involves the boson species whoserepresentation has been trivialized. Recall that in column j of the lattice, bosons fromany of the families { , . . . , j } are allowed, but we trivialize the representation of thefirst family. Hence at the top of each column we are permitted to eject type 1 particleswithout causing the trace to vanish. In the example above, we eject 1 trivial boson fromcolumn 3, 2 trivial bosons from column 2, and 1 trivial boson from column 1, since theseare the corresponding multiplicities in the composition λ .Having set up the boundary conditions of the lattice in this way, we obtain allpossible terms in the matrix product (24) by summing over all lattice paths which evolve atrix product formula for Macdonald polynomials f λ ( x , . . . , x n ) in terms of an n × r latticeas described, then (79) can be interpreted as summing over all possible configurations ofcolumn r of the lattice, with the remaining r − f µ ( x , . . . , x n ).The elements T λ,µ ( x , . . . , x n ) of the transition matrix can thus be viewed as partitionfunctions of a single column. We illustrate the calculation of T λ,µ ( x , . . . , x n ) on anexplicit example in the Appendix. We give an example of the lattice-path calculation for f δ ( x , . . . , x n ), where δ =(0 , , , , , 2) is an anti-dominant weight. This is of particular interest, since in thiscase f δ = E δ ( x , . . . , x n ; q, t ). The possible lattice configurations for this example: (2) (1) x x x x x x (2) (1) (2) (1)(2) (1) x x x x x x (2) (1) (2) (1) where we highlight in green the tiles which give rise to an x weight (these are the tileswhich have an occupied left edge). Note that in column 2 of the lattice it is possible atrix product formula for Macdonald polynomials δ + E δ ( x , . . . , x ; q, t ) = x x x x Tr[ k S ] + x x x x x Tr[ ak a † S ] + x x x x x Tr[ ak a † S ]+ x x x x x Tr[ ak a † S ] + x x x x x Tr[ ak a † S ] + x x x x x x Tr[ aak a † a † S ] , in agreement with our earlier calculation (57). 7. Symmetric Macdonald polynomials We relate our results to symmetric Macdonald polynomials. The following result isalready mentioned in [36], Lemma 3. Let λ be a dominant composition, i.e. a partition. Then the sum P λ ( x , . . . , x n ; q, t ) = X µ f µ ( x , . . . , x n ; q, t ) , (85) where the sum runs through all permutations µ of λ + , is symmetric.Proof. We need to show that T i P λ = t / P λ for all i = 1 , . . . , n − 1. From (18) and (6)we find for λ i < λ i +1 that t / T i f ...,λ i ,λ i +1 ,... = tT i f ...,λ i +1 ,λ i ,... = t (cid:0) t / − t − / ) T i (cid:1) f ...,λ i +1 ,λ i ,... = tf ...,λ i +1 ,λ i ,... + ( t − f ...,λ i ,λ i +1 ,... . (86)Combining this with (17) and (18) we thus find t / T i X µ f µ = X µ : µ i <µ i +1 ( tf s i µ + ( t − f µ ) + X µ : µ i = µ i +1 tf µ + X µ : µ i >µ i +1 f s i µ = X µ : µ i <µ i +1 tf s i µ + X µ : µ i ≤ µ i +1 tf µ = t X µ f µ . (87)The Macdonald polynomial P λ is the unique symmetric polynomial (up tonormalisation) which can be obtained by taking linear combinations of the non-symmetric Macdonald polynomials E µ , where µ is a permutation of λ . By (23), each f µ can be written as a linear combination of non-symmetric Macdonald polynomials E ν ,where ν is a permutation of µ and ν ≤ µ . It follows that P λ is a linear combination of E µ such that µ is a permutation of λ , and since P λ is symmetric, P λ and P λ must beequal up to normalisation, due to uniqueness. By this argument and Lemma 3, we havethe following theorem, atrix product formula for Macdonald polynomials Theorem 3. For λ ⊂ r n Ω λ P λ ( x , . . . , x n ; q, t ) = X µ ∈ S n · λ Tr S n Y i =1 A µ i ( x i ) , (88) where Ω λ = Ω ( r ) λ , S = S ( r ) , A µ = A ( r ) µ and the sum is over all permutations µ of λ . A corollary of Theorem 3 is that the generating function for symmetric Macdonaldpolynomials for partitions λ ⊂ r n can be written as a single matrix product. Define first A ( y , . . . , y r ; x ) = r X i =0 y i A i ( x ) . (89)Notice that [ A ( y , . . . , y r ; x ) , A ( y , . . . , y r ; x ′ )] = 0 by a similar argument as in Lemma 3.Moreover we have thatTr[ S n Y i =1 A ( y , . . . , y r ; x i )] = X λ ⊂ r n Ω λ r Y i =0 y m i ( λ ) i P λ ( x , . . . , x n ; q, t ) , (90)where again all superscripts ( r ) are suppressed and m i ( λ ) is the multiplicity of part i in λ . Special values of q include the Hall–Littlewood polynomials ( q = 0), which havebeen expressed as Bethe wave-functions in the t -boson model [45]. The structure of theexpression in [45] has some features in common with our matrix product formula, but isdistinguished by some important differences. In particular, higher rank L -matrices arenot needed in the Hall–Littlewood formula, and the construction does not involve a traceoperation. Other special values of q include Schur functions ( q = t ), Jack polynomials( q = t α , t → 1) and q = 1 results in the normalisation of the inhomogeneous multi-speciesasymmetric exclusion process with periodic boundary conditions. The homogeneouslimit of the latter is recovered by sending x i → 8. Calculation of traces So far we have explained how to calculate the matrix product (24) in terms of traces of t -oscillators, without explicitly evaluating these. We now turn to the evaluation of thetraces, using the representation (3) of the t -oscillator algebras. Let us start with theexamples encountered in equation (57):Tr[ k p ] = ∞ X m =0 ( t p ) m = 11 − t p = 1(1 − t )[ p ] . Tr[ aa † k p ]Tr[ k p ] = P ∞ m =0 ( t p ) m (1 − t m +1 )Tr[ k p ] = 1[ p + 1] , Tr[ aaa † a † k p ]Tr[ k p ] = P ∞ m =0 ( t p ) m (1 − t m +1 )(1 − t m +2 )Tr[ k p ] = [2][ p + 1][ p + 2] . (91) atrix product formula for Macdonald polynomials t -oscillators can be evaluated by simply summingup geometric series. On the other hand, as the examples become more complicated, theanswer is no longer neatly factorized. In order to evaluate the most general case, let D ℓ be the set of all Dyck paths of length 2 ℓ . It is easy to verify that all types of tracesarising in this work have the generic form Tr[D ℓ k p ] where p is an arbitrary exponent,and D ℓ represents a Dyck path of length 2 ℓ generated by the oscillators ( a means anup step, a † means a down step). Indeed, the trace of any string of t -oscillators can bebrought into this form (up to overall factors of t ) using the cyclicity of the trace and thecommutation relations (1). Let us denote D ℓ by a series of open/closed parentheses, forexample: Tr[()(()) k p ] := Tr[ aa † aaa † a † k p ] . We consider the following map: M : D ℓ → N ℓ D ℓ ( m , . . . , m ℓ ) (92)where m i is the number of parenthetic pairs ( . . . ) in D ℓ which are surrounded by i − m = 1, m = 2 and m = 1. Using this map we deduce the following lemma. Lemma 4. Tr[D ℓ k p ] = ∞ X n =0 t pn (1 − t n +1 ) m . . . (1 − t n + ℓ ) m ℓ , (93) where ( m , . . . , m ℓ ) = M (D ℓ ) . It is now useful to introduce the following operators on functions: δ t [ f ( z )] = f ( z ) − f ( tz ) , ∆ ( m ) t = z ◦ δ t ◦ · · · ◦ δ t | {z } m , m ≥ , ∆ (0) t = z. (94)The operator ∆ ( m ) t acts multiplicatively on monomials and it is easy to see that∆ ( m ) t ( z n ) = (1 − t n ) m z n +1 . (95)It therefore follows that ∞ X n =0 x n (1 − t n +1 ) m . . . (1 − t n + ℓ ) m ℓ = " ∆ ( m ℓ ) t · · · ∆ ( m ) t ∞ X n =0 x n z n +1 z =1 = (cid:20) ∆ ( m ℓ ) t · · · ∆ ( m ) t z − xz (cid:21) z =1 . (96)If we define the function ψ [ m ,...,m ℓ ] ( x ) = (cid:20) ∆ ( m ℓ ) t · · · ∆ ( m ) t z − xz (cid:21) z =1 , (97) atrix product formula for Macdonald polynomials ℓ k p ] = ψ [ m ,...,m ℓ ] ( t p ) , where ( m , . . . , m ℓ ) = M (D ℓ ) . Hence we are able to evaluate all traces that we encounter in terms of the function (97),whose definition is relatively elementary. 9. Conclusion The main result of this paper is a matrix product formula for Macdonald polynomialsin terms of deformed bosonic operators. This formula implies a new explicit way toefficiently compute these polynomials and also provides a combinatorial interpretation.Our result firmly connects the polynomial representation theory of the affine Heckealgebra with the theory of solvable lattice models using tools such as the Yang–Baxterequation, R -matrices, the Zamolodchikov–Faddeev algebra and the deformed Knizhnik–Zamolodchikov equation.The results discussed in this paper have a direct generalisation to theinhomogeneous multi-species asymmetric exclusion process with boundaries [48, 49, 46,47, 50] resulting in a matrix product formula for Koornwinder polynomials [51]. Aconnection between Koornwinder polynomials and the quantum XXZ chain, which isclosely related to the exclusion process by a similarity transformation, was made in [52].We note that a different generalisation of the multi-species asymmetric exclusionprocess, using inhomogeneous hopping parameters, was discussed in [41, 43, 42, 44]. Itwould be interesting to clarify the matrix product structure [43] in this case, as well asthe relation to integrability [53]. Acknowledgment LC has been supported by the CNRS through a Chaire d’Excellence. JdG and MWare generously supported by the Australian Research Council (ARC) and the ARCCentre of Excellence for Mathematical and Statistical Frontiers (ACEMS). We thankKayed Al Qasemi, Eric Ragoucy, Sergey Sergeev, Ole Warnaar and Paul Zinn-Justin fordiscussion. Appendix A. A further combinatorial example We illustrate the combinatorial meaning of (79) on a small rank 3 example, namely f λ ( x , x , x , x ) with λ = (3 , , , f (3 , , , can be expanded intofunctions f µ where µ is a permuation of (2 , , , L (3) vanish, we haveonly to consider the permutations ( π , π , π , π ) of (2, 1, 0, 0) such that π = 1 , π = 2. Hence we obtain the following four terms in the expansion: f λ = T λ, (2 , , , f (2 , , , + T λ, (0 , , , f (0 , , , + T λ, (2 , , , f (2 , , , + T λ, (1 , , , f (1 , , , . (A.1) atrix product formula for Macdonald polynomials x x x x (3) (2) (1) x x x x (3) (2) (1) Replacing each tile in column 3 with the operator(s) it represents, this becomes f (3 , , , = (1 − qt )(1 − q t ) x x x (cid:16) Tr[ k k k s ] f (2 , , , + Tr[ a † k k a k s ] f (0 , , , + Tr[ k k a k a † s ] f (2 , , , + Tr[ a † a k k a k a † s ] f (1 , , , (cid:17) . 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