aa r X i v : . [ m a t h . QA ] A ug FREE FIELD APPROACH TO THE MACDONALD PROCESS
SHINJI KOSHIDA
Abstract.
The Macdonald process is a stochastic process on the collection of parti-tions that is a ( q, t )-deformed generalization of the Schur process. In this paper, weapproach the Macdonald process identifying the space of symmetric functions with aFock representation of the deformed Heisenberg algebra. By using the free field real-ization of operators diagonalized by the Macdonald symmetric functions, we propose amethod of computing several correlation functions with respect to the Macdonald pro-cess. It is well-known that expectation value of several observables for the Macdonaldprocess admit determinantal expression. We find that this determinantal structure isapparent in free field realization of the corresponding operators. We also propose ageneralized Macdonald measure motivated by recent studies on generalized Macdonaldfunctions whose existence relies on the Hopf algebra structure of the Ding–Iohara–Mikialgebra. Introduction
Let Y n be the collection of partitions of n ∈ Z ≥ , and set Y := S ∞ n =0 Y n , where Y = {∅} . The Macdonald measure MM q,t is a probability measure on Y definedby [BC14, BCGS16] MM q,t ( λ ) = 1Π( X, Y ; q, t ) P λ ( X ; q, t ) Q λ ( Y ; q, t ) , λ ∈ Y . Here P λ ( X ; q, t ) is the Macdonald symmetric function of X = ( x , x , . . . ) for a partition λ and Q λ ( Y ; q, t ) is its dual symmetric function of Y = ( y , y , . . . ) (see Sect. 2 fordefinition). From the Cauchy identity, the normalization factor is computed asΠ( X, Y ; q, t ) = X λ ∈ Y P λ ( X ; q, t ) Q λ ( Y ; q, t ) = Y i,j ≥ ( tx i y j ; q ) ∞ ( x i y j ; q ) ∞ , where ( a ; q ) ∞ = Q ∞ n =0 (1 − aq n ). In the following, we suppress the parameters q and t ifthere is no ambiguity, for instance, by writing P λ ( X ) = P λ ( X ; q, t ). Precisely speaking,to obtain a genuine probability measure, we have to adopt a nonnegative specializationof the Macdonald symmetric functions whose classification was conjectured in [Ker92]and recently proved in [Mat19].As a generalization of the Macdonald measure, the N -step Macdonald process [BC14,BCGS16] for N ≥ MP Nq,t on Y N defined so that the probability Mathematics Subject Classification.
Key words and phrases.
Macdonald process, Macdonald symmetric function, Ding–Iohara–Miki al-gebra, generalized Macdonald functions. for a sequence ( λ (1) , . . . , λ ( N ) ) ∈ Y N of partitions is given by MP Nq,t ( λ (1) , . . . , λ ( N ) )(1.1) := P λ (1) ( X (1) )Ψ λ (1) ,λ (2) ( Y (1) , X (2) ) · · · Ψ λ ( N − ,λ ( N ) ( Y ( N − , X ( N ) ) Q λ ( N ) ( Y ( N ) ) Q ≤ i ≤ j ≤ N Π( X ( i ) , Y ( j ) ) . Here the transition function Ψ λ,µ ( Y, X ) is given by(1.2) Ψ λ,µ ( Y, X ) = X ν ∈ Y Q λ/ν ( Y ) P µ/ν ( X ) , λ, µ ∈ Y with P λ/ν being the skew-Macdonald symmetric functions for a skew-partition λ/ν and Q λ/µ being its dual. The case of N = 1 is just the Macdonald measure, MM q,t = MP q,t .It is known that the Macdonald process reduces to several interesting stochastic mod-els by specializing the variables and limiting the parameters and has given many ap-plications to probability theory (see [BG12, Bor14, BC14, BP14]). In particular, whenwe set q = t , the Macdonald symmetric functions reduce to the Schur functions and,correspondingly, the Macdonald process reduces to the Schur process [Oko01, OR03].The Schur process can be shown to be a determinantal point process in a simple mannerowing to the infinite-wedge realization of the Schur functions and action of the chargedfree fermion [Oko01]. The analogous field theoretical approach to the Macdonald processis, however, absent to the author’s knowledge.In the present paper, we study the Macdonald process by identifying the space ofsymmetric functions with a Fock representation of the deformed Heisenberg algebraas was suggested in [FW17]. We use the free field realization of operators that arediagonalized by the Macdonald symmetric functions [FHH +
09] to compute expectationvalue of several observables for the Macdonald process. Some of our results reproducethe ones presented in [BCGS16]. We remark that our approach is fully algebraic andformal and does not require any specialization of the variables. Therefore, our resultsapply to any models obtained by specialization of the Macdonald process.It is well-known [BC14,BCGS16] that several expectation values concerning the Mac-donald process admit determinantal expression. The initial motivation of this work wasunderstanding the origin of this determinantal structure. We have found that the deter-minantal structure gets apparent in the free field realization of operators diagonalizedby the Macdonald symmetric functions. Therefore, we could say that the determinantalstructure of the Macdonald process originates from the free field realization, but thedeterminantal structure in the free field realization is still mysterious.We also propose a certain generalization of the Macdonald measure. It is known thatthe Ding–Iohara–Miki (DIM) algebra [DI97, Mik07] plays a relevant role in the theoryof the Macdonald symmetric functions. In [AFH + Correspondence among correlation functions.
We set F = C ( q, t ). Then weregard a function f : Y → F as a random variable or an observable. REE FIELD APPROACH TO THE MACDONALD PROCESS 3
Definition 1.1.
Let f , . . . , f N : Y → F be random variables. The correlation function E Nq,t [ f [1] · · · f N [ N ]] with respect to the N -step Macdonald process is defined by E Nq,t [ f [1] · · · f N [ N ]] := X ( λ (1) ,...,λ ( N ) ) ∈ Y N f ( λ (1) ) · · · f N ( λ ( N ) ) MP Nq,t ( λ (1) , . . . , λ ( N ) ) . In case of N = 1, we simply write E q,t [ f ] := E q,t [ f [1]].We write Λ for the ring of symmetric functions over F . Then it is isomorphic to aFock representation F and its dual F † of a Heisenberg algebra (see Sect 3), where theMacdonald symmetric function P λ corresponding to λ ∈ Y is identified with h P λ | ∈ F † and its dual Q λ is identified with | Q λ i ∈ F . We introduce operators on F :(1.3) Γ( X ) ± = exp X n> − t n − q n p n ( X ) n a ± n ! , where a n , n ∈ Z \{ } are generators of the Heisenberg algebra and p n ( X ), n ≥ n -th power sum symmetric function of variable X = ( x , x , . . . ) (see Sect. 2).We write F [ Y ] := { f : Y → F } for the totality of random variables. The methodof computing correlation functions using the Macdonald difference operators has beendeveloped in [BC14, BC15, BCGS16, Dim18, GZ18]. Here we shall formulate a comple-mentary algebraic scheme to compute correlation functions. Definition 1.2.
Regarding a random variable as a spectrum, we define a mapping O : F [ Y ] → End( F ); f X λ ∈ Y f ( λ ) | P λ i h Q λ | . For a random variable f ∈ F [ Y ], we also define ψ X,Yf := Γ( Y ) + O ( f )Γ( X ) − . Then we have the following correspondence between correlation functions under aMacdonald process and ones in the Fock space.
Theorem 1.3.
Let f , . . . , f N ∈ F [ Y ] be random variables. Then their correlation func-tion with respect to the N -step Macdonald process becomes E Nq,t [ f [1] · · · f N [ N ]] = h | ψ X ( N ) ,Y ( N ) f N · · · ψ X (1) ,Y (1) f | ih | ψ X ( N ) ,Y ( N ) · · · ψ X (1) ,Y (1) | i . Here | i ∈ F and h | ∈ F † are the vacuum vectors and ∈ F [ Y ] is the unit constantfunction given by λ ) = 1 , λ ∈ Y . Theorem 1.3 is proved in Sect. 5, where we also present some applications.1.2.
Determinantal expression of operators.
The Macdonald symmetric functionsare simultaneous eigenfunctions of commuting operators including the Macdonald oper-ators. Under the isomorphism F ≃ Λ, these operators are identified with operators on F , which were studied in [Shi06, FHH +
09] as free field realization.
SHINJI KOSHIDA
We shall seek different expression of these free field realization using determinant.Let us introduce a vertex operator (1.4) η ( z ) = exp X n> − t − n n a − n z n ! exp − X n> − t n n a n z − n ! , which lies in End( F )[[ z, z − ]]. Theorem 1.4.
Let r = 1 , , . . . . The free field realization ˆ E r of the r -th Macdonaldoperator is expressed as ˆ E r = t − r r ! Z r Y i =1 dz i π √− ! det (cid:18) z i − t − z j (cid:19) ≤ i,j ≤ r : η ( z ) · · · η ( z r ): . Here the linear functional
R (cid:16)Q ri =1 dz i π √− (cid:17) takes the residue. We prove Theorem 1.4 in Sect. 4 and also show determinantal expression of otheroperators. Combining these determinantal expressions with Theorem 1.3, we derivedeterminantal formulas of correlation functions in Sect. 5.1.3.
Generalized Macdonald measure.
Let m ∈ N be fixed. In [FHS +
10, AFH + m -fold tensor product e F ⊗ m of e F := C ( q / , t / ) ⊗ F F admits aMacdonald type basis labeled by m -tuple of partitions. Under the isomorphism F ⊗ m ≃ Λ ⊗ m , the level m generalized Macdonald functions P λ ( X ), λ = ( λ (1) , . . . , λ ( m ) ) ∈ Y m ,and their dual Q λ ( X ) are defined. We define the level m generalized Macdonald measureas a probability measure GM mq,t on Y m so that GM mq,t ( λ ) ∝ P λ ( X ) Q λ ( Y ) , λ ∈ Y m . In Sect. 6, we compute expectation value of a random variable under a generalizedMacdonald measure as demonstration.We close this introduction by listing the future directions.
Application to stochastic models.
Since our results are formal and do not requireany specialization of variables, they apply to any reduction of the Macdonald process.For application to stochastic models, however, one has to specialize variables and carryout further analyses typically studying asymptotic behaviors (e.g. [FV15, Bar15]). It isnot clear so far how our results are useful for such application and more study is needed.
Further studies on generalized Macdonald measure.
We need to study general-ized Macdonald measure in application to stochastic models. For this purpose, we haveto consider positive specialization of generalized Macdonald functions to define a gen-uine probability measure. We also need to define skew generalized Macdonald functionsand combinatorial formula of their few-variable specialization.
Elliptic generalization.
In [Sai13, Sai14], the elliptic Macdonald operators were re-alized as operators on a Fock space by means of the elliptic DIM algebra. Thoughthe elliptic Macdonald symmetric functions as a basis of the Fock space have not beencaptured so far, once a Macdonald-type basis is found, a similar story as in this paperwould work in the elliptic case.
REE FIELD APPROACH TO THE MACDONALD PROCESS 5
Relation to higher spin six-vertex models.
Another pillar than the Macdonaldprocess in the field of integrable probability is a higher spin six-vertex model [BCG16,CP16, BP18] and there are attempts to understand these two on the same footing[GdGW17, Bor18]. Notably, partition functions of a higher spin six-vertex model give afamily of symmetric rational functions that are regarded as generalization of the Hall–Littlewood polynomials. It is also known [BBBF18, BBBG18] that, for some latticemodels such as a metaplectic ice model, a vertex operator acting on a Fock space worksas a transfer matrix and its matrix elements give a family of symmetric functions. Sincea Fock space and vertex operators are also basic tools in this paper, the present workcould give a new insight to this subject from a perspective of the representation theoryof quantum algebras.The present paper is organized as follows: In Sect. 2, we review basic notions ofsymmetric functions and introduce the Macdonald symmetric functions and the skew-Macdonald symmetric functions, which are needed to define the Macdonald process(1.1). In Sect. 3, we recall that the space of symmetric functions is isomorphic to aFock representation of the deformed Heisenberg algebra and see the free field realizationof operators that are diagonalized by the Macdonald symmetric functions. In Sect. 4,we rewrite the free field realizations in Sect. 3 by using determinants to prove Theorem1.4 and others. In Sect. 5, we prove theorem 1.3 and present its applications followingdeterminantal expression obtained in Sect. 4. In Sect. 6, we introduce generalizedMacdonald measure and compute expectation value of a random variable. In AppendixA, we give a proof of free field realization of a certain family of operators diagonalizedby the Macdonald symmetric functions.Throughout this paper, we use the notations[ n ] q = 1 − q n − q , [ n ] q ! = n Y k =1 [ k ] q , ( x ; q ) n = n − Y k =0 (1 − xq k ) . Acknowledgements.
The author is grateful to Makoto Katori, Tomohiro Sasamoto,Takashi Imamura, Yoshihiro Takeyama and Alexander Bufetov for fruitful discussion.This work was supported by the Grant-in-Aid for JSPS Fellows (No. 17J09658, No.19J01279) 2.
Preliminaries on symmetric functions
In this paper, we regard the parameters q and t as indeterminates and set F := C ( q, t ).Let us prepare some terminologies of symmetric functions and introduce the Macdonaldsymmetric functions. The relevant reference is [Mac95].2.1. Ring of symmetric functions.
Let Λ ( n ) = F [ x , . . . , x n ] S n be the ring of sym-metric polynomials in n variables over F . For m > n , we define a surjectionΛ ( m ) → Λ ( n ) ; ( x , . . . , x n , x n +1 , . . . , x m ) ( x , . . . , x n , , . . . , . Then, the collection { Λ ( m ) → Λ ( n ) } m>n forms a projective system. We denote itsprojective limit in the category of graded rings as Λ = lim ←− n Λ ( n ) . For a symmetricfunction F ∈ Λ, its image under the canonical surjection Λ ։ Λ ( n ) , will be denoted as SHINJI KOSHIDA F ( n ) , n ∈ Z ≥ . In the following, we write X = ( x , x , . . . ) for a set of infinitely manyvariables and use the notation Λ X if the variables are need to be specified.A sequence of integers λ = ( λ , λ , . . . ) is said to be a partition if λ ≥ λ ≥ · · · ≥ | λ | := P ∞ i =1 λ i < ∞ . The number | λ | is called the weight of λ . If | λ | = n , we saythat λ is a partition of n and write λ ⊢ n . For a partition λ , its length is defined by ℓ ( λ ) := max { i = 1 , , . . . | λ i > } . Obviously, if λ ⊢ n , then ℓ ( λ ) ≤ n and the equality is realized by λ = (1 , , . . . , | {z } n , , . . . ).For a partition λ = ( λ , λ , . . . ), the multiplicity of i ∈ Z ≥ in λ is defined by m i ( λ ) := |{ j = 1 , , . . . | λ j = i }| . Using the multiplicity, we also write the partition as λ = (1 m ( λ ) m ( λ ) · · · ).We denote the collection of partitions of n ∈ Z ≥ by Y n = { λ | λ ⊢ n } and set Y = S n ∈ Z ≥ Y n with Y = {∅} . For two partitions λ, µ ∈ Y we write λ ≥ µ if | λ | = | µ | and λ + · · · + λ n ≥ µ + · · · + µ n , n = 1 , , . . . . Then ≥ defines a partial order on Y called the dominance order.For two partitions λ, µ ∈ Y , we say that µ is included in λ if λ i ≥ µ i , i = 1 , , . . . ,hold, and write µ ⊂ λ . Their difference is called a skew-partition and denoted as λ/µ .Let us introduce some important symmetric functions. For r ∈ Z > , the r -th elemen-tary symmetric function e r ( X ) is defined by e r ( X ) := X i < ··· ) r , on which the r -th symmetric group acts by permutation of components. Then we write S S r ( λ ) forthe subgroup of S r consisting of permutations that fix λ . The monomial symmetricfunction m λ ( X ) corresponding to λ is defined as m λ ( X ) = X i < ···
Macdonald symmetric functions.
To define Macdonald symmetric polynomi-als, we introduce Macdonald difference operators. Fix n ∈ Z ≥ . Then for r = 1 , . . . , n ,the r -th Macdonald difference operator D ( n ) r acting on Λ ( n ) is defined by [Mac95, Sect.VI. 3] D ( n ) r = D ( n ) r ( q, t ) := t r ( r − / X I ⊂{ , ,...,n }| I | = r Y i ∈ Ij I tx i − x j x i − x j Y i ∈ I T q,x i , where T q,x i is the q -shift operator ( T q,x i f )( x , . . . , x n ) := f ( x , . . . , qx i , . . . , x n ). Proposition 2.1.
For λ ∈ Y such that ℓ ( λ ) ≤ n , there is a unique symmetric polynomial P ( n ) λ ( x ; q, t ) ∈ Λ n satisfying P ( n ) λ ( x ; q, t ) = m ( n ) λ ( x ) + X µ<λ c λµ m ( n ) µ ( x ) ,D ( n ) r ( q, t ) P ( n ) λ ( x ; q, t ) = e ( n ) r ( t n s λ , . . . , t n s λn ) P ( n ) λ ( x ; q, t ) , r = 1 , . . . , n. Here we set s λi = q λ i t − i , i = 1 , . . . , n . Lemma 2.2.
Fix n ∈ Z ≥ and let λ ∈ Y be such that ℓ ( λ ) ≤ n . Then, for any r ∈ Z ≥ ,we have e r ( s λ ) = r X k =0 t − nr − ( r − k +12 )( t − ; t − ) r − k e ( n ) k ( t n s λ ) . Here we understand the part / ( t − ; t − ) r − k as t − ; t − ) r − k = r − k Y i =1 − t − i = r − k Y i =1 ∞ X j =0 t − ij . This lemma immediately implies the following result.
Proposition 2.3.
Let r ∈ Z ≥ and n ≥ r . Define a difference operator on Λ ( n ) E ( n ) r := r X k =0 t − nr − ( r − k +12 )( t − ; t − ) r − k D ( n ) k . Then, for an arbitrary λ ∈ Y such that ℓ ( λ ) ≤ n , we have E ( n ) r P ( n ) λ ( x ; q, t ) = e r ( s λ ) P ( n ) λ ( x ; q, t ) . In particular, the projective limit E r = lim ←− n E ( n ) r exists. Correspondingly, the Macdonald symmetric functions are characterized by the follow-ing properties:
Theorem 2.4.
Let λ ∈ Y be a partition. The corresponding Macdonald symmetricfunction P λ ( X ; q, t ) ∈ Λ is uniquely characterized by the following properties: P λ ( X ; q, t ) = m λ ( X ) + X µ<λ c λµ m µ ( X ) , c λµ ∈ F ,E P λ ( X ; q, t ) = e ( s λ ) P λ ( X ; q, t ) . SHINJI KOSHIDA
Moreover, the Macdonald symmetric functions also diagonalize the higher operators sothat E r P λ ( X ; q, t ) = e r ( s λ ) P λ ( X ; q, t ) , r = 2 , , . . . . Since, by definition, a Macdonald symmetric function is triangulated by monomialsymmetric functions, the collection { P λ : λ ∈ Y } forms a basis of Λ. Example . Let us write down the difference operator E ( n )1 . It reads E ( n )1 = t − n D ( n )1 + t − n − − t − = t − n n X i =1 n Y j =1 j = i tx i − x j x i − x j T q,x i + ∞ X i =0 t − n − i − . Remark . The following operator is also frequently used: E ( n ) := E ( n )1 − ∞ X i =1 t − i = t − n n X i =1 n Y j =1 j = i tx i − x j x i − x j T q,x i − n X i =1 t − i . The projective limit E = lim ←− n E ( n ) also exists and is diagonalized by the Macdonaldsymmetric functions so that EP λ ( X ; q, t ) = ∞ X i =1 ( q λ i − t − i P λ ( X ; q, t ) , λ ∈ Y . Definition using an inner product.
The Macdonald symmetric functions arealso characterized as an orthogonal basis of Λ with respect to an inner product definedbelow. We write the inner product as h· , ·i q,t : Λ × Λ → F and define it as [Mac95, Sect.VI. 2] h p λ , p µ i q,t = z λ ( q, t ) δ λ,µ , λ, µ ∈ Y , where we set z λ ( q, t ) = z λ ℓ ( λ ) Y i =1 − q λ i − t λ i , z λ = ∞ Y i =1 m i ( λ )! i m i ( λ ) , λ ∈ Y . Theorem 2.7.
The Macdonald symmetric functions P λ , λ ∈ Y are characterized by thefollowing properties: P λ = m λ + X µ<λ c λµ m µ , c λµ ∈ F , h P λ , P µ i q,t = 0 , λ = µ. We set Q λ := h P λ ,P λ i q,t P λ . Then the collection { Q λ } λ ∈ Y is the dual basis of { P λ } λ ∈ Y with respect to the inner product h· , ·i q,t .2.4. Other operators.
We also consider the following operators: Let r, n ∈ Z ≥ anddefine an operator on Λ ( n ) by H ( n ) r := X ν ∈ ( Z ≥ ) n | ν | = r Y ≤ i Here we wrote ν = ( ν , . . . , ν n ) and | ν | = P ni =1 ν i . These operators are also diagonal-ized by the Macdonald polynomials. To describe the eigenvalues, we set g r ( · ; q, t ) := Q ( r ) ( · ; q, t ), r = 1 , , . . . . Note that we have [Mac95, Chapter VI, (2.8), (5.5)](2.1) Y i ≥ ( tx i u ; q ) ∞ ( x i u ; q ) ∞ = exp X n> − t n − q n p n ( X ) n u n ! = ∞ X r =0 g r ( X ; q, t ) u r , with g ( · ; q, t ) = 1. Proposition 2.8 ( [FHH + 09, Proposition 3.24]) . For a partition λ ∈ Y such that ℓ ( λ ) ≤ n , we have H ( n ) r P ( n ) λ ( x ; q, t ) = g ( n ) r ( t n s λ , . . . , t n s λn ; q, t ) P ( n ) λ ( x ; q, t ) , for all r = 1 , , . . . . Notice the following fact: Lemma 2.9. For all r, n ∈ Z > and λ ∈ Y such that ℓ ( λ ) ≤ n , we have g r ( s λ ; q, t )= ( − r t − nr q ( r )( q ; q ) r r X l =0 ( − l q − ( l ) q − l ( r − l ) ( q − l + r − ; q ) l g ( n ) l ( t n s λ , . . . , t n s λn ; q, t ) . Proof. By property (2.1) of symmetric functions g r ( X ; q, t ), r = 1 , , . . . , we have ∞ X r =0 g r ( s λ ; q, t ) u r = Y i ≥ ( q λ i t − i +1 u ; q ) ∞ ( q λ i t − i u ; q ) ∞ . When we suppose that ℓ ( λ ) ≤ n , it yields ∞ X r =0 g r ( s λ ; q, t ) u r = n Y i =1 ( q λ i t − i +1 u ; q ) ∞ ( q λ i t − i u ; q ) ∞ Y i ≥ n +1 ( t − i +1 u ; q ) ∞ ( t − i u ; q ) ∞ = n Y i =1 ( q λ i t n − i +1 t − n u ; q ) ∞ ( q λ i t n − i t − n u ; q ) ∞ · ( t − n u ; q ) ∞ = ∞ X r,s =0 ( − s q ( s )( q ; q ) s g ( n ) r ( t n s λ , . . . , t n s λn ) t − n ( r + s ) u r + s = ∞ X r =0 r X l =0 ( − r − l q ( r − l )( q ; q ) r − l g ( n ) l ( t n s λ , . . . , t n s λn ) t − nr u r . Here we used the expansion ( x ; q ) ∞ = ∞ X s =0 q ( s )( q ; q ) s ( − x ) s . Now note that q ( r − l )( q ; q ) r − l = q ( r )( q ; q ) r q − ( l ) q − l ( r − l ) ( q r − l +1 ; q ) l . Then comparing coefficients of u r , r = 1 , , . . . , we obtain the expected identities. (cid:3) Therefore, we have Theorem 2.10. If we set G ( n ) r := ( − r t − nr q ( r )( q ; q ) r r X l =0 ( − l q − ( l ) q − l ( r − l ) ( q − l + r − ; q ) l H ( n ) l , the projective limit G r = lim ←− n G ( n ) r exists for r ∈ Z ≥ and is diagonalized by the Mac-donald symmetric functions so that G r P λ ( X ) = g r ( s λ ; q, t ) P λ ( X ) , λ ∈ Y . Skew-Macdonald symmetric functions. We next define the skew-Macdonaldfunctions. Let X = ( x , x , . . . ) and Y = ( y , y , . . . ) be two sets of variables and supposethat they are combined to be a single set of variables ( X, Y ) = ( x , x , . . . , y , y , . . . ).Then we can think of the Macdonald symmetric functions P λ ( X, Y ) of these variables.The skew-Macdonald function P λ/µ for a skew-partition λ/µ is defined by P λ ( X, Y ) = P µ ∈ Y P λ/µ ( X ) P µ ( Y ). Similarly, we define Q λ/µ by Q λ ( X, Y ) = P µ ∈ Y Q λ/µ ( X ) Q µ ( Y ).3. Free field realization Fock representation. Let h = (cid:16)L n ∈ Z \{ } F a n (cid:17) ⊕ F c be the deformed HeisenbergLie algebra [Jin94] defined by[ a m , a n ] = m − q | m | − t | m | δ m + n, c, m, n ∈ Z \{ } , [ c, h ] = 0 . We decompose the Heisenberg Lie algebra so that h = h > ⊕ F c ⊕ h < , where h > := M n> F a n , h < := M n< F a n are Lie subalgebras. A one-dimensional representation F | i of h ≥ := h > ⊕ F c is definedby a n | i = 0 , n > , c | i = | i . The induced representation is the Fock representation of h : F := U ( h ) ⊗ U ( h ≥ ) F | i ≃ U ( h < ) ⊗ F F | i . Here, for a Lie algebra g , U ( g ) is its universal enveloping algebra. For a partition λ = ( λ , λ , . . . ) ∈ Y , we set | λ i := a − λ a − λ · · · | i . Then the Fock space has a basis {| λ i : λ ∈ Y } .The dual Fock space F † is also constructed by induction. Let F h | be a one-dimensionalright representation of h ≤ = h < ⊕ F c defined by h | a n = 0 , n < , h | c = h | . Then the dual Fock space is obtained by F † = F h | ⊗ U ( h ≤ ) U ( h ) ≃ F h | ⊗ F U ( h > ) . For a partition λ = ( λ , λ , . . . ) ∈ Y , we set h λ | = h | a λ a λ · · · . Then the collection {h λ | : λ ∈ Y } forms a basis of F † .We define an F -bilinear paring h·|·i : F † × F → F by the properties h | i = 1 and h v | a n · | w i = h v | · a n | w i , h v | ∈ F † , | w i ∈ F , n ∈ Z \{ } . REE FIELD APPROACH TO THE MACDONALD PROCESS 11 Lemma 3.1. Let λ, µ ∈ Y be partitions. Then h λ | µ i = δ λ,µ z λ ( q, t ) . Proof. It follows from the definition of the Heisenberg algebra h and Fock representations F and F † . Indeed, unless λ = µ , one of a λ i ’s is applied on the vacuum | i to make theinner product vanish. If λ = µ , we have a λ · · · a λ ℓ ( λ ) a − λ · · · a − λ ℓ ( λ ) = z λ ( q, t ) + · · · ∈ U ( h ) , where the terms expressed by an ellipsis annihilate either | i or h | . (cid:3) Proposition 3.2. The Fock space and the dual Fock space are isomorphic to the spaceof symmetric functions Λ by the assignments ι : F → Λ; | λ i 7→ p λ ,ι † : F † → Λ; h λ | 7→ p λ . Moreover, these assignments are compatible with the inner products so that the followingdiagram is commutative: F † ⊗ F ι † ⊗ ι / / h·|·i " " ❋❋❋❋❋❋❋❋❋ Λ ⊗ Λ h· , ·i q,t | | ②②②②②②②②② F . Proof. The first half is obvious from consideration on bases of each space. The secondhalf follows from Lemma 3.1. (cid:3) Proposition 3.3. Set Γ( X ) ± := exp X n> − t n − q n p n ( X ) n a ± n ! as we did in (1.3). Then the mappings ι and ι † are also expressed as ι ( | v i ) = h | Γ( X ) + | v i , | v i ∈ F ,ι † ( h v | ) = h v | Γ( X ) − | i , h v | ∈ F † . Proof. It follows from standard computation that we show below. Introducing a param-eter α ∈ C , we have(3.1) ddα (cid:0) Γ( X ) α + a − n Γ( X ) − α + (cid:1) = p n ( X ) , n > + ( X ) ± α = exp ± α X n> − t n − q n p n ( X ) n a n ! . Integrating (3.1) from α = 0 to α = 1, we haveΓ( X ) + a − n Γ( X ) − = a − n + p n ( X ) , n > X ) − − a n Γ( X ) − = a n + p n ( X ) , n > They imply that h | Γ( X ) + a − n | v i = p n ( X ) h | Γ( X ) + | v i , | v i ∈ F , n > , h v | a n Γ( X ) − | i = p n ( X ) h v | Γ( X ) − | i , h v | ∈ F † , n > . Since h | Γ( X ) + | i = h | Γ( X ) − | i = 1, we conclude that ι ( | v i ) = h | Γ( X ) + | v i , | v i ∈ F and ι † ( h v | ) = h v | Γ( X ) − | i , h v | ∈ F † . (cid:3) Remark . It can also be shown that ι ( a n | v i ) = n − q n − t n ∂∂p n ι ( | v i ) , | v i ∈ F , n > ,ι † ( h v | a − n ) = n − q n − t n ∂∂p n ι † ( h v | ) , h v | ∈ F † , n > . For the Macdonald symmetric function P λ ∈ Λ and its dual Q λ ∈ Λ associated witha partition λ ∈ Y , we set | P λ i := ι − ( P λ ) ∈ F , | Q λ i := ι − ( Q λ ) ∈ F , h P λ | := ( ι † ) − ( P λ ) ∈ F † , h Q λ | := ( ι † ) − ( Q λ ) ∈ F † . Then it follows from Proposition 3.2 that h Q λ | P µ i = h P λ | Q µ i = δ λ,µ . These propertiesand Proposition 3.3 verify the following. Proposition 3.5. We have h | Γ( X ) + = X λ ∈ Y P λ ( X ) h Q λ | , Γ( X ) − | i = X λ ∈ Y P λ ( X ) | Q λ i . Remark . We can see that the computation of h | Γ( Y ) + Γ( X ) − | i reproduces theCauchy identity. Indeed, on one hand, it reads h | Γ( Y ) + Γ( X ) − | i = X λ ∈ Y P λ ( X ; q, t ) Q λ ( Y ; q, t ) . On the other hand, a standard computation givesΓ( Y ) + Γ( X ) − = exp X n> − t n − q n p n ( X ) p n ( Y ) n ! Γ( X ) − Γ( Y ) + = Π( X, Y ; q, t )Γ( X ) − Γ( Y ) + . This reproduces the Cauchy identity X λ ∈ Y P λ ( X ; q, t ) Q λ ( Y ; q, t ) = h | Γ( Y ) + Γ( X ) − | i = Π( X, Y ; q, t ) . The skew-Macdonald symmetric functions are also expressed as matrix elements. Proposition 3.7. Let λ/µ be a skew-partition. Then the corresponding skew-Macdonaldsymmetric function has the following expressions. P λ/µ ( X ) = h Q µ | Γ( X ) + | P λ i = h P λ | Γ( X ) − | Q µ i . Also, the dual skew-Macdonald symmetric function is expressed as Q λ/µ ( X ) = h P µ | Γ( X ) + | Q λ i = h Q λ | Γ( X ) − | P µ i . REE FIELD APPROACH TO THE MACDONALD PROCESS 13 Proof. Let us set Γ( X, Y ) + = Γ( X ) + Γ( Y ) + . Then we have P λ ( X, Y ) = h | Γ( X, Y ) + | P λ i = X µ ∈ Y h | Γ( X ) + | P µ i h Q µ | Γ( Y ) + | P λ i = X µ ∈ Y h Q µ | Γ( Y ) + | P λ i P µ ( X ) . Here we used the identity Id F = P λ ∈ Y | P λ i h Q λ | . From the definition of the skew-Macdonald functions (Subsec. 2.5), we conclude that P λ/µ ( Y ) = h Q µ | Γ( Y ) + | P λ i .Similarly, if we set Γ( X, Y ) − = Γ( X ) − Γ( Y ) − , then we have P λ ( X, Y ) = h P λ | Γ( X, Y ) − | i = X µ ∈ Y h P λ | Γ( Y ) − | Q µ i P µ ( X ) . Therefore, it follows that P λ/µ ( Y ) = h P λ | Γ( Y ) − | Q µ i . (cid:3) Free field realization of operators. First, let us fix our terminologies. Definition 3.8. Let T ∈ End(Λ) be an operator on Λ. We call the operatorˆ T := ι − T ι ∈ End( F )the free field realization of T . Definition 3.9. Let z be a formal variable. An End( F )-valued formal power series V ( z )of the following form is said to be a vertex operator: V ( z ) = V ( z ) − V ( z ) + ∈ End( F )[[ z, z − ]] ,V ( z ) + = exp X n> γ n a n z − n ! , γ n ∈ F , n > ,V ( z ) − = exp X n> γ − n a − n z n ! , γ − n ∈ F , n > Remark . A vertex operator does not converge in U ( h )[[ z, z − ]], but makes sense inEnd( F )[[ z, z − ]]. Definition 3.11. Let V i ( z i ) ∈ End( F )[[ z i , z − i ]], i = 1 , , . . . , r be vertex operators.Their normally-ordered product : V ( z ) · · · V r ( z r ) : ∈ End( F )[[ z i , z − i | i = 1 , . . . , r ]] isdefined as : V ( z ) · · · V r ( z r ): := V ( z ) − · · · V r ( z r ) − V ( z r ) + · · · V r ( z r ) + . In this subsection, we see the free field realization of operators introduced in Sect.2 that are diagonalized by the Macdonald symmetric functions. We begin with theMacdonald operators E r , r = 1 , , , . . . . Recall a vertex operator in (1.4): η ( z ) = exp X n> − t − n n a − n z n ! exp − X n> − t n n a n z − n ! . Theorem 3.12 ( [Shi06, FHH + . Let r = 1 , , . . . . The following operator gives thefree field realization of E r . ˆ E r = t − r ( r +1) / ( t − ; t − ) r Z r Y i =1 dz i π √− z i ! Y ≤ i The following operators on F are the free field realization of G r ( q − , t − ) : ˆ G r ( q − , t − ) = ( − r q − ( r )( q − ; q − ) r Z r Y i =1 dz i π √− z i ! Y ≤ i Determinantal expression In this section, we derive alternative expression of the free field realizations presentedin the previous Section 3 by using determinants. As a preliminary, we introduce theoperation of symmetrization. REE FIELD APPROACH TO THE MACDONALD PROCESS 15 Definition 4.1. For a function f ( z , . . . , z n ) of n variables z , . . . , z n , the symmetriza-tion is defined by Sym[ f ( z , . . . , z n )] := 1 n ! X σ ∈ S n f ( z σ (1) , . . . , z σ ( n ) ) . The following lemma plays a key role in this section. Lemma 4.2. We have Sym Y ≤ i The Hall–Littlewood polynomial for the empty partition reads [Mac95, ChapterIII, (1.4)] X σ ∈ S n σ Y i Proof of Theorem 1.4. Recall that the constant term of a multivariable function is in-variant under symmetrization. Therefore we haveˆ E r = t − r ( r +1) / ( t − ; t − ) r Z r Y i =1 dz i π √− z i ! Sym Y ≤ i In similar manners, we can also derive the following expressions. Theorem 4.3. For r = 1 , , . . . , we have ˆ E r ( q − , t − ) = t r r ! Z r Y i =1 dz i π √− ! det (cid:18) z i − tz j (cid:19) ≤ i,j ≤ r : ξ ( z ) · · · ξ ( z r ): . Theorem 4.4. For r = 1 , , . . . , we have ˆ G r = ( − r r ! Z r Y i =1 dz i π √− ! det (cid:18) z i − qz j (cid:19) ≤ i,j ≤ r : η ( z ) · · · η ( z r ): . Theorem 4.5. For r = 1 , , . . . , we have ˆ G r ( q − , t − ) = ( − r r ! Z r Y i =1 dz i π √− ! det (cid:18) z i − q − z j (cid:19) ≤ i,j ≤ r : ξ ( z ) · · · ξ ( z r ): . The following property will be used in the next Section 5. REE FIELD APPROACH TO THE MACDONALD PROCESS 17 Lemma 4.6. Let r = 1 , , . . . and γ ∈ F and set a functional Z D rγ z := Z r Y i =1 dz i π √− ! det (cid:18) z i − γz j (cid:19) : F [[ z i , z − i | i = 1 , . . . , r ]] → F . Then it is invariant under a uniform scale transformation and the uniform inversion.Namely, if we set w i = αz i , i = 1 , . . . , r, α ∈ F or w i = z − i , i = 1 , . . . , r, then we have Z D rγ z = Z D rγ w. Proof. The invariance under a scale transformation is obvious. We consider the case ofinversion: w i = z − i , i = 1 , . . . , n . The operation taking the residues in z i is written as Z r Y i =1 dz i π √− ! = Z r Y i =1 dw i π √− ! r Y i =1 w − i . The determinant givesdet (cid:18) z i − γz j (cid:19) ≤ i,j ≤ r = det (cid:18) w i w j w j − γw i (cid:19) ≤ i,j ≤ r = r Y i =1 w i det (cid:18) w i − γw j (cid:19) ≤ i,j ≤ r . Then the desired invariance is proved. (cid:3) Proof of Theorem 1.3 and appications Proof of Theorem 1.3. We first compute the following matrix element: h P µ | ψ X,Yf | Q ν i = h P µ | Γ( Y ) + O ( f )Γ( X ) − | Q ν i = X λ ∈ Y f ( λ ) P λ/ν ( X ) Q λ/µ ( Y ) . Inserting the identity operator Id F = P λ ∈ Y | Q λ i h P λ | , we have h | ψ X ( N ) ,Y ( N ) f N · · · ψ X (1) ,Y (1) f | i = X ν (1) ,...,ν ( N − ∈ Y h | ψ X ( N ) ,Y ( N ) f N | Q ν ( N − i h P ν ( N − | ψ X ( N − ,Y ( N − f N − | Q ν ( N − i× h P ν (1) | ψ X (1) ,Y (1) f | i = X ν (1) ,...,ν ( N − ∈ Y X λ (1) ,...,λ ( N ) ∈ Y f N ( λ ( N ) ) P λ ( N ) /ν ( N − ( X ( N ) ) Q λ ( N ) / ∅ ( Y ( N ) ) × f N − ( λ ( N − ) P λ ( N − /ν ( N − ( X ( N − ) Q λ ( N − /ν ( N − ( Y ( N − ) · · · · · ·× f ( λ (1) ) P λ (1) / ∅ ( X (1) ) Q λ (1) /ν (1) ( Y (1) )= X λ (1) ,...,λ ( N ) ∈ Y f ( λ (1) ) · · · f N ( λ ( N ) ) P λ (1) ( X (1) )Ψ λ (1) ,λ (2) ( Y (1) , X (2) ) · · · × Ψ λ ( N − ,λ ( N ) ( Y ( N − , X ( N ) ) Q λ ( N ) ( Y ( N ) ) . Here we used the definition of the transition function (1.2). In the case when f , . . . , f N =1, the above result gives the normalization factor: h | ψ X ( N ) ,Y ( N ) · · · ψ X (1) ,Y (1) | i = Y ≤ i ≤ j ≤ N Π( X ( i ) , Y ( j ) ) . Therefore, we have h | ψ X ( N ) ,Y ( N ) f N · · · ψ X (1) ,Y (1) f | ih | ψ X ( N ) ,Y ( N ) · · · ψ X (1) ,Y (1) | i = X λ (1) ,...,λ ( N ) ∈ Y f ( λ (1) ) · · · f N ( λ ( N ) ) MP Nq,t ( λ (1) , . . . , λ ( N ) ) , which is just the correlation function E Nq,t [ f [1] · · · f N [ N ]].5.2. Applications. We give applications of Theorem 1.3. As preliminary, we introducean expectation value of a generating function of random variables. Definition 5.1. Let f n : Y → F , n = 0 , , . . . be random variables and let F ( u ) = P ∞ n =0 f n u n with u being a formal variable be a generating function of them. Then theexpectation value of F ( u ) with respect to the Macdonald measure is given by E q,t [ F ( u )] := ∞ X n =0 E q,t [ f n ] u n ∈ F [[ u ]] . We next introduce the formal Fredholm determinant. REE FIELD APPROACH TO THE MACDONALD PROCESS 19 Definition 5.2. Let K ( z, w ) ∈ F [[ z, z − , w, w − ]] be a formal power series in two vari-ables and let u be another formal variable. Then we define the formal Fredholm deter-minant det( I + uK ) ∈ F [[ u ]] bydet( I + uK ) := 1 + ∞ X r =1 u r r ! Z r Y i =1 dz i π √− ! det[ K ( z i , z j )] ≤ i,j ≤ r if it exists.The first example of observables we consider is e σr : Y → F ; e σr ( λ ) := e r ( σ λ ) , r = 1 , , . . . , where e r ( X ) is the r -th elementary symmetric function and the specialization is givenby σ λ = ( q λ i t − i +1 ; i = 1 , , . . . ). Let us also introduce a generating function as follows. E σ ( u ) = E σ ( · , u ) := ∞ X r =0 e σr ( · ) u r ; E σ ( λ, u ) = Y i ≥ (1 + q λ i t − i +1 u ) , λ ∈ Y , with e σ = 1. Theorem 5.3. Let N ∈ { , , . . . } and r , . . . , r N ∈ { , , . . . } . The correlation functionof e σr , . . . , e σr N with respect to the N -step Macdonald process becomes E Nq,t [ e σr [1] · · · e σr N [ N ]]= 1 Q Nα =1 r α ! Z N Y α =1 r α Y i =1 dw ( α ) i π √− ! det w ( α ) i − t − w ( α ) j ! ≤ i,j ≤ r α × Y ≤ α ≤ β ≤ N r β Y i =1 H ( w ( β ) i ; X ( α ) ) − ! r α Y i =1 H (( tw ( α ) i ) − ; Y ( β ) ) − ! × Y ≤ α<β ≤ N W ( w ( α ) ; w ( β ) ) . Here we set H ( w ; X ) = Y i ≥ − tx i w − x i w , and W ( w ( α ) ; w ( β ) ) = r α Y i =1 r β Y j =1 (1 − w ( β ) j /w ( α ) i )(1 − qt − w ( β ) j /w ( α ) i )(1 − qw ( β ) j /w ( α ) i )(1 − t − w ( β ) j /w ( α ) i ) . Proof. We use Theorem 1.3. Since e r ( σ λ ) = t r e r ( s λ ), we have from Theorem 1.4, for r = 1 , , . . . , O ( e σr ) = t r ˆ E r = 1 r ! Z r Y i =1 dz i π √− ! det (cid:18) z i − t − z j (cid:19) ≤ i,j ≤ r : η ( z ) · · · η ( z r ): . Let r , . . . , r N ∈ { , , . . . } We shall compute the unnormalized correlation function h | ψ X ( N ) ,Y ( N ) e σrN · · · ψ X (1) ,Y (1) e σr | i = 1 Q Nα =1 r α ! Z N Y α =1 r α Y i =1 dz ( α ) i π √− ! det z ( α ) i − t − z ( α ) j ! ≤ i,j ≤ r α × h | Γ( Y ( N ) ) + η ( z ( N ) )Γ( X ( N ) ) − · · · Γ( Y (1) ) + η ( z (1) )Γ( X (1) ) − | i , where we set η ( z ( α ) ) =: η ( z ( α )1 ) · · · η ( z ( α ) r α ) : , α = 1 , . . . , N . To compute the matrixelement, we write η ( z ( α ) ) = η ( z ( α ) ) − η ( z ( α ) ) + , α = 1 , . . . , N, where η ( z ( α ) ) + := exp − X n> − t n n a n r α X i =1 ( z ( α ) i ) − n ! ,η ( z ( α ) ) − := exp X n> − t − n n a − n r α X i =1 ( z ( α ) i ) n ! . Then we can see thatΓ( Y ( α ) ) + η ( z ( α ) )Γ( X ( α ) ) − = exp X n> − t − n n p n ( Y ( α ) ) r α X i =1 ( z ( α ) i ) n ! exp − X n> − t n n p n ( X ( α ) ) r α X i =1 ( z ( α ) i ) − n ! × η ( z ( α ) ) − Γ( Y ( α ) ) + Γ( X ( α ) ) − η ( z ( α ) ) + = Π( X ( α ) , Y ( α ) ) r α Y i =1 H (( z ( α ) i ) − ; X ( α ) ) − H ( t − z ( α ) i ; Y ( α ) ) − × η ( z ( α ) ) − Γ( X ( α ) ) − Γ( Y ( α ) ) + η ( z ( α ) ) + . Noting that the vertex operator η ( z ) exhibits the following operator product expansion(OPE): η ( z ) η ( w ) = (1 − w/z )(1 − qt − w/z )(1 − qw/z )(1 − t − w/z ) : η ( z ) η ( w ): . Then we can obtain the following formula:Γ( Y ( β ) ) + η ( z ( β ) ) + η ( z ( α ) ) − Γ( X ( α ) ) − = Π( X ( α ) , Y ( β ) ) W ( z ( β ) , z ( α ) ) r α Y i =1 H ( t − z ( α ) i ; Y ( β ) ) − r β Y i =1 H (( z ( β ) i ) − ; X ( α ) ) − × η ( z ( α ) ) − Γ( X ( α ) ) − Γ( Y ( β ) ) + η ( z ( β ) ) + . REE FIELD APPROACH TO THE MACDONALD PROCESS 21 Combining the above formulas we can compute the matrix element as h | Γ( Y ( N ) ) + η ( z ( N ) )Γ( X ( N ) ) − · · · Γ( Y (1) ) + η ( z (1) )Γ( X (1) ) − | i = Y ≤ i ≤ j ≤ N Π( X ( i ) , Y ( j ) ) Y ≤ α ≤ β ≤ N r α Y i =1 H ( t − z ( α ) i ; Y ( β ) ) − ! r β Y i =1 H (( z ( β ) i ) − ; X ( α ) ) − ! × Y ≤ α<β ≤ N W ( z ( β ) , z ( α ) ) . Therefore, we have E Nq,t [ e σr [1] · · · e σr N [ N ]]= 1 Q Ni =1 r i ! Z N Y α =1 r α Y i =1 dz ( α ) i π √− ! det z ( α ) i − t − z ( α ) j ! ≤ i,j ≤ r α × Y ≤ α ≤ β ≤ N r α Y i =1 H ( t − z ( α ) i ; Y ( β ) ) − ! r β Y i =1 H (( z ( β ) i ) − ; X ( α ) ) − ! × Y ≤ α<β ≤ N W ( z ( β ) , z ( α ) ) . Finally we adopt a transformation of integral variables so that w ( α ) i = ( z ( α ) i ) − , i =1 , . . . , r α , α = 1 , . . . , N . With help of Lemma 4.6 and the property W ( z − , . . . , z − m ; w − , . . . , w − n ) = W ( w , . . . , w n ; z , . . . , z m ) , we obtain E Nq,t [ e σr [1] · · · e σr N [ N ]]= 1 Q Ni =1 r i ! Z N Y α =1 r α Y i =1 dw ( α ) i π √− ! det w ( α ) i − t − w ( α ) j ! ≤ i,j ≤ r α × Y ≤ α ≤ β ≤ N r β Y i =1 H ( w ( α ) i ; X ( α ) ) − ! r α Y i =1 H (( tw ( α ) i ) − ; Y ( β ) ) − ! × Y ≤ α<β ≤ N W ( w ( α ) , w ( β ) ) . Then the proof is complete. (cid:3) In particular, when N = 1, we have the following. Corollary 5.4. Set K σ ( z, w ) := 1 z − t − w H ( z ; X ) − H (( tw ) − ; Y ) − . Then we have (5.1) E q,t [ E σ ( u )] = det( I + uK σ ) . Proof. When N = 1, Thorem 5.3 reduces to E q,t [ e σr ] = 1 r ! Z r Y i =1 dw i π √− ! det[ K σ ( w i , w j )] ≤ i,j ≤ r , r = 1 , , . . . . Therefore, the desired result follows from Definitions 5.1 and 5.2. (cid:3) Remark . In case that r = · · · = r N = 1, the result of Theorem 5.3 is essentiallyequivalent to that of [BCGS16, Theorem 6.1]. Remark . The left hand side of Eq. (5.1) is intrinsically dependent on two parameters q and t , but in the right hand side of Eq. (5.1), it is manifest that it is independent of q . The next observables we consider are defined by e ρr : Y → F ; e ρr ( λ ) := e r ( ρ λ ) , r = 1 , , . . . , where we set ρ λ = ( q − λ i t i − ; i = 1 , , . . . ). They are just obtained from e σr , r = 1 , , . . . by inverting parameters q and t . We also write the generating function of them with aformal variable u as E ρ ( u ) = E ρ ( · , u ) = ∞ X r =0 e ρr ( · ) u r ; E ρ ( λ, u ) = Y i ≥ (1 + q − λ i t i − u ) , λ ∈ Y , with e ρ = 1. Theorem 5.7. Let N ∈ { , , . . . } and r , . . . , r N ∈ { , , . . . } . The correlation functionof e ρr , . . . , e ρr N with respect to the N -step Macdonald process becomes E Nq,t [ e ρr [1] · · · e ρr N [ N ]]= 1 Q Nα =1 r α ! Z N Y α =1 r α Y i =1 dw ( α ) i π √− ! det w ( α ) i − tw ( α ) j ! ≤ i,j ≤ r α × Y ≤ α ≤ β ≤ N r β Y i =1 H ( w ( β ) i ; X ( α ) ) ! r α Y i =1 H (( qw ( α ) i ) − ; Y ( β ) ) ! × Y ≤ α<β ≤ N f W ( w ( α ) ; w ( β ) ) . Here we set f W ( w ( α ) ; w ( β ) ) = r α Y i =1 r β Y j =1 (1 − w ( β ) j /w ( α ) i )(1 − q − tw ( β ) j /w ( α ) i )(1 − q − w ( β ) j /w ( α ) i )(1 − tw ( β ) j /w ( α ) i ) . Proof. The proof is very similar to that of Theorem 1.3. The observable e ρr is just ob-tained from e σr by inverting the parameters q and t . Since P λ ( X ; q, t ) = P λ ( X ; q − , t − )[Mac95, Chapter VI, (4.14.iv) ], and from Theorem 4.3, we have O ( e ρr ) = t − r ˆ E r ( q − , t − )= 1 r ! Z r Y i =1 dz i π √− ! det (cid:18) z i − tz j (cid:19) ≤ i,j ≤ r : ξ ( z ) · · · ξ ( z r ): . REE FIELD APPROACH TO THE MACDONALD PROCESS 23 For r , . . . , r N ∈ { , , . . . } , the unnormalized correlation function reads h | ψ X ( N ) ,Y ( N ) e ρrN · · · ψ X (1) ,Y (1) e ρr | i = 1 Q Nα =1 r α ! Z N Y α =1 r α Y i =1 dz ( α ) i π √− ! det z ( α ) i − tz ( β ) j ! ≤ i,j ≤ r α × h | Γ( Y ( N ) ) + ξ ( z ( N ) )Γ( X ( N ) ) − · · · Γ( Y (1) ) + ξ ( z (1) )Γ( X (1) ) − | i , where we set ξ ( z ( α ) ) =: ξ ( z ( α )1 ) · · · ξ ( z ( α ) r α ) : , α = 1 , . . . , N . We again write ξ ( z ( α ) ) = ξ ( z ( α ) ) − ξ ( z ( α ) ) + , α = 1 , . . . , N , where ξ ( z ( α ) ) + = exp X n> − t n n ( t/q ) n/ a n r α X i =1 ( z ( α ) i ) − n ! ,ξ ( z ( α ) ) − = exp − X n> − t − n n ( t/q ) n/ a − n r α X i =1 ( z ( α ) i ) n ! . To compute the matrix element, we first note the following formula:Γ( Y ( α ) ) + ξ ( z ( α ) )Γ( X ( α ) ) − = exp − X n> − t − n n ( t/q ) n/ p n ( Y ( α ) ) r α X i =1 ( z ( α ) i ) n ! × exp X n> − t n n ( t/q ) n/ p n ( X ( α ) ) r α X i =1 ( z ( α ) i ) − n ! × ξ ( z ( α ) ) − Γ( Y ( α ) ) + Γ( X ( α ) ) − ξ ( z ( α ) ) + = Π( X ( α ) , Y ( α ) ) r α Y i =1 H ( t / q − / ( z ( α ) i ) − ; X ( α ) ) H ( t − / q − / z ( α ) i ; Y ( α ) ) × ξ ( z ( α ) ) − Γ( X ( α ) ) − Γ( Y ( α ) ) + ξ ( z ( α ) ) + . From the OPE ξ ( z ) ξ ( w ) = (1 − w/z )(1 − q − tw/z )(1 − q − w/z )(1 − tw/z ) : ξ ( z ) ξ ( w ): , we also haveΓ( Y ( β ) ) + ξ ( z ( β ) ) + ξ ( z ( α ) ) − Γ( X ( α ) ) − = Π( X ( α ) , Y ( β ) ) r α Y i =1 H ( t − / q / z ( α ) i ; Y ( β ) ) r β Y i =1 H ( t / q − / ( z ( β ) i ) − ; X ( α ) ) × f W ( z ( β ) , z ( α ) ) ξ ( z ( α ) ) − Γ( X ( α ) ) − Γ( Y ( β ) ) + ξ ( z ( β ) ) + . Therefore, the correlation function becomes E Nq,t [ e ρr [1] · · · e ρr N [ N ]]= 1 Q Nα =1 r α ! Z N Y α =1 r α Y i =1 dz ( α ) i π √− ! det z ( α ) i − tz ( α ) j ! ≤ i,j ≤ r α × Y ≤ α ≤ β ≤ N r β Y i =1 H ( t / q − / ( z ( β ) i ) − ; X ( α ) ) ! r α Y i =1 H ( t − / q / z ( α ) i ; Y ( β ) ) ! × Y ≤ α<β ≤ N f W ( z ( β ) , z ( α ) ) . When we perform a transformation of integrable variables so that w ( α ) i := t / q − / ( z ( α ) i ) − , i = 1 , . . . , r α , α = 1 , . . . , N , with help of Lemma 4.6, we obtain the desired result. (cid:3) Again, in the case when N = 1, we obtain the following result in the same manner asfor Corollary 5.4. Corollary 5.8. Set K ρ ( z, w ) := 1 z − tw H ( z ; X ) H (( qw ) − ; Y ) . Then we have E q,t [ E ρ ( u )] = det( I + uK ρ ) . Remark . The result in Theorem 5.7 can be found in [BCGS16, Theorem 1.1].We also consider other random variables defined by g sr ( λ ; q, t ) := g r ( s λ ; q, t ) , s λ = ( q λ i t − i ; i = 1 , , . . . ) , λ ∈ Y , r = 1 , , . . . . We also introduce their generating function F sq,t ( u ) = F sq,t ( · , u ) = ∞ X r =0 g sr ( · ; q, t ) u r ; F sq,t ( λ, u ) = Y i ≥ ( q λ i t − i +1 u ; q ) ∞ ( q λ i t − i u ; q ) ∞ , with g s ( q, t ) = 1. Theorem 5.10. Let N ∈ { , , . . . } and r , . . . , r N ∈ { , , . . . } . The correlation func-tion of g sr ( q, t ) , . . . , g sr N ( q, t ) with respect to the N -step Macdonald process becomes E Nq,t [ g sr ( q, t )[1] · · · g sr N ( q, t )[ N ]]= ( − P Nα =1 r α Q Nα =1 r α ! Z N Y α =1 r α Y i =1 dw ( α ) i π √− ! det w ( α ) i − qw ( α ) j ! ≤ i,j ≤ r α × Y ≤ α ≤ β ≤ N r β Y i =1 H ( w ( β ) i ; X ( α ) ) − ! r α Y i =1 H (( tw ( α ) i ) − ; Y ( β ) ) − ! × Y ≤ α<β ≤ N W ( w ( α ) ; w ( β ) ) . REE FIELD APPROACH TO THE MACDONALD PROCESS 25 Proof. From Theorem 2.10 and Thorem 4.4, we have O ( g sr ( q, t )) = ˆ G r = ( − r r ! Z r Y i =1 dz i π √− ! det (cid:18) z i − qz j (cid:19) ≤ i,j ≤ r : η ( z ) · · · η ( z r ): . The essential part of the proof is computation of the matrix element h | Γ( Y ( N ) ) + η ( z ( N ) )Γ( X ( N ) ) − · · · Γ( Y (1) ) + η ( z (1) )Γ( X (1) ) − | i , but this was given in the proof of Theorem 5.3. Therefore, we obtain the desire result. (cid:3) In case of N = 1, we have Corollary 5.11. Set K sg ( z, w ) := 1 z − qw H ( z ; X ) − H (( tw ) − ; Y ) − . Then we have E q,t [ F sq,t ( u )] = det( I − uK sg ) . The final series of observables g ˜ ρr ( q, t ) = g ˜ ρr ( · ; q, t ), r = 1 , , . . . is defined by g ˜ ρr ( λ ; q, t ) = g r (˜ ρ λ ; q, t ) , ˜ ρ = ( q − λ i +1 t i − ; i = 1 , , . . . ) . Their generating function reads F ˜ ρq,t ( u ) = F ˜ ρq,t ( · , u ) = ∞ X r =0 g ˜ ρr ( · ; q, t ) u r ; F ˜ ρq,t ( λ, u ) = Y i ≥ ( q − λ i +1 t i u ; q ) ∞ ( q − λ i +1 t i − u ; q ) ∞ , with g ˜ ρ ( q, t ) = 1. Theorem 5.12. Let N ∈ { , , . . . } and r , . . . , r N ∈ { , , . . . } . The correlation func-tion of g ˜ ρr ( q, t ) , . . . , g ˜ ρr N ( q, t ) with respect to the N -step Macdonald process becomes E Nq,t [ g ˜ ρr ( q, t )[1] · · · g ˜ ρr N ( q, t )[ N ]]= ( − P Nα =1 r α Q Nα =1 r α ! Z N Y α =1 r α Y i =1 dw ( α ) i π √− ! det w ( α ) i − q − w ( α ) j ! ≤ i,j ≤ r α × Y ≤ α ≤ β ≤ N r β Y i =1 H ( w ( β ) i ; X ( α ) ) ! r α Y i =1 H (( qw ( α ) i ) − ; Y ( β ) ) ! × Y ≤ α<β ≤ N ˜ W ( w ( α ) ; w ( β ) ) . Proof. The operators ˆ G r ( q − , t − ), r = 1 , , . . . are diagonalized by | P λ i , λ ∈ Y so thatˆ G r ( q − , t − ) | P λ i = g r ( s − λ ; q − , t − ) | P λ i , s − λ = ( q − λ i t i ; i = 1 , , . . . ) . Lemma 5.13. For r = 1 , , . . . , we have g r ( X ; q − , t − ) = ( qt − ) r g r ( X ; q, t ) . Proof. From the property (2.1), it follows that X r ≥ g r ( X ; q, t ) u r = exp X n> − t − n − q − n p n ( X ) n ( tq − u ) n ! = X r ≥ g r ( X ; q − , t − )( tq − u ) r . Comparison of the coefficients of u r , r = 1 , , . . . gives the desired result. (cid:3) From the above Lemma, we have g r ( s − λ ; q − , t − ) = g r (˜ ρ λ ; q, t ), r = 1 , , . . . , whichimplies that O ( g ˜ ρr ( q − , t − )) = ˆ G r ( q − , t − ) . Using the expression in Theorem 4.5 and following computation in the proof of Theorem5.7, we obtain the formula in Theorem 5.12. (cid:3) In case of N = 1, we have Corollary 5.14. Set K ˜ ρg ( z, w ) := 1 z − q − w H ( z ; X ) H (( qw ) − ; Y ) . Then we have E q,t [ F ˜ ρq,t ( u )] = det( I − uK ˜ ρg ) . Corollary 5.14 admits a nontrivial q -Whittaker ( t → 0) limit. The generating function F ˜ ρq,t ( u ) reduces at the q -Whittaker limit to F ˜ ρq, ( λ, u ) = 1( q − λ +1 u ; q ) ∞ , and the kernel function becomes(5.2) K ˜ ρg ( z, w ) | t =0 = 1 z − q − w Y i ≥ − x i z − y i / ( qz ) . Let us informally state the result at the q -Whittaker limit: Corollary 5.15. At the q -Whittaker limit t → , we have E q, (cid:20) q − λ +1 u ; q ) ∞ (cid:21) = det( I − uK ˜ ρg | t =0 ) . Remark . A similar observable has been considered in [BCFV15, Theorem 3.3],while our result does not require any specialization of variables and the kernel function(5.2) seems to be simpler.6. Generalized Macdonald measure In this section, we propose a generalization of Macdonald measure using the repre-sentation theory of the DIM algebra [DI97, Mik07]. We begin with introducing the DIMalgebra following [DI97, AFH + g ( z ) = G + ( z ) G − ( z ) ∈ F [[ z ]] , G ± ( z ) = (1 − q ± z )(1 − t ∓ z )(1 − q ∓ t ± z ) . The DIM algebra U is a unital associative algebra over F generated by currents x ± ( z ) = X n ∈ Z x ± n z − n , ψ ± ( z ) = X ± n ∈ Z ≥ ψ ± n z − n REE FIELD APPROACH TO THE MACDONALD PROCESS 27 and an invertible central element γ / subject to relations ψ ± ( z ) ψ ± ( w ) = ψ ± ( w ) ψ ± ( z ) ,ψ + ( z ) ψ − ( w ) = g ( γw/z ) g ( γ − w/z ) ψ − ( w ) ψ + ( z ) ,ψ + ( z ) x ± ( w ) = g ( γ ∓ / w/z ) ∓ x ± ( w ) ψ + ( z ) ,ψ − ( z ) x ± ( w ) = g ( γ ∓ / w/z ) ± x ± ( w ) ψ − ( z ) , [ x + ( z ) , x − ( w )] = (1 − q )(1 − t − )1 − qt − ( δ ( γ − z/w ) ψ + ( γ / w ) − δ ( γz/w ) ψ − ( γ − / w )) ,G ∓ ( z/w ) x ± ( z ) x ± ( w ) = G ± ( z/w ) x ± ( w ) x ± ( z ) . Here we set δ ( z ) = P n ∈ Z z n as the formal delta distribution.The DIM algebra U is a formal Hopf algebra with coproduct ∆ defined by∆( γ / ) = γ / ⊗ γ / , ∆( ψ ± ( z )) = ψ ± ( γ ± / z ) ⊗ ψ ± ( γ ∓ / z ) , ∆( x + ( z )) = x + ( z ) ⊗ ψ ( γ / z ) ⊗ x + ( γ (1) z ) , ∆( x − ( z )) = x − ( γ (2) z ) ⊗ ψ + ( γ / z ) + 1 ⊗ x − ( z ) . Here we used the notation γ / = γ / ⊗ γ / = 1 ⊗ γ / .To consider a representation of U on a Fock space e F := C ( q / , t / ) ⊗ F F , we introduce,in addition to vertex operators η ( z ) and ξ ( z ), the following ones ϕ ± ( z ) := exp ∓ X n> − t ± n n (1 − ( t/q ) n )( t/q ) − n/ a ± n z ∓ n ! . Then the assignment ρ : U → End( e F ) defined by ρ ( γ / ) = ( t/q ) / , ρ ( ψ ± ( z )) = ϕ ± ( z ) , ρ ( x + ( z )) = η ( z ) , ρ ( x − ( z )) = ξ ( z )gives a level one representation of U on e F . As we saw in Theorem 3.12, the zero mode x +0 acts essentially as the first Macdonald operator on F so that ρ ( x +0 ) | P λ i = ε λ | P λ i , ε λ = 1 + ( t − ℓ ( λ ) X i =1 ( q λ i − t − i , λ ∈ Y . Using the formal Hopf algebra structure of U , we can equip the m -fold tensor product e F ⊗ m , m ∈ N with an action of U . We set ∆ (1) = Id, ∆ (2) = ∆ and, inductively,∆ ( m ) = (Id ⊗ · · · ⊗ Id ⊗ ∆) ◦ ∆ ( m − , m ∈ N . Then the assignment ρ ( m ) := ( ρ ⊗ · · · ⊗ ρ ) ◦ ∆ ( m ) : U → End( e F ⊗ m )gives an level m representation of U on e F ⊗ m .We write, for T ∈ End( e F ) and i = 1 , . . . , m , T ( i ) := Id ⊗ · · · ⊗ Id ⊗ i ˇ T ⊗ Id ⊗ · · · ⊗ Id , and set Γ ( X ) ± = m Y i =1 Γ( X ( i ) ) ( i ) ± , X = ( X (1) , . . . , X ( m ) ) . Then the isomorphisms ι ⊗ m : e F ⊗ m → e Λ X (1) ⊗ · · · ⊗ e Λ X ( m ) and ( ι † ) ⊗ m : ( e F † ) ⊗ m → e Λ X (1) ⊗ · · · ⊗ e Λ X ( m ) , where e Λ = C ( q / , t / ) ⊗ F ⊗ Λ, are identified with ι ⊗ m = h | Γ ( X ) + , ( ι † ) ⊗ m = Γ ( X ) − | i , where we set | i = | i ⊗ · · · ⊗ | i and h | = h | ⊗ · · · ⊗ h | .We write an m -tuple of partitions as λ = ( λ (1) , . . . , λ ( m ) ) ∈ Y m . An analogue of thedominance order on Y m is defined so that, for λ , µ ∈ Y m , we say that λ ≥ µ if | λ (1) | + · · · | λ ( j − | + i X k =1 λ ( j ) k ≥ | µ (1) | + · · · + | µ ( j − | + i X k =1 µ ( j ) k holds for all i ≥ j = 1 , . . . , m . For a monomial symmetric function m λ , λ ∈ Y , wewrite h m λ | = ( ι † ) − ( m λ ) and set h m λ | := h m λ (1) | ⊗ · · · ⊗ h m λ ( m ) | ∈ ( F † ) ⊗ m , λ ∈ Y m . The following proposition was presented in [AFH + 11] (see also [Ohk17,FOS19,MM19]for recent studies). Proposition 6.1 ( [AFH + . For an m -tuple of partitions λ ∈ Y m , a vector h P λ | ∈ ( e F † ) ⊗ m is uniquely determined by the following properties: h P λ | = h m λ | + X µ < λ c λµ h m µ | , c λµ ∈ C ( q / , t / ) , h P λ | X +0 = ε λ h P λ | , ε λ = m X i =1 ε λ ( i ) , where we set X +0 := ρ ( m ) ( x +0 ) . The collection {h P λ | | λ ∈ Y m } forms a basis of ( e F † ) ⊗ m . Definition 6.2. Let {| Q λ i | λ ∈ Y m } be the basis of e F ⊗ m dual to {h P λ | | λ ∈ Y m } so that h P λ | Q µ i = δ λµ , λ , µ ∈ Y m . We also set P λ := ( ι † ) ⊗ m ( h P λ | ) ∈ Λ ⊗ m and Q λ := ι ⊗ m ( | Q λ i ) ∈ Λ ⊗ m , λ ∈ Y m . Remark . Differently from the usual Macdonald symmetric functions, P λ and Q λ arenot proportional to each other.It immediately follows that Proposition 6.4. We have the following expansions: Γ ( X ) − | i = X λ ∈ Y m P λ ( X ) | Q λ i , h | Γ ( X ) + = X λ ∈ Y m Q λ ( X ) h P λ | . As a corollary of this proposition, we obtain a kind of the Cauchy identity: Corollary 6.5. We have X λ ∈ Y m P λ ( X ) Q λ ( Y ) = Π ( m ) ( X , Y ) := m Y i =1 Π( X ( i ) , Y ( i ) ) . REE FIELD APPROACH TO THE MACDONALD PROCESS 29 Proof. Computation of h | Γ ( Y ) + Γ ( X ) − | i in two ways proves the desired result. On one hand, we have h | Γ ( Y ) + Γ ( X ) − | i = X λ ∈ Y m P λ ( X ) Q λ ( Y ) , while, on the other hand, we also have h | Γ ( Y ) + Γ ( X ) − | i = m Y i =1 h | Γ( Y ( i ) ) + Γ( X ( i ) ) − | i = Π ( m ) ( X , Y ) . (cid:3) Now we define the level m generalized Macdonald measure on Y m . Definition 6.6. The level m generalized Macdonald measure is a probability measureon Y m so that the weight of λ ∈ Y m is given by GM mq,t ( λ ) := 1Π ( m ) ( X , Y ) P λ ( X ) Q λ ( Y ) . As a demonstration, let us compute the expectation value of a function ε • : Y m → F defined by GE mq,t [ ε • ] := X λ ∈ Y m ε λ GM mq,t ( λ ) . Theorem 6.7. Set M ( z ; X ) := Y k ≥ (1 − zx k )(1 − q − zx k )(1 − q − tzx k )(1 − t − zx k ) , X = ( x , x , . . . ) . Then, we have GE mq,t [ ε • ] = Z dz π √− z m X i =1 i − Y j =1 M ( p − ( j +1) / z ; Y ( i ) ) × H ( p ( i − / z − ; X ( i ) ) − H ( t − p − ( i − / z ; Y ( i ) ) − , where we set p = q/t .Proof. We compute the quantity h | Γ ( Y ) + X +0 Γ ( X ) − | i in two ways. On one hand, it is shown that h | Γ ( Y ) + X +0 Γ ( X ) − | i = Π ( m ) ( X , Y ) GE mq,t [ ε • ]from the definition of level m generalized Macdonald functions presented in Proposition6.1 and Definition 6.2 and expansions in Proposition 6.4.On the other hand, we can also write X +0 = Z dz π √− z X + ( z ) , X + ( z ) = ρ ( m ) ( x + ( z )) = m X i =1 ˜Λ i ( z ) , where we set˜Λ i ( z ) = ϕ − ( p − / z ) ⊗ ϕ − ( p − / z ) ⊗ · · · ⊗ ϕ − ( p − (2 i − / z ) ⊗ i ˇ η ( p − ( i − / z ) ⊗ Id ⊗ · · · ⊗ Idfor i = 1 , . . . , m . In can verified that Γ( X ) + and ϕ − ( z ) exhibit OPEΓ( X ) + ϕ − ( z ) = Y k ≥ (1 − p − / x k z )(1 − t − p / x k z )(1 − p / x k z )(1 − t − p − / x k z ) ϕ − ( z )Γ( X ) + . Then a similar argument as in Sect. 5 gives the desire result. (cid:3) Appendix A. Proof of Theorem 3.15 Fix r ∈ Z ≥ . What we show is the identity h | Γ( X ) + ˆ G r ( q − , t − ) = G r ( q − , t − ) h | Γ( X ) + , which is equivalent to h | Γ n ( X ) + ˆ G r ( q − , t − ) = G ( n ) r ( q − , t − ) h | Γ n ( X ) + , n = 1 , , . . . , where Γ n ( X ) + = exp X m> − t m − q m p ( n ) m ( X ) m a m ! is the n -variable reduction of Γ( X ) + .The following lemma can be checked by standard computation: Lemma A.1. We have Γ n ( X ) + ξ ( z ) = n Y i =1 − t / q − / x i z − t − / q − / x i z ξ ( z )Γ n ( X ) + . By using this, we can see that h | Γ n ( X ) + ˆ G r ( q − , t − )= ( − r q − ( r )( q ; q ) r Z (cid:18) dz i π √− z i (cid:19) Y ≤ i Correspondingly, we set H ( n ) , ( µ ) r ( q − , t − )(A.2):= X ν ∈ ( Z ≥ ) n | ν | = r Y ≤ i 0) of the following one: Theorem A.2. For µ ∈ Z n and r = 1 , , . . . , we have G ( µ ) r ( q − , t − )= t rn q r | µ | r X l =0 ( − l q ( l ) q l ( r − l ) ( q − r + l − ; q − ) l H ( n ) , ( µ ) l ( q − , t − ) h | Γ n ( X ) + . To prove this theorem, we prepare lemmas. The first one can be checked by a simplecalculation. Lemma A.3. We have h | ξ ( t / q / x − k )Γ n ( X ) + = T q − ,x k h | Γ n ( X ) + . We also have Lemma A.4. Let ν = ( ν , . . . , ν n ) ∈ Z n . Then n X k =1 ( q − ν k − Y i = k x k − q − ν i x i x k − x i = q −| ν | − . Proof. By comparing residues and behavior at z → ∞ , we have n Y i =1 z − q − ν i x i z − x i = n X k =1 (1 − q − ν k ) x k z − x k Y i = k x k − q − ν i x i x k − x i + 1 . Then, setting z = 0, we can see the desired result. (cid:3) Proof of Theorem A.2. We first integrate out the variable z in Eq. (A.1). Then theresidues at z = t / q / x − i , i = 1 , . . . , n and z = ∞ contribute. (To make thiscomputation easier to see, it might be convenient to transform the variables so that w i = z − i , i = 1 , . . . , r .) Consequently, we obtain G ( µ ) r ( q − , t − )= Z r Y i =2 dz i π √− z i ! n X k =1 (1 − tq µ k ) Y ≤ i 1. To use this induction hypothesis, wemake some preliminaries. For ν = ( ν , . . . , ν n ) ∈ ( Z ≥ ) n and µ = ( µ , . . . , µ n ) ∈ Z n , set h ( µ ) ν ( q − , t − ) = Y ≤ i Observe that h ( µ + ǫ k ) ν ( q − , t − ) = 1 − t − q − µ k − ν k − t − q − µ k n Y i =1 i = k x k − t − q − µ k − ν i x i x k − t − q − µ k x i h ( µ ) ν ( q − , t − )(A.3)and T q − ,x k h ( µ ) ν ( q − , t − )(A.4)= 1 − q − ν k − − t − q − µ k − ν k n Y i =1 i = k t − q − µ k +1 x i − x k x i − t − q − µ i x i x k − q − ν i x i x k − t − q − µ k − ν i +1 x i h ( µ ) ν + ǫ k ( q − , t − ) T q − ,x k . By the induction hypothesis, we have G ( µ ) r ( q − , t − ) = " t ( r − n q ( r − | µ | +1) r − X l =0 ( − l q ( l ) q l ( r − l +1) ( q − r + l ; q − ) l × n X k =1 (1 − tq µ k ) n Y i =1 i = k x k − tq µ i x i x k − x i T q − ,x k H ( n ) , ( µ + ǫ k ) l ( q − , t − )+ t rn q r | µ | r − X l =0 ( − l q ( l ) q l ( r − l +1) ( q − r + l ; q − ) l × H ( n ) , ( µ ) l ( q − , t − ) h | Γ n ( X ) + . Let us set X := n X k =1 (1 − tq µ k ) n Y i =1 i = k x k − tq µ i x i x k − x i T q − ,x k H ( n ) , ( µ + ǫ k ) l ( q − , t − ) . Then, using Eqs. (A.3) and (A.4), we can verify that X = t n q | µ | X ν ∈ ( Z ≥ ) n | ν | = l n X k =1 ( q − ν k − − n Y i =1 i = k x k − q − ν i x i x k − x i h ( µ ) ν + ǫ k ( q − , t − ) T q − ,x k n Y i =1 T ν i q − ,x i = t n q | µ | X ν ∈ ( Z ≥ ) n | ν | = l +1 n X k =1 ( q − ν k − n Y i =1 i = k x k − q − ν i x i x k − x i h ( µ ) ν ( q − , t − ) n Y i =1 T ν i q − ,x i . Here we use Lemma A.4 to obtain X = t n q | µ | ( q − l − − H ( n ) , ( µ ) l +1 ( q − , t − ) . Therefore G ( µ ) r ( q − , t − ) = " t ( r − n q ( r − | µ | +1) r − X l =0 ( − l q ( l ) q l ( r − l +1) ( q − r + l ; q − ) l × t n q | µ | ( q − l − − H ( n ) , ( µ ) l +1 ( q − , t − )+ t rn q r | µ | r − X l =0 ( − l q ( l ) q l ( r − l +1) ( q − r + l ; q − ) l × H ( n ) , ( µ ) l ( q − , t − ) h | Γ n ( X ) + . It can be checked that this coincides with t rn q r | µ | r X l =0 ( − l q ( l ) q l ( r − l ) ( q − r + l − ; q − ) l H ( n ) , ( µ ) l ( q − , t − ) h | Γ n ( X ) + . Then the proof is complete. (cid:3) References [AFH + 11] H. Awata, B. Feigin, A. Hoshino, M. Kanai, J. Shiraishi, and S. Yanagida. Notes on Ding-Iohara algebra and AGT conjecture, 2011. arXiv:1106.4088.[Bar15] G. Barraquand. A phase transition for q -TASEP with a few slower particles. Stoch. Process.Their Appl. , 125:2674–2699, 2015.[BBBF18] B. Brubaker, V. Buciumas, D. Bump, and S. Friedberg. 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