Factorizations of symmetric Macdonald polynomials

We prove many factorization formulas for highest weight Macdonald polynomials indexed by particular partitions called quasistaircases. As a consequence we prove a conjecture of Bernevig and Haldane stated in the context of the fractional quantum Hall theory.


Introduction
Jack polynomials have many applications in physics, in particular in statistical physics and quantum physics, due to their relation to the many-body problem. In particular, fractional quantum Hall states of particles in the lowest Landau levels are described by such polynomials [7,6,8]. In that context, some properties, called clustering properties, are highly relevant. A clustering property can occur for some negative rational parameters of a Jack polynomial and means that the Jack polynomial vanishes when s distinct clusters of k + 1 equal variables are formed. Using tools of algebraic geometry, Berkesch-Zamaere et al proved several clustering properties [5] including some special cases conjectured by Bervenig and Haldane [7]. Coming from theoretical physics, the study of these properties raises very interesting problems in combinatorics and representation theory of the affine Hecke algebras. More precisely, the problem is studied in the realm of Macdonald polynomials which form a (q, t)-deformation of the Jack polynomials related to the double affine Hecke algebras and the results are recovered by making degenerate the parameters q and t. Instead of stating the results in terms of clustering properties, we prefer to state them in terms of factorizations. Indeed, clustering properties are shown to be equivalent to very elegant formulas involving factorizations of Macdonald polynomials. For instance, many such factorizations have been already investigated in [4,9,3,14]. In particular, this paper is the sequel of [14] in which two of the authors prove factorizations for rectangular Macdonald polynomials. In this paper, we investigate factorizations for more general partitions, called quasistaircases.
The paper is organized as follows. In section 2 we recall essential prerequisite on Macdonald polynomials. In section 3 we give a brief account on the physics motivations coming from the fractional Quantum Hall theory. In Section 4 we investigate some factorizations involved for generic values of (q, t). Section 5 is devoted to the special cases of specializations of the type t α q β = 1 and, in particular, to the consequences on spectral vectors. In Section 6, we deduce factorizations from the results of Feigen et al [16] and, in Section 7, we prove more general results which are consequences of the highest weight condition of some quasistaircase Macdonald polynomials proved in [19]. In the last section (Section 8), we first illustrate our results by proving a conjecture stated by Bernevig and Haldane [6] and also we show many other examples of factorizations that do not follow from our formulas.

Background
This paper is focused in the study of four variants of the Macdonald polynomials: symmetric, nonsymmetric, shifted symmetric and shifted nonsymmetric. Before getting into the results, we introduce these polynomials as well as some useful notation. All the results contained in this section are well known results showed in several papers (see eg [2,1,10,13,20,22,23,30,32]). The results of [24,25,26] are extensively used throughout our paper.

Partitions and Vectors
A partition λ = (λ 1 , . . . , λ ) of n is a weakly decreasing sequence of positive integers such that i λ i = n. The length of a partition λ is (λ) = max{i : λ i > 0}.
If we consider vectors instead of partitions, the notation is as follows: v = [v [1], . . . , v [N ]] is a vector of length N . Note that for vectors, the zero parts are taken into account in the length of the vector. We denote by v + the unique non increasing partition which is a permutation of v. We can consider the following standardization of v, std v : we label with integer from 0 to N − 1 the positions in v from the smallest entries to the largest one and from right to left. We define the reciprocal vector of v as v q,t = t stdv [1] q v [1] , . . . , t stdv [N ] q v [N ] , and the If there is no ambiguity, the indices q and t are omitted.
The dominance order defined on the partitions is naturally extended to vectors almost with the same definition: u v if and only if either u + D v + or u + = v + and u D v .
Note that this dominance order is defined initially only for vectors with the same norm. We can straightforwardly extend it for any vectors by adding the condition u ≺ v when |u| < |v|.

Affine Hecke Algebra
Let N ≥ 2 be an integer, t and q be two independent parameters, and X = {x 1 , . . . , x N } be an alphabet of formal variables. We consider the right operators T i acting on Laurent polynomials in the variables x j by where s i is the elementary transposition permuting the variables x i and x i+1 . For instance, In fact, the operators T i are the unique operators that commute with multiplication by symmetric functions in x i and x i+1 satisfying (2).
Consider also the affine operator τ defined by The operators T i satisfy the relations of the Hecke algebra of the symmetric group: Then, together with the multiplication by variables x i and the affine operator τ , they generate the affine Hecke algebra of the symmetric group:

Symmetric functions and virtual alphabets
For the sake of simplicity, we will use Λ-ring notation for specializations of symmetric functions, see [25]. By specialization we mean a morphism of algebra from Sym to a commutative algebra. Since we manipulate only finite alphabets, specializing an alphabet is equivalent to send each letters to a value. Notice that this is no longer the case for infinite alphabets for which the theory is more complicated.
More precisely, we adopt the following convention stated in terms of power sums. For any variable x, alphabets X and Y and scalar α, we set p n (x) = x n , p n (X + Y) = p n (X) + p n (Y), p n (XY) = p n (X)p n (Y), p n (αY) = α · p n (Y).
With this notation 1 − q n 1 − q corresponds to the alphabet {1, q, . . . , q n−1 }. We set also X k = {x 1 , . . . , x k } and Y a,b = {y a , y a+1 , . . . , y b }, for a ≤ b. If X and Y are two alphabets, we will denote by the resultant of X and Y. Since this is a symmetric function in X and Y separately (but not in X ∪ Y), we can use the notation above. Hence one has

Macdonald Polynomials and Variants
In this section, we set up the definitions and the notation for the Macdonald polynomials for the different variants that appear in the paper. We also define these variants for the Jack polynomials.
Throughout this paper, the following notation is relevant and very useful. Let P (X; q, t) and Q(X; q, t) be two polynomials. We say that P (X; q, t) ( * ) = Q(X; q, t) if the equality holds up to a scalar factor consisting of powers of q and t.

Non symmetric (shifted) Macdonald polynomials
The (q, t)-version of the Cherednik operators are the operators defined by The non symmetric Macdonald polynomials (E v ) v∈N N are the unique basis of simultaneous eigenfunctions of the (q, t)-version of the Cherednik operators such that E v The corresponding spectral vectors are given by the spectral vector, .
We consider also the following variant of the ξ i operators, the Knop-Cherednik operators: Note that the polynomial E v can be recovered as a limit from M v : This follows from the following fact: The differences of the Cherednik operators and the Knop-Cherednik operators are known as the Dunkl operators, D i = Ξ i − ξ i . We say that a polynomial is singular if it is in the kernel of D i , for each i.
Then, the symmetric Macdonald polynomials, P λ , are defined as the eigenfunctions of ξ.
Similarly, we can consider the operator Ξ = i Ξ i . Then, the symmetric shifted Macdonald polynomials, M S λ , are defined as the eigenfunctions of Ξ.
The eigenvalue corresponding to the partition λ is, in both cases, λ q −1 ,t −1 . We say that a polynomial satisfy the highest weight condition if it is in the kernel of Ξ − ξ.

Jack polynomials
We define the Jack polynomials, J α v , as a limit of the Macdonald polynomials, P v , with q = t α and t → 1. This definition applies for the four versions of Macdonald polynomials that appear in this paper.

The Yang-Baxter graph
In [24], A. Lascoux described how to compute the non symmetric shifted Macdonald polynomials M v using the Yang-Baxter graph. This computation is based on the following result.
where v.s i is the vector obtained from v by exchanging the values v[i] and v[i + 1].
This result provides a method to compute the polynomials M v following the Yang-Baxter graph associated to the vector v, starting with the zero vector 0 N and M 0 N = 1. We illustrate how to do it for the non symmetric shifted Macdonald polynomials with an example. On one side, we have the following sequence for the vectors: It corresponds to the following sequence in the non symmetric shifted polynomials: The non symmetric (non shifted) Macdonald polynomials are obtained following an almost similar algorithm where the affine action is substituted by E vΦ = E v τ x N . For instance, The symmetric (shifted and non-shifted) are hence obtained by applying the symmetrizing , to the polynomials labeled by a decreasing vector. Also Jack polynomials are obtained following a Yang-Baxter graph with degenerated intertwining operators.

Vanishing properties
The shifted polynomials in all their versions (non symmetric Macdonald, symmetric Macdonald, non symmetric Jack, symmetric Jack) can be defined alternatively by interpolation. Indeed, one shows with the help of the Yang-Baxter graph that the shifted non symmetric Macdonald polynomials are characterized up to a global coefficient by the equations for any vector u satisfying |u| ≤ |v| and u = v. By symmetrization, one shows that the shifted symmetric Macdonald polynomials are characterized up to a global coefficient by for any decreasing partition µ. Also, vanishing properties of shifted symmetric and non symmetric Jack polynomials are obtained by making equations (3) and (4) degenerate.
3 Clustering properties of Jack polynomial and the quantum Hall effect

A gentle history of the quantum Hall effect
The quantum Hall effect is a phenomenon involving a collection of electrons restricted to move in a two-dimensional space and subject to a strong magnetic field.
The classical Hall effect was discovered by Edwin Hall in 1879 [18] and is a direct consequence of the motion of electrons in a magnetic field. More precisely, it comes from the fact that the magnetic field causes electrons to move in circles. Let us recall quickly the calculation. This phenomena is known under the name cyclotron effect and is deduced from the equations of the motion for a particle of mass m and charge −e in a z-directed magnetic field of intensity B: The general solution, x(t) = x 0 − r sin(ω B t + φ) and y(t) = y 0 + r cos(ω B t + φ), describes a circle. The parameters x 0 , y 0 , r and φ are chosen arbitrary, while ω B = eB m is a linear function of B and is called the cyclotron frequency. Taking into account an electric field E generating the current together with a linear friction term modeled by the scattering time τ , the motion equations become This model is known under the name of Drude model [11,12] and consists in considering the electrons as classical balls. Assuming that the velocity is constant, it can be written as: The current density J is related to the velocity by the equality J = −ne v, n denoting the number of charged particles. Hence, denotes the conductivity. We see that there are two components to the resistivity: the off-diagonal component (Hall resistivity) ρ xy = mω B e 2 n , which does not depend on τ but is linear in B, and the diagonal component (longitudinal resistivity) ρ xx = m e 2 nτ , which does not depend on B and tends to 0 when the scattering time τ tends to ∞. In 1980, Von Klitzing et al. [21] realized measurements of the Hall voltage of a two-dimensional electron gas with a silicon metaloxide-semiconductor field-effect transistor and showed the Hall resistivity has fixed values. The exhibited phenomena is called the integer quantum Hall effect 1 . Both the Hall resistivity and longitudinal resistivity have a behavior which highlights a quantum phenomena at the mesoscopic scale. The Hall resistivity is no longer a linear function of B but sits on a plateau for a range of magnetic field before jumping to the next one. These plateaus are centered on a values B ν = r q n ν , where r q = 2π e is the quantum resistivity, depending on a parameter ν ∈ Z and the Hall resistivity takes the values ρ xy = rq ν . The longitudinal resistivity vanishes when ρ xy sits on a plateau and spikes when ρ xy jumps to the next one.
The fractional quantum Hall effect was discovered by Tsui et al [35]. They observed that, as the disorder is decreased, the integer Hall plateaus become less prominent and other plateaus emerge for fractional values of ν 2 . The difference between the integer quantum Hall effect 8 Figure 1: Classical Hall effect and the fractional quantum Hall effect is that, to explain the second, physicists need to take into account the interactions between the particles. The interaction between the electrons make the problem interesting from a mathematical point of view. The theoretical approach was pioneered by Laughlin [28] for ν = 1 2m+1 . Since the Hamiltonian is very difficult to diagonalize, he proposed directly a wave function fulfilling several properties motivated by physical insight. The Laughlin wave function overlaps more than 99% with the true ground state. From the observations of Tsui et al and the work of Laughlin, more than 80 plateaus have been observed for various filling fractions. The description of the wavefunctions is one of the challenges of the study. It is in this context that Jack polynomials appear.

Quantum Hall wavefunctions
The fractional quantum Hall effect appears in many configurations of the gas. Indeed, the Hall voltage can be generated by the motion of the particles but also by quasiparticles or quasiholes. Quasiparticles and quasiholes are virtual particles that occur when the matter behaves as if contained different weakly interacting particles. The charges of these virtual particles are fractions of the electron charge and their masses are also different. But, in all the cases, for fermion gases, the wave function must take the form where φ is a (polynomial) symmetric function, and B = eB is the magnetic length, which is a characteristic length scale governing quantum phenomena in a magnetic field. This expression is obtained assuming that the system is in the lowest Landau level, the single particle wave functions take the form φ(z) = z m e −|z| 2 4 2 B , and that all the particles play the same role. This last condition is a quite puzzling point. Indeed, since the particles are placed in a finite portion of the plane, they can not play the same role because the interactions must take into account the distance between the particles and the sides of the sample. Hence, the symmetry comes from an approximation when assuming that the number of particles tends to infinity. The Haldane approach [17] for this theory consists in placing the particles on a sphere. The position of a particle is specified by spinor coordinates u = cos( 1 2 θ)e i 1 2 ψ and v = sin( 1 2 θ)e i 1 2 ψ . When the radius tends to infinity the two approaches coincide and the wavefunctions in the spherical geometry are used to compute approximation for the plane geometry via stereographic projection. All the operators and wave function introduced by Haldane have been translated by this way in the plane geometry. In what follow, we consider the Haldane point of view after stereographic projection.
The Laughlin wave function [28] models the simplest FQH states which occurs for ν = 1 m . This wave function is given by Notice that φ Laughlin can be seen as the stereographic projection of Haldane wavefunction stated in terms of spinor coordinates by In the Laughlin states no quasi-particle or quasihole are involved. From a mathematical point of view the absence of quasi-particle and quasi-hole is interpreted in terms of differential operators as follows: We consider the operators E n := i z n i ∂ ∂z i and we set L + = E 0 and The parameter N φ is interpreted in the spherical geometry by the fact that the sphere surrounds a monopole with charge N φ . The absence of quasiparticle is characterized by L + φ = 0 (HW: highest weight condition) while the absence of quasihole is characterized by L − φ = 0 (LW: lowest weight condition). Noticing that [E m , E n ] = (m − n)E n+m−1 , we find that if P is a polynomial satisfying the HW and LW The HW condition means that the polynomial is invariant by translations. A fast computation shows that φ Laughlin is both a HW and a LW state. Other interesting wavefunctions have been exhibited. For instance, the Moore-Read (Pfaffian) state [29] is where Pf denotes the Pfaffian. Surprisingly it describes the FQH for ν = 1. To understand the difference with the integer quantum Hall effect, physicists introduced two values k and r such that ν = k r . The parameter k means that the function vanishes for k+1 particles together but not for k and the parameter r is the order of the zeros. For instance, in the Laughlin state we have r = 2m and k = 1, while for the Moore-Read state we have r = 2 and k = 2. In [7,6], Bernevig and Haldane showed how to associate to each Hall state an occupation number configuration. The occupation number configuration is a vector n φ such that n φ [i] is the number of particles in the ith lowest Landau level orbital (i ≥ 0). For a Laughlin state Instead to use the occupation number configuration, we will use a decreasing partition λ φ such that the multiplicity of the part i in . . , 4, 4, 2, 2, 0, 0 (N needs to be even for Moore-Read states). We see that N φ is the biggest part in λ φ . Reader interested by fractional quantum Hall theory can refer to [34] for a complete picture on the topic.

FQHT and Jack polynomials
Some of the trial wave functions proposed to describe the FQHE are Jack functions. This is the case of the simplest one, This was first noticed by Bernevig and Haldane [7]. They obtained this equality by proving that φ m Laughlin is an eigenstate of the Laplace-Beltrami operator This is particularly interesting to remark that the main argument of the proof comes from clustering properties. Indeed, the Laughlin wavefunction, considered as a polynomial in z i (for some i ∈ {1, . . . , N }) has a multiplicity 2m root at z i = z j for any j = i. So it vanishes under the action of the operator D L,2m In the same paper [7], a similar (but a little more complicated) reasoning is used to study the Moore-Read state described in [29]: φ 0 M R (6). They proved that Other examples are treated in [7]. For instance, the Z p parafermionic states where N = pN and S denotes the symmetrizing operator. This example generalizes (9) and is a special case of a Read-Rezayi state for ν = p 2 [31]. A more complicated example is involved at ν = 2 5 and refereed to as "Gaffnian" [33]. This wavefunction is also proved to be a Jack polynomial [7] φ G (z 1 , . . . , z N ) with N = 2N .
In the aim to provide tools for the understanding of FQH states, Bernevig and Haldane investigated clustering properties of Jack polynomials, [6]. In particular, they exhibited a family of highest weight Jack polynomials in N variables that vanish when s distinct clusters of k + 1 particles are formed. The corresponding partitions depends on 4 parameters (the parameter β depends on N and on 3 other parameters, and is implicit in [6]) λ β k,r,s = [(βr + s(r − 1) + 1) k , . . . , (s(r − 1) + 1) k , 0 n 0 ] with n 0 = (k + 1)s − 1 and N = βk + n 0 . Notice that, in this case, the flux (i.e. the maximal degree in each variable) equals Bernevig and Haldane investigated three kinds of clustering properties that occur when k + 1 and s − 1 are coprime: 1. First clustering property They considered s − 1 clusters of k + 1 particles The other particles (variables) remain free. For such a specialization, the Jack polynomial J − k+1 2. Second clustering property They considered a cluster of n 0 = (k + 1)s − 1 particles when each z i tends to Z. More specifically, for highest weight Jack polynomials, one has the following explicit formula: (z n 0 +1 +· · ·+z N ). (12) For instance, 3. Third clustering property It is obtained by forming s − 1 clusters of 2k + 1 particles For instance, The aim of our paper is to show how the material described in [14] can help in this context. In particular, we focus on the second clustering property for HW polynomials.
To be more complete, the wavefunctions are not all Jack polynomials but many of them can be obtained from Jack polynomials by acting by an operator modeling the adding of a quasiparticle or a quasihole (see e.g. [8]).

The interest of shifted Macdonald polynomials
In [7] (i.e. for s = 1), Bernevig and Haldane proved the clustering properties on HW Jack polynomials using a result of B. Feigin et al [15] together with Lassalle binomial formulas for Jack polynomials [27]. Lassalle binomial formula are used to describe the action of the operator L + on a Jack polynomial. When s > 1, the partitions do not fulfill some admissibility conditions of B. Feigin et al. [15], and so, the equations are just conjectured from extensive numerical computations. For the purpose of manipulating these identities properly, we must leave the framework of homogeneous Jack polynomials. First, clustering properties deal with vanishing properties. So shifted Jack polynomials should be more appropriate for these problems. Nevertheless, the multiplicities of the roots of the polynomials are difficult to manage. The idea consists in (q, t)-deforming these identities in such a way that they involve products of distinct factors. For instance, a factor (z With such a deformation, it is also easier to manipulate the eigenspaces which are smaller (see [9] for the example related to φ Laughlin ). In consequence, we follow the strategy initiated in [14], which consists to manipulate shifted Macdonald polynomials in the aim to prove the identities. The recipe is as follows: • We find a Macdonald version of the conjecture and we state it in terms of vanishing properties.
• We prove that the Macdonald polynomial involved is a highest weight polynomial (i.e. in the kernel of a q-deformation of L + ). When it is possible, this property comes from [16] (Macdonald version of [15]), while for the other cases we apply the results of [19], which are based on the Lassalle binomial formula for Macdonald polynomials [27].
• In this case, the shifted Macdonald polynomial equals the homogeneous Macdonald polynomials.
• We deduce the equality from vanishing properties of the shifted Macdonald polynomial.
• We recover the identity on Jack by sending q to 1.
Notice that in [19], one of the authors with Thierry Jolicoeur found some families of polynomials which have not been considered in [6]. Indeed, Bernevig and Haldane missed that the family λ β k,r,s can be extended by adding a parameter corresponding to the multiplicity of the largest part which can be smaller than k. Also, some other Macdonald polynomials do not specialize to a Jack for the considered specialization of (q, t).
We detail all of that in the next sections.
Recall that ( * ) = means that the equality holds up to a scalar factor.
Proof. Recall the affine step for non symmetric Macdonald polynomials (resp. Shifted Macdonald polynomials) If we apply this step N times, we obtain respectively, Hence, if we apply the affine step N times again, we obtain respectively, By induction, starting with v = [λ 1 − λ N , λ 2 − λ N , . . . , λ N −1 − λ N , 0] and applying the affine step N λ N times, one finds Since the polynomials x λ N i q k − 1 are symmetric, they commute with the action of the symmetrizing operator S and the result is obtained by applying the symmetrizing operator to E λ and M λ .
Remark 4.2 Notice that one has an alternative proof for the second result. One has to examine the vanishing properties of Let If [λ N , . . . , λ N ] ⊂ µ then this means that µ N < λ N . Then, the factor (x N − q µ N ) in (14) vanishes for x N = q µ N . This proves that the two polynomials have the same vanishing properties and so, that they are equal.
Proof. Consider the polynomial This polynomial vanishes for [x 1 , · · · , x N −k ] = [q µ 1 t N −k−1 , . . . , q µ N −k t 0 ], for any µ = λ. Moreover, |µ| ≥ |λ|, since  properties of M S 3200 implies that M S 3200 (x 1 , x 2 , t 2 , t, 1) vanishes for the following values of (x 1 , x 2 ): (q 5 t 4 , t 3 ), (q 4 t 4 , qt 3 ), (q 4 t 4 , t 3 ), (q 3 t 4 , qt 3 ), (q 2 t 4 , q 2 t 3 ), (q 3 t 4 , t 3 ), (q 2 t 4 , qt 3 ), (q 2 t 4 , t 3 ), (qt 4 , qt 3 ), (qt 4 , t 3 ) and (t 4 , t 3 ). Since, M S 3200 (x 1 , x 2 , t 2 , t, 1) is a degree 5 symmetric polynomial in two variables, these vanishing properties completely characterize it up to a global factor. Indeed, there are exactly 12 independent symmetric functions of degree at most 5 in 2 variables. The basis of the space spanned by these functions is generated by the polynomials M S 50 (x 1 , x 2 ; q, t), M S 41 (x 1 , x 2 ; q, t), M S 32 (x 1 , x 2 ; q, t), . . . , M S 00 (x 1 , x 2 ). The polynomial M S 3200 (x 1 , x 2 , t 2 , t, 1) is symmetric and so, is a linear combination of the 12 polynomials above. It follows that a series of 11 vanishing properties is sufficient to produce a system of linear equations characterizing the coefficients of this combination. we deduce that the polynomials M S 32000 (X 2 + t 2 + t + 1) and M S 32 (t −3 X 2 ) are proportional. These two polynomials have low degree and so, are easy to compute by the help of the Yang-Baxter graph. One finds M S 32 (X 2 ; q, t) and M S 3200 (X 2 + t 2 + t + 1; q, t) Corollary 4.5 Denoting by λ i the number of parts of λ lower or equal to i, and by m λ the multiplicity of the maximal part in λ, we have Proof. We prove the property by induction on N + |λ|. We have to consider two cases: 1. If λ N > 0, then by Proposition 4.1, we have , . . . , λ N −1 , λ N and by induction: But λ j = λ j+λ N . Hence, as expected.
As expected, the last polynomial is proportional to 5 Specializations of the type t α q β = 1 and quasi-staircase partitions
Notice that the reason for the definition of ω 1 is given in the following: Lemma 5.2 Suppose t α q γ = 1, t = u b , and q = ω 1 u −a . Then, α = p ( + 1) and γ = p (s − 1), for some p ∈ N.
Notice that Proposition 5.5 can be alternatively stated as follows.
Corollary 5.7 Suppose that µ ⊆ λ = QS( , k; s, r; β) is a partition such that µ is a permutation of λ . Then, • the intersection of the eigenspace of ξ with eigenvalue λ q −1 ,t −1 , and the space generated by P µ , with µ ⊆ λ, has dimension 1, • the intersection of the eigenspace of Ξ with eigenvalue λ q −1 ,t −1 , and the space generated by M S µ , with µ ⊆ λ, has dimension 1.
Proof. It is easy to see that for a specialization of type (t, q) = (u b , ω 1 u a ) the following four assertions are equivalent: 1. µ is a permutation of λ , Hence, Proposition 5.5 allows us to complete the proof.
This concludes the proof. for any ≥ 1, 0 ≤ k < , s ≥ 2 and β ≥ 1. We also assume that the parameters q and t specialize as where g = gcd( + 1, s − 1) and ω s−1 g 1 is a primitive gth root of the unity.

Wheel condition and admissible partitions
In this section we recall the main results of [16]. A symmetric polynomial P (x 1 , . . . , x N ) satisfies the (s, )-wheel condition if . . , tq s−1 } implies that P (x 1 , . . . , x N ) = 0. It is easy to check that the set of the symmetric polynomials satisfying the wheel condition is an ideal. This ideal is denoted by J ,s N in [16]. A ( , s, N )admissible partition is a partition λ = [λ 1 , . . . , λ N ] satisfying λ i − λ i+ ≥ s for any i = 1, . . . , N − . The following theorem summarizes two results of [16].
• The space J ,s N is stable under the action of L + q,t .
is homogeneous. And so, it is equal to P 20 (x 1 , x 2 ; q = 1 u , u 2 ) up to a multiplicative factor.

Proposition 6.4 implies
Iterating, one finds Once again applying Proposition 6.2 and Proposition 4.3, one gets the following result.

Beyond the wheel conditions
First we recall that, for some specialization of (q, t), the quasistaircase polynomials satisfy the highest weight condition (results contained in [19]). Using an argument of dimension of eigenspaces, we show that the shifted Macdonald are homogeneous. We also continue to consider the specialization (t, q) = u satisfying that g = gcd( + 1, s − 1) and ω s−1 g 1 is a gth primitive root of the unity. Recall the following result of [19].
Remarking that L + q,t = (1 − q) D i = (1 − q) ( ξ i − Ξ i ), we find that the shifted symmetric polynomial M S QS( ,k;s,r;β) is an eigenfunction of ξ i having the same eigenvalue as P QS( ,k;s,r;β) . But from Corollary 5.7 the corresponding eigenspace has dimension 1. This proves the following result.
Proof. From equality (24), we obtain The use of Theorem 6.6 completes the proof.
The formula (24) is more general than those conjectured in [6] for two reasons. First, we consider quasistaircase partitions QS( , k; s, r; β), with 0 ≤ k < (in [7,6] only the case k = 0 was investigated). It should be interesting to know if some of these polynomials can be interpreted as wave functions in FQHT. Also, when ω 1 = 1, the Macdonald polynomial does not degenerate to a Jack when u tends to 1.
More generally, the following equality is obtained from Theorem 7.3:
Notice that this polynomial does not satisfy the highest weight condition. This suggests that there exists a Macdonald version of the result of [5], Theorem 1.1. A precise statement remains to be formulated.

Other clustering/factorizations properties
The first and third clustering conjecture suggest that there exist many ways to factorize highest weight Macdonald polynomials by specializing the variables x 1 , x 2 , . . . , x N . Let us illustrate this remark by giving an example.
The precise statements, the proofs, and the connection with the factorizations of symmetric Macdonald polynomials remain to be investigated.