Open AccessFeature PaperArticle

Factorizations of Symmetric Macdonald Polynomials

Laura Colmenarejo

^1,*,†

Charles F. Dunkl

and

Jean-Gabriel Luque

Department of Mathematics and Statistics, York University, 4700 Keele St, Toronto, ON M3J 1P3, Canada

Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA

Laboratoire d’Informatique, du Traitement de l’Information et des Systèmes (LITIS), Université de Rouen, 685 Avenue de l’Université, 76800 Saint-Étienne-du-Rouvray, France

Author to whom correspondence should be addressed.

^†

Current address: Max Planck Institute for Mathematics at Leipzig, 04103 Leipzig, Germany.

Symmetry 2018, 10(11), 541; https://doi.org/10.3390/sym10110541

Submission received: 2 October 2018 / Revised: 17 October 2018 / Accepted: 20 October 2018 / Published: 24 October 2018

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Abstract

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

Keywords:

quantum Hall theory; Macdonald polynomials; Jack polynomials; clustering properties; highest weight polynomials; quasistaircases factorizations

1. Introduction

Jack polynomials have many applications in physics, especially in statistical physics and quantum physics due to their relation to the many-body problem. In particular, fractional quantum Hall (FQH) states of particles in the lowest Landau levels are described by such polynomials [1,2,3]. 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 distinct clusters of equal variables are formed. Using tools of algebraic geometry, Berkesch-Zamaere et al. [4] proved several clustering properties including some special cases conjectured by Bervenig and Haldane [1]. Coming from theoretical physics, the study of these properties raises very interesting problems in combinatorics and representation theory of the affine Hecke algebra. 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 algebra and the results are recovered by making degenerate the parameters q and t. Recently, a very similar technique has been used in [5] to prove general formulas for perturbative correlators in basic matrix models that can be interpreted as the Schur-preservation property of Gaussian measures. In this case, the substitution of Schur functions by Macdonald polynomials defines a deformation of the matrix model. For our case, 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 already been investigated in [6,7,8,9]. In particular, this paper is the sequel of [9] in which two of the authors proved factorizations for rectangular Macdonald polynomials. In this paper, we investigate factorizations for more general partitions, called quasistaircase partitions.

The paper is organized as follows. In Section 2, we recall essential prerequisites on Macdonald polynomials. In Section 3, we give a brief account on the physics motivations coming from the FQH effect 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^{a} q^{b} = 1

and, in particular, to the consequences on spectral vectors. In Section 6, we deduce factorizations from the results of Feigen et al. [10] and, in Section 7, we prove more general results which are consequences of the highest weight (HW) condition of some Macdonald polynomials indexed by quasistaircase partitions proved in [11]. Section 8 is devoted to illustrate our results by proving a conjecture stated by Bernevig and Haldane [2] and also we show many other examples of factorizations that do not follow from our formulas.

2. Background

This paper is focused on 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, e.g., [12,13,14,15,16,17,18,19,20]). The results of [21,22,23] are extensively used throughout our paper.

2.1. Partitions and Vectors

A partition

λ = [λ_{1}, \dots, λ_{ℓ}]

of n is a weakly decreasing sequence of nonnegative integers such that

| λ | : = \sum_{i} λ_{i} = n

. The length of a partition

λ

is the number of nonzero parts,

ℓ (λ) = \max {i : λ_{i} > 0}

. We denote 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 consider a dominance order on the partitions: Let

λ

and

μ

be partitions, we say that

λ ⪯_{D} μ

if and only if, for all i,

λ_{1} + \dots + λ_{i} \leq μ_{1} + \dots + μ_{i}

We also consider vectors, denoting by

v = [v [1], \dots, v [N]]

a vector of length

ℓ (v) = N

and with nonnegative entries. We denote by

| v |

the sum of the entries of v. For each vector v, there is a unique partition,

v^{+}

, which is a permutation of v. The dominance order defined on partitions extends to vectors with the same definition. In addition, we define another order: For u and v vectors, we say that

u ⪯ v

if and only if either

u^{+} ⪯_{D} v^{+}

(u^{+} = v^{+}

and

u ⪯_{D} v)

. Initially, this order is only defined for vectors such that

| u | = | v |

, and we extend it straightforwardly for any vectors by adding the condition

u ≺ v

when

| u | < | v |

We define the standardization of v,

{std}_{v}

, as the vector obtained by labeling with the integers from 0 to

N - 1

the positions in v from the smallest entries to the largest one and from right to left. Moreover, we define the reciprocal vector of v as

〈 v 〉 = [t^{{std}_{v} [1]} q^{v [1]}, \dots, t^{{std}_{v} [N]} q^{v [N]}]

, and the reciprocal sum of v as

⟅ v ⟆ = \sum_{i} 〈 v 〉 [i]

. If there is ambiguity, the indices q and t are added. Note that in the particular case of considering a partition

λ

⟅ λ ⟆ = \sum_{i} t^{N - i} q^{λ_{i}}

For example, consider the vector

v = [1, 2, 2, 0, 1, 1]

of length 6. Then,

std (v) = [3, 5, 4, 0, 2, 1]

〈 v 〉 = [q t^{3}, q^{2} t^{5}, q^{2} t^{4}, 1, q t^{2}, q t]

, and

⟅ v ⟆ = 1 + q t + q t^{2} + q t^{3} + q^{2} t^{4} + q^{2} t^{5}

2.2. Affine Hecke Algebra

We present a brief introduction to the affine Hecke algebra

H_{N} (q, t)

and the Demazure–Lusztig operators

T_{i}

. For more details, see [24,25,26,27].

Let

N \geq 2

be an integer, q and t be two independent parameters, and

X = {x_{1}, \dots, x_{N}}

be a set of formal variables. We consider the right operators

T_{i}

, for

1 \leq i \leq N - 1

, acting on Laurent polynomials in the variables

X

\begin{matrix} T_{i} = t + (s_{i} - 1) \frac{t x_{i + 1} - x_{i}}{x_{i + 1} - x_{i}}, \end{matrix}

where

s_{i}

is the elementary transposition permuting the variables

x_{i}

and

x_{i + 1}

. For instance,

1 T_{i} = t

and

x_{i + 1} T_{i} = x_{i}

. 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 these last two equations. Consider also the affine operator

τ

defined by

f (x_{1}, \dots, x_{N}) τ = f (\frac{x_{N}}{q}, x_{1}, \dots, x_{N - 1}) .

The operators

T_{i}

satisfy the relations of the Hecke algebra of the symmetric group:

\begin{matrix} T_{i} T_{i + 1} T_{i} = T_{i + 1} T_{i} T_{i + 1}, for all i, T_{i} T_{j} = T_{j} T_{i} for | i - j | > 1, (T_{i} - t) (T_{i} + 1) = 0, for all i . \end{matrix}

Then, together with the multiplication by variables

x_{i}

and the affine operator

τ

, they generate the affine Hecke algebra of the symmetric group:

H_{N} (q, t) = C (q, t) [x_{1}^{\pm}, \dots, x_{N}^{\pm}, T_{1}^{\pm}, \dots, T_{N_{1}}^{\pm}, τ] .

2.3. Symmetric Functions and Virtual Alphabets

For the sake of simplicity, we use

Λ

-ring notation for specializations of symmetric polynomials (see [22]). A finite alphabet is a finite set of formal variables and we denote it by

X = {x_{1}, x_{2}, \dots, x_{n}}

X = x_{1} + x_{2} + \dots + x_{n}

. By specialization, we mean a morphism of algebras and, since we manipulate only finite alphabets, specializing an alphabet is equivalent to sending each letter 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 the basis of power sums symmetric polynomials. For any variable x, alphabets

X

and

Y

and scalar

α

, we set

\begin{matrix} \begin{matrix} p_{n} (x) = x^{n}, \\ p_{n} (X \pm Y) = p_{n} (X) \pm p_{n} (Y), \\ p_{n} (X Y) = p_{n} (X) p_{n} (Y), \\ p_{n} (α Y) = α \cdot p_{n} (Y) . \end{matrix} \end{matrix}

We set also the following notation:

X_{k} = {x_{1}, \dots, x_{k}}

and

Y_{a, b} = {y_{a}, y_{a + 1}, \dots, y_{b}}

, for

a \leq b

. For two alphabets

X

and

Y

, the resultant of

X

and

Y

is defined as

R (X; Y) = \prod_{(x, y) \in X \times Y} (x - y),

which is a symmetric polynomial in

X

and

Y

separately (but not in

X \cup Y

). Hence,

\begin{matrix} R (X + X^{'}; Y) = R (X; Y) R (X^{'}; Y) and R (X; Y + Y^{'}) = R (X; Y) R (X; Y^{'}) . \end{matrix}

Throughout this paper, the following notation is relevant and very useful. Let

P (X; q, t)

and

Q (X; q, t)

be two polynomials in

X

with coefficients in q and t. We say that

P (X; q, t) \overset{(*)}{=} Q (X; q, t)

if the equality holds up to a scalar factor consisting of powers of q and t.

2.4. Macdonald Polynomials and Variants

In this section, we set up the definitions and the notation for the different variants of Jack and Macdonald polynomials that appear in this paper. For that, we define several vectors and the basis of eigenfunctions associated. First, we define the two variants of nonsymmetric Macdonald polynomials, indexed by vectors.

Definition 1.

The

(q, t)

-version of the Cherednik operators are the operators defined by

\begin{matrix} ξ_{i} : = t^{1 - i} T_{i - 1} \dots T_{1} τ T_{N - 1}^{- 1} \dots T_{i}^{- 1}, f o r 1 \leq i \leq N . \end{matrix}

The nonsymmetric Macdonald polynomials

{\{E_{v}\}}_{v \in N^{N}}

are the unique basis of simultaneous eigenfunctions of the

(q, t)

-version of the Cherednik operators such that

E_{v} \overset{(*)}{=} x^{v} + \sum_{u ≺ v} α_{u, v} x^{u}

. The corresponding vector of eigenvalues is called spectral vector and its ith entry is

S p e c {(v)}_{i} = \frac{1}{{〈 v 〉}_{i}}

We consider also the following variant of the

ξ_{i}

operators.

Definition 2.

The Knop–Cherednik operators are defined by

\begin{matrix} Ξ_{i} : = t^{1 - i} T_{i - 1} \dots T_{1} τ (1 - \frac{1}{x_{N}}) T_{N - 1}^{- 1} \dots T_{i}^{- 1} + \frac{1}{x_{i}}, f o r 1 \leq i \leq N . \end{matrix}

The nonsymmetric shifted Macdonald polynomials

{\{M_{v}\}}_{v \in N^{N}}

are the unique basis of simultaneous eigenfunctions of the Knop–Cherednik operators such that

M_{v} \overset{(*)}{=} x^{v} + \sum_{u ≺ v} α_{u, v} x^{u}

As in the case of the nonsymmetric Macdonald polynomials, the eigenvalues make up the spectral vector

Spec (v)

Note that the polynomial

E_{v}

can be recovered as a limit from

M_{v}

\begin{matrix} E_{v} (x_{1}, \dots, x_{N}) = \lim_{a \to 0} a^{| v |} M_{v} (\frac{x_{1}}{a}, \dots, \frac{x_{N}}{a}), \end{matrix}

which follows from the relation

M_{v} (x_{1}, \dots, x_{N}) =^{(*)} E_{v} (x_{1}, \dots, x_{N}) + \sum_{u ≺ v} β_{u, v} E_{u}

Now, we define both variants for symmetric Macdonald polynomials, indexed by partitions, by considering the sum of the operators defined above.

The Debiard–Sekiguchi–Macdonald operator is the operator defined as

ξ = \sum_{i} ξ_{i}

. The symmetric Macdonald polynomials,

P_{λ}

, are defined as the eigenfunctions of

ξ

. Similarly, we can consider the operator

Ξ = \sum_{i} Ξ_{i}

and the symmetric shifted Macdonald polynomials,

M S_{λ}

, are defined as the eigenfunctions of

Ξ

. In both cases, the eigenvalues are

{⟅ λ ⟆}_{q^{- 1}, t^{- 1}}

We define the symmetric Jack polynomials,

J_{λ}^{α}

, as the limit of the Macdonald polynomials,

P_{λ}

, with

q = t^{α}

and

t \to 1

. The other versions of Jack polynomials can be also defined as a limit, but we focus our attention on the symmetric variant.

Two families of polynomials are relevant for our study and rely on the difference of the operators presented above. 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 all i. We say that a polynomial is a HW polynomial if it is in the kernel of

Ξ - ξ

2.5. Computing Macdonald Polynomials Using the Yang–Baxter Graph

In [21], Lascoux described how to compute the nonsymmetric shifted Macdonald polynomials

M_{v}

using the Yang–Baxter graph. This computation is based on the following result.

Proposition 1.

Let v be a vector.

If $v [i] < v [i + 1]$ , $M_{v . s_{i}} = M_{v} (T_{i} + \frac{1 - t}{1 - \frac{〈 v 〉 [i + 1]}{〈 v 〉 [i]}})$ , where $v . s_{i}$ is the vector obtained from v by exchanging the values $v [i]$ and $v [i + 1]$ .
$M_{v Φ} = M_{v} τ (x_{N} - 1)$ , where $v Φ = [v [2], \dots, v [N], v [1] + 1]$ . We refer to this step as affine step.

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 nonsymmetric shifted Macdonald polynomials with an example. On the one side, we have the following sequence for the vectors:

\begin{matrix} [000] \overset{Φ}{\to} [001] \overset{s_{2} = (23)}{\to} [010] \overset{s_{1} = (12)}{\to} [100] \overset{Φ}{\to} [002] \overset{s_{2} = (23)}{\to} [020] \overset{Φ}{\to} [201] . \end{matrix}

Notice that we simplify the notation by writing

[000]

for the vector

[0, 0, 0]

. Since our examples do not contain numbers larger than nine, there is no misunderstanding.

The sequence above corresponds to the following sequence on the polynomials:

\begin{matrix} M_{[000]} \overset{τ (x_{3} - 1)}{\to} M_{[001]} \overset{T_{2} + \frac{1 - t}{1 - q t^{2}}}{\to} M_{[010]} \overset{T_{1} + \frac{1 - t}{1 - q t}}{\to} M_{[100]} \overset{τ (x_{3} - 1)}{\to} M_{[002]} \overset{T_{2} + \frac{1 - t}{1 - q^{2} t^{2}}}{\to} M_{[020]} \overset{τ (x_{3} - 1)}{\to} M_{[201]} . \end{matrix}

The nonsymmetric Macdonald polynomials are obtained following an almost similar algorithm where the affine action is substituted by

E_{v Φ} = E_{v} τ x_{N}

. For instance, using the same sequence as before,

\begin{matrix} E_{[000]} \overset{τ x_{3}}{\to} E_{[001]} \overset{T_{2} + \frac{1 - t}{1 - q t^{2}}}{\to} E_{[010]} \overset{T_{1} + \frac{1 - t}{1 - q t}}{\to} E_{[100]} \overset{τ x_{3}}{\to} E_{[002]} \overset{T_{2} + \frac{1 - t}{1 - q^{2} t^{2}}}{\to} E_{[020]} \overset{τ x_{3}}{\to} E_{[201]} . \end{matrix}

The symmetric variants of the Macdonald polynomials are hence obtained by applying the symmetrizing operator

S = \sum_{σ \in S_{N}} T_{σ}

, where

T_{σ} = T_{i_{1}} \dots T_{i_{k}}

σ = s_{i_{1}} \dots s_{i_{k}}

is a shortest decomposition of

σ

in elementary transpositions, to the polynomials labeled by a decreasing vector. In addition, Jack polynomials are obtained following a Yang–Baxter graph with degenerated intertwining operators.

2.6. Vanishing Properties

The shifted variants for Jack and Macdonald polynomials can be defined alternatively by interpolation. Indeed, one shows with the help of the Yang–Baxter graph that the shifted nonsymmetric Macdonald polynomials are characterized up to a global coefficient by the equations

M_{v} (〈 u 〉) = 0

, for any vector u satisfying

| u | \leq | v |

and

u \neq v

. That is, by specializing the variables of

X

to the entries of the vector

〈 u 〉

in the nonsymmetric Macdonald polynomial

M_{v}

By symmetrization, the shifted symmetric Macdonald polynomials are characterized up to a global coefficient by

M S_{μ} (〈 λ 〉) = 0

, for any partition

μ

satisfying

| μ | \leq | λ |

and

μ \neq λ

In addition, vanishing properties of shifted symmetric and nonsymmetric Jack polynomials are obtained by making the equations above degenerate.

3. Clustering Properties of Jack Polynomials and the Quantum Hall Effect

3.1. 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 [28] 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:

\begin{matrix} \{\begin{matrix} m \frac{d^{2}}{d t^{2}} x = - e B \frac{d}{d t} y, \\ m \frac{d^{2}}{d t^{2}} y = e B \frac{d}{d t} x . \end{matrix} \end{matrix}

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} = \frac{e B}{m}

is a linear function of B and is called the cyclotron frequency. Taking into account an electric field

\vec{E}

generating the current together with a linear friction term modeled by the scattering time

τ

, the motion equations become

\begin{matrix} m (\frac{d}{d t}) \vec{v} = - e \vec{v} \land \vec{B} - \frac{m}{τ} \vec{v} - e \vec{E} . \end{matrix}

This model is known under the name of Drude model [29,30] and consists in considering the electrons as classical balls. Assuming that the velocity is constant, it can be written as:

\begin{matrix} (\begin{matrix} \frac{m}{2} & - e B \\ e B & \frac{m}{2} \end{matrix}) \vec{v} = \vec{E} . \end{matrix}

The current density

\vec{J}

is related to the velocity by the equality

\vec{J} = - n e \vec{v}

, where n denotes the number of charged particles. Hence,

\vec{E} = ρ \vec{J}

where

\begin{matrix} ρ = \frac{m}{e^{2} n τ} (\begin{matrix} 1 & ω_{B} τ \\ - ω_{B} τ & 1 \end{matrix}) \end{matrix}

denotes the conductivity.

We see that there are two components to the resistivity: the off-diagonal component (Hall resistivity, see Figure 1)

ρ_{x y} = \frac{m ω_{B}}{e^{2} n}

, which does not depend on

τ

but is linear in B, and the diagonal component (longitudinal resistivity)

ρ_{x x} = \frac{m}{e^{2} n τ}

, which does not depend on B and tends to 0 when the scattering time

τ

tends to ∞.

In 1980, Von Klitzing et al. [31] realized measurements of the Hall voltage of a two-dimensional electron gas with a silicon metal-oxide-semiconductor field-effect transistor and showed the Hall resistivity has fixed values. The exhibited phenomenon is called the integer quantum Hall effect, see Figure 2 (This figure was obtained by replacing the vertical scale

R_{H}

ν

in [32] (Figure 4.1, Section 4.4) and it is under Creative Commons Attribution-Noncommercial-Share Alike 3.0 Generic License).

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 plateaux are centered on values

B_{ν} = r_{q} \frac{n}{ν}

, where

r_{q} = \frac{2 π ℏ}{e}

is the quantum resistivity, depending on a parameter

ν \in Z

and the Hall resistivity takes the value

ρ_{x y} = \frac{r_{q}}{ν}

. The longitudinal resistivity vanishes when

ρ_{x y}

sits on a plateau and spikes when

ρ_{x y}

jumps to the next one.

The FQH effect was discovered by Tsui et al. [33]. They observed that, as the disorder is decreased, the integer Hall plateaux become less prominent and other plateaux emerge for fractional values of

ν

, see Figure 3 (This figure was drawn by modifying a picture from [34]).

The difference between the integer quantum Hall effect and the FQH 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 [35] for

ν = \frac{1}{2 m + 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 plateaux have been observed for various filling fractions. The description of the wave functions is one of the challenges of the study. It is in this context that Jack polynomials appear.

3.2. Quantum Hall Wave Functions

The FQH effect appears in many configurations of the gas. Indeed, the Hall voltage can be generated by the motion of the particles but also by quasi-particles or quasi-holes. Quasi-particles and quasi-holes are virtual particles that occur when matter behaves as if it contains different weakly interacting particles. The charges of these virtual particles are fractions of the electron charge and their masses are also different. However, in all cases, for fermion gases, the wave function must take the form

\begin{matrix} Φ (z_{1}, \dots, z_{N}) = ϕ (z_{1}, \dots, z_{N}) \prod_{i < j} (z_{i} - z_{j}) exp \{- \sum_{i = 1}^{N} \frac{- | z_{i} |^{2}}{4 ℓ_{B}^{2}}\}, \end{matrix}

where

ϕ

is a symmetric polynomial, and

ℓ_{B} = \sqrt{\frac{ℏ}{e B}}

is the magnetic length, which is a characteristic length scale governing quantum phenomena in a magnetic field (see, e.g., [36] for details). This expression is obtained assuming that the system is in the lowest Landau level, where the single particle wave functions take the form

ϕ (z) = z^{m} e^{\frac{- {| z |}^{2}}{4 ℓ_{B}^{2}}}

, 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 cannot 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 [37] for this theory consists in placing the particles on a sphere. The position of a particle is specified by spinor coordinates

u = cos (\frac{1}{2} θ) e^{i \frac{1}{2} ψ}

and

v = sin (\frac{1}{2} θ) e^{i \frac{1}{2} ψ}

. When the radius tends to infinity the two approaches coincide and the wave functions in the spherical geometry are used to compute approximation for the plane geometry via stereographic projection. All operators and wave function introduced by Haldane have been translated by this way in the plane geometry. In the following, we consider the Haldane point of view after stereographic projection.

The Laughlin wave function [35] models the simplest FQH states which occurs for

ν = \frac{1}{m}

. This wave function is given by

\begin{matrix} ϕ_{L a u g h l i n}^{m} (z_{1}, \dots, z_{N}) : = \prod_{i < j} {(z_{i} - z_{j})}^{2 m} . \end{matrix}

Notice that

ϕ_{L a u g h l i n}

can be seen as the stereographic projection of Haldane wave function stated in terms of spinor coordinates by

\begin{matrix} ϕ_{H a l d a n e}^{m} = \prod_{i < j} {(u_{i} v_{j} - u_{j} v_{i})}^{2 m} . \end{matrix}

In the Laughlin states no quasi-particle or quasi-hole is 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} : = \sum_{i} z_{i}^{n} \frac{\partial}{\partial z_{i}}

and we set

L^{+} = E_{0}

and

L^{-} = N_{ϕ} \sum_{i} z_{i} - E_{2}

where

N_{ϕ} = 2 \frac{\deg ϕ}{N}

. 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 quasi-particle is characterized by

L^{+} ϕ = 0

(HW condition) while the absence of quasi-hole is characterized by

L^{-} ϕ = 0

(lowest weight (LW) 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 conditions then

E_{1} P = \frac{1}{2} [E_{2}, E_{0}] P = - \frac{1}{2} E_{0} E_{2} P = \frac{1}{2} N_{ϕ} N P = \deg (P) P

, that is P is an homogeneous polynomial. The HW condition means that the polynomial is invariant by translations. A fast computation shows that

ϕ_{L a u g h l i n}

is both a HW and a LW state.

Other interesting wave functions have been exhibited. For instance, the Moore–Read (Pfaffian) state [38] is

\begin{matrix} ϕ_{M R}^{0} = \prod_{i < j} (z_{i} - z_{j}) Pf (\frac{1}{z_{i} - z_{j}}), \end{matrix}

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 = \frac{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 = 2 m

and

k = 1

, while for the Moore–Read state we have

r = 2

and

k = 2

. In [1,2], 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 \geq 0

). For a Laughlin state

ϕ_{L a u g h l i n}^{m}

, we have

n_{ϕ_{L a u g h l i n}^{m}} = [1, 0^{2 m - 1}, 1, 0^{2 m - 1}, \dots]

. For the Moore–Read state, we have

n_{ϕ_{M R}^{0}} = [2, 0, 2, 0, 2, \dots]

. The number

N_{ϕ}

is the greatest integer i such that

n_{ϕ} [i] \neq 0

. Instead of using the occupation number configuration, we use a decreasing partition

λ_{ϕ}

such that the multiplicity of the part i in

λ_{ϕ}

equals

n_{ϕ} [i]

. For instance

λ_{ϕ_{L a u g h l i n}^{m}} = [(N - 1) m, \dots, m, 0]

and

λ_{ϕ_{M R}^{0}} = [2 (\frac{N}{2} - 1), 2 (\frac{N}{2} - 1), \dots, 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 FQH effect theory can refer to [36] for a complete picture on the topic.

3.3. 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,

\begin{matrix} ϕ_{L a u g h l i n}^{m} (z_{1}, \dots, z_{N}) \overset{(*)}{=} J_{[2 (N - 1) m, 2 (N - 2) m, \dots, 0]}^{(\frac{- 2}{2 m - 1})} (z_{1}, \dots, z_{N}) . \end{matrix}

This was first noticed by Bernevig and Haldane [1]. They obtained this equality by proving that

ϕ_{L a u g h l i n}^{m}

is an eigenstate of the Laplace–Beltrami operator

\begin{matrix} H^{(α)} = \sum_{i} {(z_{i} \frac{\partial}{\partial z_{i}})}^{2} + \frac{1}{α} \sum_{i < j} \frac{z_{i} + z_{j}}{z_{i} - z_{j}} (z_{i} \frac{\partial}{\partial z_{j}} - z_{j} \frac{\partial}{\partial z_{i}}) . \end{matrix}

It is particularly interesting to remark that the main argument of the proof comes from clustering properties. Indeed, the idea is that the Laughlin wave function, considered as a polynomial in

z_{i}

for some

i \in {1, \dots, N}

, has a multiplicity

2 m

root at

z_{i} = z_{j}

for any

j \neq i

. Therefore, it vanishes under the action of the operator

D_{i}^{L, 2 m}

where

D_{i}^{L, r} = \frac{\partial}{\partial z_{i}} - r \sum_{j \neq i} \frac{1}{z_{i} - z_{j}}

. Since

ϕ_{L a u g l i n}^{m}

is in the kernel of

\sum_{i} z_{i} D_{i}^{L, 1} z_{i} D_{i}^{L, r} = H^{(- \frac{2}{2 m - 1})} - \frac{m}{6} N (N - 1) (N + 1 + 6 m (N - 1))

, it is an eigenfunction of

H^{(- \frac{2}{2 m - 1})}

. Hence,

ϕ_{L a u g h l i n}^{m}

is identified with

J_{[2 (N - 1) m, 2 (N - 2) m, \dots, 0]}^{(\frac{- 2}{2 m - 1})} (z_{1}, \dots, z_{N})

by considering its dominant monomial.

In the same paper [1], a similar (but a little more complicated) reasoning is used to study the Moore–Read state

ϕ_{M R}^{0}

described in [38]. They proved that

\begin{matrix} ϕ_{M R}^{0} (z_{1}, \dots, z_{N}) \overset{(*)}{=} J_{[2 (\frac{N}{2} - 1), 2 (\frac{N}{2} - 1), \dots, 4, 4, 2, 2, 0, 0]}^{(- 3)} (z_{1}, \dots, z_{N}) . \end{matrix}

Other examples are treated in [1]. For instance, a generalization of the previous example is given by the

Z_{p}

parafermionic states

\begin{matrix} ϕ_{R R}^{0} = S (\prod_{k = 1}^{N} \prod_{(k - 1) N^{'} \leq i < j \leq k N^{'}} {(z_{i} - z_{j})}^{2}) \overset{(*)}{=} J_{[{(2 (N^{'} - 1))}^{p}, \dots, 2^{p}, 0^{p}]}^{- (p + 1)} (z_{1}, \dots, z_{N}), \end{matrix}

where

N = p N^{'}

and

S

denotes the symmetrizing operator. This example is a special case of a Read–Rezayi state for

ν = \frac{p}{2}

[39]. A more complicated example is involved at

ν = \frac{2}{5}

and refereed to as “Gaffnian” [40]. This wave function is also proved to be a Jack polynomial [1]

\begin{matrix} ϕ_{G} (z_{1}, \dots, z_{N}) \overset{(*)}{=} J_{[3 (N^{'} - 1), 3 (N^{'} - 1), 3 (N^{'} - 2), 3 (N^{'} - 2), \dots, 3, 3, 0, 0}^{(\frac{- 3}{4})} (z_{1}, \dots, z_{N}), \end{matrix}

with

N = 2 N^{'}

In the aim to provide tools for the understanding of FQH states, Bernevig and Haldane investigated clustering properties of Jack polynomials, [2]. In particular, they exhibited a family of HW Jack polynomials in N variables that vanish when s distinct clusters of

k + 1

particles are formed. The corresponding partitions are

λ_{k, r, s}^{β} = [{(β r + s (r - 1) + 1)}^{k}, \dots, {(s (r - 1) + 1)}^{k}, 0^{n_{0}}]

with

n_{0} = (k + 1) s - 1

and

N = β k + n_{0}

. Notice that, in this case, the flux (i.e., the maximal degree in each variable) equals

\begin{matrix} N_{ϕ} = β r + s (r - 1) + 1 = \frac{r}{k} (N - k - (k + 1) (s - 1)) + (r - 1) (s - 1) . \end{matrix}

Bernevig and Haldane investigated three kinds of clustering properties that occur when

k + 1

and

s - 1

are coprime:

First clustering property: $s - 1$ clusters of $k + 1$ particles and one cluster of k particles, with the remaining particles free.
This situation is formalized by setting $Z_{1} = z_{1} = \dots = z_{k + 1}$ , $Z_{2} = z_{k + 1} = \dots = z_{2 (k + 1)}$ ,⋯, $Z_{s - 1} = z_{(s - 2) (k + 1) + 1} = \dots = z_{(s - 1) (k + 1)}$ , and $Z_{F} = z_{(s - 1) (k + 1)} = \dots = z_{s (k + 1) - 1}$ . Then, the Jack polynomial $J_{λ_{k, r, s}^{β}}^{- \frac{k + 1}{r - 1}} ((k + 1) (Z_{1} + \dots + Z_{s - 1}) + k Z_{F} + z_{s (k + 1)} + \dots + z_{N})$ behaves as $\prod_{i = s (k + 1)}^{N} {(Z_{F} - z_{i})}^{r}$ when each $z_{i}$ , with $i = s (k + 1), \dots, N$ , tends to $Z_{F}$ . For instance, we have

$\begin{matrix} J_{53}^{(- 2)} (2 Z_{1} + Z_{F} + z_{3} + z_{4}) = {(Z_{F} - z_{4})}^{2} {(Z_{F} - z_{3})}^{2} P (Z_{1}, Z_{F}, z_{3}, z_{4}), with \\ P (Z_{1}, Z_{F}, z_{3}, z_{4}) = 144 {(z_{3} - z_{4})}^{2} (z_{3} z_{4} + Z_{F} z_{4} + Z_{F} z_{3} - 2 Z_{1} z_{4} - 2 Z_{1} z_{3} - 2 Z_{1} Z_{F} + 3 {Z_{1}}^{2}) . \end{matrix}$
Second clustering property: A cluster of $n_{0} = (k + 1) s - 1$ particles.
To formalize this situation, we set $z_{1} = \dots = z_{(k + 1) s - 1} = Z$ . Then, the Jack polynomial $J_{λ_{k, r, s}}^{(- \frac{k + 1}{r - 1})} (n_{0} Z + z_{n_{0} + 1} + \dots + z_{N})$ behaves as $\prod_{i = s (k + 1)}^{N} {(Z - z_{i})}^{(r - 1) s + 1}$ , when each $z_{i}$ tends to Z. More specifically, for HW Jack polynomials,

$\begin{matrix} J_{λ_{k, r, s}^{β}}^{(- \frac{k + 1}{r - 1})} (n_{0} Z + z_{n_{0} + 1} + \dots + z_{N}) \overset{(*)}{=} \prod_{i = s (k + 1)}^{N} {(Z - z_{i})}^{(r - 1) s + 1} J_{λ_{k, r, 1}^{β - 1}}^{(- \frac{k + 1}{r - 1})} (z_{n_{0} + 1} + \dots + z_{N}) . \end{matrix}$

(1)

For instance,

$\begin{matrix} J_{53}^{(- 2)} (3 Z + z_{4} + z_{5}) = - 144 {(Z - z_{4})}^{3} {(Z - z_{5})}^{3} J_{2}^{(- 2)} (z_{4} + z_{5}) . \end{matrix}$
Third clustering property: $s - 1$ clusters of $2 k + 1$ particles.
By setting $s - 1$ sets of variables as follows: $Z_{1} = z_{1} = \dots = z_{2 k + 1}$ ,…, $Z_{s - 1} = z_{(s - 2) (2 k + 1) + 1} = \dots = z_{(s - 1) (2 k + 1)}$ , the HW Jack $J_{λ_{k, r, s}^{β}}$ satisfies

$\begin{matrix} J_{λ_{k, r, s}^{β}}^{(- \frac{k + 1}{r - 1})} ((2 k + 1) (Z_{1} + \dots + Z_{s - 1}) + z_{(s - 1) (2 k + 1) + 1} + \dots + z_{N}) \overset{(*)}{=} \\ \prod_{1 \leq i < j \leq s - 1} {(Z_{i} - Z_{j})}^{k (3 r - 2)} \prod_{i = 1}^{s - 1} \prod_{ℓ = (s - 1) (2 k + 1) + 1}^{N} {(Z_{i} - z_{ℓ})}^{2 r - 1} \cdot J_{λ_{k, r}^{β - s + 1}} (z_{(s - 1) (2 k + 1) + 1} + \dots + z_{N}) . \end{matrix}$

For instance,

$\begin{matrix} J_{64}^{(- 2)} (3 (Z_{1} + Z_{2}) + z_{7}) = - 3456 {(Z_{1} - Z_{2})}^{4} {(Z_{1} - z_{7})}^{3} {(Z_{2} - z_{7})}^{3} . \end{matrix}$

The aim of our paper is to show how the material described in [9] can help in this context. In particular, we focus on the second clustering property for HW polynomials. Although the wave functions are not all Jack polynomials, many of them can be obtained from Jack polynomials by acting by an operator modeling the adding of a quasi-particle or a quasi-hole (see, e.g., [3]).

3.4. The Interest of Shifted Macdonald Polynomials

In [1], Bernevig and Haldane looked at the case

s = 1

and proved the clustering properties on HW Jack polynomials using a result of B. Feigin et al. [41] together with Lassalle binomial formulas for Jack polynomials [42]. 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. [41], 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. One reason is because clustering properties deal with vanishing properties. Therefore, shifted Jack polynomials should be more appropriate for these problems. Nevertheless, the multiplicities of the roots of the polynomials are difficult to manage. Therefore, 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_{i} - z_{j})}^{n}

should become

(z_{i} - z_{j}) (z_{i} - q z_{j}) \dots (z_{i} - q^{n - 1} z_{j})

. With such a deformation, it is also easier to manipulate the eigenspaces which are smaller (see [7] for the example related to

ϕ_{L a u g h l i n}

). Consequently, we follow the strategy initiated in [9], which consists to manipulate shifted Macdonald polynomials in the aim to prove the identities. The recipe, which could be used for other problems, is as follows:

Step 1:: Find a Macdonald version of the conjecture and we state it in terms of vanishing properties.
Step 2:: Prove that the Macdonald polynomial involved is a HW polynomial (i.e., in the kernel of a q-deformation of $L^{+}$ ).
This property might come from [10] (Macdonald version of [41]), or by applying the results of [11], which are based on the Lassalle binomial formula for Macdonald polynomials [42].
Step 3:: In the last case of Step 2, the shifted Macdonald polynomial equals the homogeneous Macdonald polynomials.
Step 4:: Consequently, we deduce the equality from vanishing properties of the shifted Macdonald polynomial and we recover the identity on Jack by setting $t = q^{α}$ and q to 1.

Notice that, in [11], one of the authors with Thierry Jolicoeur found some families of polynomials which have not been considered in [2]. Indeed, Bernevig and Haldane missed that the family

λ_{k, r, s}^{β}

can be extended by adding a parameter corresponding to the multiplicity of the largest part which can be smaller than k. In addition, some other Macdonald polynomials do not specialize to a Jack for the considered specialization of

(q, t)

. This is explained in the next sections.

4. Factorizations for Generic $(q, t)$ Parameters

In this section, we study several factorizations of Macdonald polynomials for different partitions and specializations of the variables, keeping q and t as generic parameters.

4.1. Saturated Partitions

We start by looking at the partitions ending with zeros. From the partitions perspective, adding zeros at the end of a partition does not make a big difference. From the Yang–Baxter graph, the starting vector is longer and the computation might require extra steps permuting entries. Let us see how this relates to the Macdonald polynomials by starting with the definition.

Definition 3.

Fix an integer

N > 0

. We say that a partition

λ = [λ_{1}, \dots, λ_{N}]

is saturated if

λ_{N} > 0

The following result shows a general factorization for the Macdonald polynomials indexed by saturated partitions.

Proposition 2.

λ = [λ_{1}, \dots, λ_{N}]

is saturated, then:

$P_{λ} (x_{1}, \dots, x_{N}; q, t) = {(x_{1} \dots x_{N})}^{λ_{N}} P_{[λ_{1} - λ_{N}, \dots, λ_{N - 1} - λ_{N}, 0]} (q^{- λ_{N}} x_{1}, \dots, q^{- λ_{N}} x_{N}; q, t)$ .
$M S_{λ} (x_{1}, \dots, x_{N}; q, t) \overset{(*)}{=} \prod_{k = 0}^{λ_{N} - 1} \prod_{i = 1}^{N} (x_{i} - q^{k}) M S_{[λ_{1} - λ_{N}, \dots, λ_{N - 1} - λ_{N}, 0]} (q^{- λ_{N}} x_{1}, \dots, q^{- λ_{N}} x_{N}; q, t)$ .

Proof.

We prove both statements at the same time. Recall the affine step for nonsymmetric Macdonald polynomials (respectively, Shifted Macdonald polynomials) from Proposition 1

\begin{matrix} E_{v Φ} = E_{v} τ x_{N} and M_{v Φ} = M_{v} τ (x_{N} - 1) . \end{matrix}

If we apply this step N times, we obtain

\begin{matrix} E_{[v [1] + 1, v [2] + 1, \dots, v [N] + 1]} & = & \prod_{i = 1}^{N} x_{i} \times E_{v} (\frac{x_{1}}{q}, \frac{x_{2}}{q}, \dots \frac{x_{N}}{q}); and, \\ M_{[v [1] + 1, v [2] + 1, \dots, v [N] + 1]} & = & \prod_{i = 1}^{N} (x_{i} - 1) \times M_{v} (\frac{x_{1}}{q}, \frac{x_{2}}{q}, \dots \frac{x_{N}}{q}) . \end{matrix}

Hence, if we apply the affine step N times again, we obtain

\begin{matrix} E_{[v [1] + 2, v [2] + 2, \dots, v [N] + 2]} & = & \prod_{i = 1}^{N} \frac{x_{i}^{2}}{q} \times E_{v} (\frac{x_{1}}{q^{2}}, \frac{x_{2}}{q^{2}}, \dots \frac{x_{N}}{q^{2}}); and, \\ M_{[v [1] + 2, v [2] + 2, \dots, v [N] + 2]} & = & \prod_{i = 1}^{N} (x_{i} - 1) (\frac{x_{i}}{q} - 1) \times M_{v} (\frac{x_{1}}{q^{2}}, \frac{x_{2}}{q^{2}}, \dots \frac{x_{N}}{q^{2}}) . \end{matrix}

By induction, starting with

v = [λ_{1} - λ_{N}, λ_{2} - λ_{N}, \dots, λ_{N - 1} - λ_{N}, 0]

and applying the affine step

N λ_{N}

times, one finds

\begin{matrix} E_{λ} & \overset{(*)}{=} & \prod_{i = 1}^{N} x_{i}^{λ_{N}} \times E_{v} (\frac{x_{1}}{q^{λ_{N}}}, \frac{x_{2}}{q^{λ_{N}}}, \dots \frac{x_{N}}{q^{λ_{N}}}); and, \\ M_{λ} & = & \prod_{i = 1}^{N} \prod_{k = 0}^{λ_{N} - 1} (\frac{x_{i}^{λ_{N}}}{q^{k}} - 1) \times M_{v} (\frac{x_{1}}{q^{λ_{N}}}, \frac{x_{2}}{q^{λ_{N}}}, \dots \frac{x_{N}}{q^{λ_{N}}}) . \end{matrix}

Since the polynomials

\prod_{i = 1}^{N} x_{i}^{λ_{N}}

and

\prod_{i = 1}^{N} \prod_{k = 0}^{λ_{N} - 1} (\frac{x_{i}^{λ_{N}}}{q^{k}} - 1)

are symmetric, they commute with the action of the symmetrizing operator

S

and the result is obtained by applying

S

E_{λ}

and

M_{λ}

. □

The first result is classical. Indeed, following the Yang–Baxter graph and applying the affine step several times, we have that for any vector v with

m = \min_{i} {v [i]} > 0

\begin{matrix} E_{v} (X_{N}) = E_{[v_{1} - m, \dots, v_{N} - m]} (X_{N}) {(τ x_{N})}^{N λ_{N}} = E_{[v_{1} - m, \dots, v_{N} - m]} (\frac{x_{N}}{q}, x_{1}, \dots, x_{N - 1}) x_{N} {(τ x_{N})}^{N λ_{N} - 1} \\ = E_{[v_{1} - m, \dots, v_{N} - m]} (\frac{x_{N - 1}}{q}, \frac{x_{N}}{q}, x_{1}, \dots, x_{N - 2}) x_{N - 1} x_{N} {(τ x_{N})}^{N λ_{N} - 2} = \dots = \\ \overset{(*)}{=} E_{[v_{1} - m, \dots, v_{N} - m]} (\frac{x_{1}}{q^{λ_{N}}}, \frac{x_{2}}{q^{λ_{N}}}, \dots \frac{x_{N}}{q^{λ_{N}}}) {(x_{1} \dots x_{N})}^{λ_{N}} . \end{matrix}

Moreover,

P_{λ}

is obtained from

E_{λ^{-}}

, where

λ^{-} = [λ_{N}, \dots, λ_{1}]

, by applying the symmetrization operator

S_{N}

. Since

{(x_{1} \dots x_{N})}^{λ_{N}}

is symmetric, it commutes with

S_{N}

. Therefore,

P_{λ}

is proportional to

P_{[λ_{1} - λ_{N}, \dots, λ_{N - 1} - λ_{N}, 0]} (X_{N}) \cdot {(x_{1} \dots x_{N})}^{λ_{N}}

and we deduce our result from the fact that the dominant coefficient of

P_{λ}

is 1. Furthermore, it is the simultaneous eigenfunction for the Cherednik operators

ξ_{i}

, whose spectral vector is

[t^{N - 1} q^{k}, \dots, q^{k}]

, and it is also an eigenfunction for the operator

\sum_{i} ξ_{i}

with eigenvalues

t^{N - 1} q^{k} + \dots + q^{k}

For the second result, there is an alternative proof based on the examination of the vanishing properties of

\begin{matrix} \prod_{k = 0}^{λ_{N} - 1} \prod_{i = 1}^{N} (x_{i} - q^{k}) M S_{[λ_{1} - λ_{N}, \dots, λ_{N - 1} - λ_{N}, 0]} (q^{- λ_{N}} X_{N}; q, t) . \end{matrix}

(2)

Let

μ \neq λ

, with

| μ | \leq | λ |

. If

[λ_{1} - λ_{N}, \dots, λ_{N - 1} - λ_{N}] \subset μ

, by the vanishing properties of

M S_{[λ_{1} - λ_{N}, \dots, λ_{N - 1} - λ_{N}, 0]}

\begin{matrix} M S_{[λ_{1} - λ_{N}, \dots, λ_{N - 1} - λ_{N}, 0]} (q^{- λ_{N}} q^{μ_{1}} t^{N - 1} + \dots + q^{- λ_{N}} q^{μ_{N - 1}}; q, t) \\ = M S_{[λ_{1} - λ_{N}, \dots, λ_{N - 1} - λ_{N}, 0]} (〈 μ_{1} - λ_{N}, \dots, μ_{N} - λ_{N} 〉) = 0 . \end{matrix}

[λ_{1} - λ_{N}, \dots, λ_{N - 1} - λ_{N}] ⊄ μ

, then this means that

μ_{N} < λ_{N}

. Then, the factor

(x_{N} - q^{μ_{N}})

in (2) vanishes for

x_{N} = q^{μ_{N}}

. This proves that the two polynomials have the same vanishing properties and so, that they are equal.

4.2. Standard Specializations for the Variables

Apart from considering partitions ending with at least one zero, we consider now a particular specialization for the variables.

Definition 4.

For a partition

λ = [λ_{1}, \dots, λ_{N - k}, 0^{k}]

with

k \geq 1

, the λ-standard specialization consists in setting

x_{N - j} = t^{j}

, for

j = 0, \dots, k - 1

The

λ

-standard specialization provides another interesting factorization.

Proposition 3.

\begin{matrix} M S_{[λ_{1}, \dots, λ_{N - k}, 0^{k}]} (X_{N - k} + \frac{1 - t^{k}}{1 - t}; q, t) \overset{(*)}{=} M S_{[λ_{1}, \dots, λ_{N - k}]} (t^{- k} X_{N - k}; q, t) \overset{(*)}{=} \\ \prod_{j = 0}^{λ_{N - k} - 1} \prod_{i = 1}^{N - k} (x_{i} - t^{k} q^{j}) M S_{[λ_{1} - λ_{N - k}, \dots, λ_{N - k - 1} - λ_{N - k}, 0]} (q^{- λ_{N - k}} t^{- k} X_{N - k}; q, t) . \end{matrix}

Proof.

Consider the polynomial

\begin{matrix} P (x_{1}, \dots, x_{N - k}) = M S_{[λ_{1}, \dots, λ_{N - k}, 0^{k}]} (X_{N - k} + \frac{1 - t^{k}}{1 - t}; q, t) . \end{matrix}

This polynomial vanishes for

[x_{1}, \dots, x_{N - k}] = [q^{μ_{1}} t^{N - k - 1}, \dots, q^{μ_{N - k}} t^{0}]

, for any

μ \neq λ

. Moreover,

| μ | \geq | λ |

, since

〈 λ_{1}, \dots, λ_{N - k}, 0^{k} 〉 = [q^{λ_{1}} t^{N - 1}, \dots, q^{λ_{N - k}} t^{N - k}, t^{k - 1}, \dots, 1]

. Therefore,

P (x_{1}, \dots, x_{N - k})

has the same vanishing properties as

M S_{[λ_{1}, \dots, λ_{N - k}]} (t^{- k} X_{N - k}; q, t)

. This proves the first equality and, as a direct consequence of Proposition 2, we get the second equality. □

Example 1.

We illustrate the principle of the proof with

λ = [32000]

. By its vanishing properties,

M S_{[32000]} (x_{1}, x_{2}, t^{2}, t, 1; q, t)

vanishes for the following values of

(x_{1}, x_{2})

(q^{5} t^{4}, t^{3})

(q^{4} t^{4}, q t^{3})

(q^{4} t^{4}, t^{3})

(q^{3} t^{4}, q t^{3})

(q^{2} t^{4}, q^{2} t^{3})

(q^{3} t^{4}, t^{3})

(q^{2} t^{4}, q t^{3})

(q^{2} t^{4}, t^{3})

(q t^{4}, q t^{3})

(q t^{4}, t^{3})

and

(t^{4}, t^{3})

Since

M S_{[32000]} (x_{1}, x_{2}, t^{2}, t, 1; q, t)

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}; q, t)

. The polynomial

M S_{[32000]} (x_{1}, x_{2}, t^{2}, t, 1; q, t)

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.

The following table compares the vanishing properties characterizing

M S_{[32]}

by listing the partitions μ of length

ℓ (μ) = 2

with

| μ | \leq 5

, together with their reciprocal vector:

$〈 50 〉$	$〈 41 〉$	$〈 32 〉$
$[q^{5} t, 1]$	$[q^{4} t, q]$	×
$〈 40 〉$	$〈 31 〉$	$〈 22 〉$
$[q^{4} t, 1]$	$[q^{3} t, q]$	$[q^{2} t, q^{2}]$
$〈 30 〉$	$〈 21 〉$
$[q^{3} t, 1]$	$[q^{2} t, q]$
$〈 20 〉$	$〈 11 〉$
$[q^{2} t, 1]$	$[q t, q]$
$〈 10 〉$
$[q t, 1]$
$〈 00 〉$
$[t, 1]$

We deduce that the polynomials

M S_{[32000]} (X_{2} + t^{2} + t + 1; q, t)

and

M S_{[32]} (t^{- 3} X_{2}; q, t)

are proportional. Since these two polynomials have low degree, they are easy to compute with the help of the Yang–Baxter graph:

\begin{matrix} M S_{[32]} (X_{2}; q, t) & \overset{(*)}{=} & (q^{2} t + q^{2} - x_{1} - x_{2}) (x_{2} - 1) (x_{1} - 1) (- x_{2} + q) (- x_{1} + q), a n d \\ M S_{[32000]} (X_{2} + t^{2} + t + 1; q, t) & \overset{(*)}{=} & (t^{3} - x_{2}) (t^{3} - x_{1}) (q t^{3} - x_{2}) (q t^{3} - x_{1}) (q^{2} t^{3} + q^{2} t^{4} - x_{1} - x_{2}) . \end{matrix}

Corollary 1.

For a partition λ,

\begin{matrix} M S_{λ} (X_{m_{λ}} + ⟅ λ_{m_{λ} + 1}, \dots, λ_{N - 1}, λ_{N} ⟆; q, t) \overset{(*)}{=} \prod_{j = 0}^{\max λ - 1} \prod_{i = 1}^{m_{λ}} (x_{i} - q^{j} t^{{⌊ λ ⌋}_{j}}) . \end{matrix}

Proof.

We prove the property by induction on

N + | λ |

. First, we consider the case

λ_{N} > 0

. By Proposition 2, we have

\begin{matrix} M S_{λ} (X_{m_{λ}} + ⟅ λ_{m_{λ} + 1}, \dots, λ_{N - 1}, λ_{N} ⟆; q, t) \overset{(*)}{=} \prod_{j = 0}^{λ_{N} - 1} \prod_{i = 1}^{m_{λ}} (x_{i} - q^{j}) \times \\ M S_{[λ_{1} - λ_{N}, \dots, λ_{N - 1} - λ_{N}, 0]} (q^{- λ_{N}} (X_{m_{λ}} + ⟅ [λ_{m_{λ} + 1}, \dots, λ_{N - 1}, λ_{N}] ⟆); q, t) . \end{matrix}

Setting

\tilde{λ} = [λ_{1} - λ_{N}, \dots, λ_{N - 1} - λ_{N}, 0]

, we have

⟅ \tilde{λ} ⟆ = q^{- λ_{N}} ⟅ λ_{m_{λ} + 1}, \dots, λ_{N - 1}, λ_{N} ⟆

and, by induction:

\begin{matrix} M S_{\tilde{λ}} (q^{- λ_{N}} X_{m_{λ}} + ⟅ \tilde{λ} ⟆; q, t) \overset{(*)}{=} \prod_{j = 0}^{\max \tilde{λ} - 1} \prod_{i = 1}^{m_{\tilde{λ}}} (x_{i} - q^{j + λ_{N}} t^{{⌊ \tilde{λ} ⌋}_{j}}) . \end{matrix}

(3)

Using that

{⌊ \tilde{λ} ⌋}_{j} = {⌊ λ ⌋}_{j + λ_{N}}

\begin{matrix} \prod_{j = 0}^{λ_{N} - 1} \prod_{i = 1}^{m_{λ}} (x_{i} - q^{j}) \prod_{j = 0}^{\max \tilde{λ} - 1} \prod_{i = 1}^{m_{\tilde{λ}}} (x_{i} - q^{j + λ_{N}} t^{{⌊ \tilde{λ} ⌋}_{j}}) \\ = \prod_{j = 0}^{λ_{N} - 1} \prod_{i = 1}^{m_{λ}} (x_{i} - q^{j}) \prod_{j = λ_{N}}^{\max λ - 1} \prod_{i = 1}^{m_{λ}} (x_{i} - q^{j} t^{{⌊ λ ⌋}_{j}}) = \prod_{j = 0}^{\max λ - 1} \prod_{i = 1}^{m_{λ}} (x_{i} - q^{j} t^{{⌊ λ ⌋}_{j}}), \end{matrix}

as expected.

λ_{N} = 0

, then we set

λ = [λ_{1}, \dots, λ_{N - k}, 0^{k}]

, with

λ_{N - k} \neq 0

, and, by Proposition 3, we obtain

\begin{matrix} M S_{[λ_{1}, \dots, λ_{N - k}, 0^{k}]} (X_{N - k} + \frac{1 - t^{k}}{1 - t}; q, t) \overset{(*)}{=} M S_{[λ_{1}, \dots, λ_{N - k}]} (t^{- k} X_{N - k}; q, t) . \end{matrix}

We conclude the proof by induction and applying (3) since

λ_{N - k} \neq 0

. □

We conclude this section with an example.

Example 2.

Consider

λ = [664331110]

; we alternatively use Propositions 2 and 3 for computing

M S_{664331110} (X_{2} + ⟅ 4331110 ⟆; q, t)

. In each step, the expression we obtain is the product of a polynomial in the

X

alphabet with coefficients in

C (q, t)

and a symmetric shifted Macdonald polynomial indexed by a smaller partition. The process finishes when

M S_{664331110} (X_{2} + ⟅ 4331110 ⟆; q, t)

is described as a polynomial in the

X

alphabet with coefficients in

C (q, t)

, up to some coefficient of the form

q^{a} t^{b}

\begin{matrix} \begin{matrix} M S_{664331110} (X_{2} + ⟅ 4331110 ⟆; q, t) \\ ↕ P r o p o s i t i o n 3 \\ M S_{66433111} (t^{- 1} X_{2} + ⟅ 433111 ⟆; q, t) \\ ↕ P r o p o s i t i o n 2 \\ (t^{- 1} x_{1} - 1) (t^{- 1} x_{2} - 1) M S_{55322000} (t^{- 1} q^{- 1} X_{2} + ⟅ 322000 ⟆; q, t) \\ ↕ P r o p o s i t i o n 3 \\ (t^{- 1} x_{1} - 1) (t^{- 1} x_{2} - 1) M S_{55322} (t^{- 4} q^{- 1} X_{2} + ⟅ 322 ⟆; q, t) \\ ↕ P r o p o s i t i o n 2 \\ \prod_{i = 1}^{2} (t^{- 1} x_{i} - 1) (t^{- 4} q^{- 1} x_{i} - 1) (t^{- 4} q^{- 1} x_{i} - q) M S_{33100} (t^{- 4} q^{- 3} X_{2} + ⟅ 100 ⟆; q, t) \\ ↕ P r o p o s i t i o n 3 \end{matrix} \end{matrix}

\begin{matrix} \begin{matrix} ↕ P r o p o s i t i o n 3 \\ \prod_{i = 1}^{2} (t^{- 1} x_{i} - 1) (t^{- 4} q^{- 1} x_{i} - 1) (t^{- 4} q^{- 1} x_{i} - q) M S_{331} (t^{- 6} q^{- 3} X_{2} + ⟅ 1 ⟆; q, t) \\ ↕ P r o p o s i t i o n 2 \\ \prod_{i = 1}^{2} (t^{- 1} x_{i} - 1) (t^{- 4} q^{- 1} x_{i} - 1) (t^{- 4} q^{- 1} x_{i} - q) (t^{- 6} q^{- 3} x_{i} - 1) M S_{220} (t^{- 6} q^{- 4} X_{2} + ⟅ 0 ⟆; q, t) \\ ↕ P r o p o s i t i o n 3 \\ \prod_{i = 1}^{2} (t^{- 1} x_{i} - 1) (t^{- 4} q^{- 1} x_{i} - 1) (t^{- 4} q^{- 1} x_{i} - q) (t^{- 6} q^{- 3} x_{i} - 1) M S_{22} (t^{- 7} q^{- 4} X_{2}; q, t) \\ ↕ P r o p o s i t i o n 2 \\ \prod_{i = 1}^{2} (t^{- 1} x_{i} - 1) (t^{- 4} q^{- 1} x_{i} - 1) (t^{- 4} q^{- 1} x_{i} - q) (t^{- 6} q^{- 3} x_{i} - 1) (t^{- 7} q^{- 4} x_{i} - 1) (t^{- 7} q^{- 4} x_{i} - q) . \end{matrix} \end{matrix}

As expected, the last polynomial is proportional to

\prod_{i = 1}^{2} (x_{i} - t) (x_{i} - q t^{4}) (x_{i} - q^{2} t^{4}) (x_{i} - q^{3} t^{6})

(x_{i} - q^{4} t^{7}) (x_{i} - q^{5} t^{7})

5. Specializations of the Type $t^{a} q^{b} = 1$ and Quasistaircase Partitions

In this section, we focus in the quasistaircase partitions for a concrete specialization of the parameters q and t.

Definition 5.

Consider the integers

1 \leq k \leq ℓ

s \geq 2

, and

β, r \geq 0

, such that

r \frac{ℓ + 1}{s - 1}

is an integer. We define the quasistaircase partition as the following partition

\begin{matrix} Q S (ℓ, k; s, r; β) : = [{((β + 1) s + r)}^{k}, {(β s + r)}^{ℓ}, \dots, {(s + r)}^{ℓ}, 0^{r \frac{ℓ + 1}{s - 1} + ℓ}] . \end{matrix}

5.1. Admissible Specializations

We start defining the specialization that we consider as in [10].

Definition 6.

(s, ℓ)

-admissible specialization is one of the form

(t, q) = (u^{\frac{s - 1}{g}}, ω_{1} u^{- \frac{ℓ + 1}{g}})

where

g = \gcd (ℓ + 1, s - 1)

and

ω_{1}^{\frac{s - 1}{g}}

is a primitive gth root of the unity.

From now on in this section, and unless specifying otherwise, these parameters defining the quasistaircase partition satisfy the conditions stated below and the parameters q and t are specialized.

Example 3.

Let us illustrate the notion of

(s, ℓ)

-admissible specialization by giving a few examples and counter-examples.

$(t, q) = (u, u^{- 3})$ is $(2, 2)$ -admissible.
$(t, q) = (u^{2}, u^{- 5})$ and $(t, q) = (u^{2}, - u^{- 5})$ are $(3, 4)$ -admissible.
$(t, q) = (u, - u^{- 2})$ is $(3, 3)$ -admissible while $(t, q) = (u, u^{- 2})$ is not $(3, 3)$ -admissible.
$(t, q) = (u, e^{\frac{2 i π}{3}} u^{- 2})$ and $(t, q) = (u, e^{\frac{2 i π}{3}} u^{- 2})$ are $(4, 5)$ -admissible but $(t, q) = (u, u^{- 2})$ is not.
$(t, q) = (u, i u^{- 2})$ and $(t, q) = (u, - i u^{- 2})$ are $(5, 7)$ -admissible, while $(t, q) = (u, u^{- 2})$ and $(t, q) = (u, - u^{- 2})$ are not $(5, 7)$ -admissible.

We finish this subsection with the following result that highlights the relevance of

ω_{1}

Lemma 1.

Suppose

t^{α} q^{γ} = 1

t = u^{b}

, and

q = ω_{1} u^{- a}

, with

\gcd (a, b) = 1

. Then,

α = p (ℓ + 1)

and

γ = p (s - 1)

, for some

p \in N

Proof.

By hypothesis

ω_{1}^{γ} u^{α b - γ a} = 1

. From

\gcd (a, b) = 1

, it follows that

α = c a

and

γ = c b

, for some

c \in N

. Thus,

{(ω_{1}^{b})}^{c} = 1

and

c = p g

, for some

p \in N

, because

ω_{1}^{b}

is a primitive gth root of unity. Hence,

α = p g a = p (ℓ + 1)

and

γ = p g b = p (s - 1)

. □

5.2. On the Reciprocal Sum $⟅ Q S (ℓ, k; s, r; β) ⟆$

In this section, we prove that the intersection of the eigenspace of

ξ

, with eigenvalue

⟅ Q S (ℓ, k; s, r; β) ⟆_{q^{- 1}, t^{- 1}}

, and the space generated by

P_{μ}

, with

μ \subseteq Q S (ℓ, k; s, r; β)

, has dimension 1.

We start with the following technical lemma.

Lemma 2.

Let λ be the partition

λ = [{(β s + r)}^{k}, {((β - 1) s + r)}^{ℓ}, \dots, {(s + r)}^{ℓ}, 0^{ℓ + α}]

, where α is an integer such that

0 \leq α \leq r \frac{ℓ + 1}{s - 1}

. Consider another partition

μ \subseteq λ

such that there exists i satisfying

〈 μ 〉 [i] = 〈 λ 〉 [k]

. Then,

i = k

and λ and μ have the same k first entries, i.e.,

μ [j] = λ [j]

, for

j = 1, \dots, k

Proof.

First, we notice that

ω_{1}^{μ [i]} u^{\frac{1}{g} ((N - i) (s - 1) - μ [i] (ℓ + 1))} = ω_{1}^{β s + r} u^{\frac{1}{g} ((s - 1) (N - k) - (β s + r) (ℓ + 1)}

, which means that

(i, μ [i])

lies on the line

\begin{matrix} y = β s + r + \frac{s - 1}{ℓ + 1} (k - x), \end{matrix}

(4)

and that

k - i = (μ [i] - (β s + r))) \frac{ℓ + 1}{s - 1}

. Since

μ \subset λ

μ [i] \leq β s + r

and, so,

k \leq i

λ [i] > 0

, then

i \leq β ℓ + k

and the point

(i, λ [i])

lies on or below the line

\begin{matrix} y = β s + r + \frac{s}{ℓ} (k - x) . \end{matrix}

(5)

Hence,

\begin{matrix} 0 \geq μ [i] - λ [i] \geq \frac{s + ℓ}{ℓ (ℓ + 1)} (i - k), \end{matrix}

(6)

That is,

i \leq k

and, so,

i = k

Now, suppose that

i > k + (β - 1) ℓ

. In this case,

μ [i] = 0

and, from Equation (4), we obtain

\begin{matrix} i = k + (β s + r) \frac{ℓ + 1}{s - 1} = r \frac{ℓ + 1}{s - 1} + β + \frac{s}{s - 1} (ℓ + 1) + k > r \frac{ℓ + 1}{s - 1} + β ℓ + k . \end{matrix}

Then, i is strictly greater than the size of

μ

, which is impossible.

The only remaining possibility is

k = i

, and so

μ [k] = λ [k]

. Since

μ \subset λ

, we deduce also that

μ [1] = λ [1]

, …,

μ [k] = λ [k]

. □

Example 4.

In Figure 4, we illustrate the proof of Lemma 2 for the parameters

s = 4

ℓ = 4

k = 3

β = 5

and

r = 3

. The shaded area illustrates the inequality in Equation (6), which is defined by the lines in Equations (4) (top) and (5) (bottom). Below it, we include the partition λ.

The following result shows that the reciprocal sum characterizes the quasistaircase partitions.

Proposition 4.

Consider a partition μ such that

μ \subseteq Q S (ℓ, k; s, r; β)

and

⟅ μ ⟆ = ⟅ Q S (ℓ, k; s, r; β) ⟆

. Then,

μ = Q S (ℓ, k; s, r; β)

Proof.

Denote by

λ

the quasistaircase partition,

λ = Q S (ℓ, k; s, r; β)

. Then, there exists i such that

〈μ〉 [i] = 〈λ〉 [k]

. Otherwise,

μ

is completely contained in

λ

and

⟅ μ ⟆ = ⟅ Q S (ℓ, k; s, r; β) ⟆

is impossible.

By Lemma 2,

i = k

and

〈μ〉 [j] = 〈λ〉 [j]

, for

1 \leq j \leq k

. Now, define two new partitions

λ^{'}

and

μ^{'}

as the partitions obtained by deleting the first k entries of

λ

and

μ

, respectively. That is,

λ^{'} [j] = λ [j + k]

and

μ^{'} [j] = μ [j + k]

, for

j \geq 1

. Then, either

λ^{'} = Q S (l, l; s, r, β - 1)

, and the above argument is applied to

α^{'}

and

μ^{'}

, or else

λ^{'} = [0^{ℓ + α}]

, where

α

is defined in the statement of Lemma 2, and therefore

μ^{'} = [0^{ℓ + α}]

. The proof is complete by noticing that the process must stop because

λ

has only finitely many steps. □

Example 5.

The following table contains all the reciprocal sums ⟅μ⟆ associated to the partitions

μ \subseteq [420]

⟅420⟆	⟅320⟆	⟅220⟆	⟅110⟆
$q^{4} t^{2} + q^{2} t + 1$	$q^{3} t^{2} + q^{2} t + 1$	$q^{2} t^{2} + q^{2} t + 1$	$q t^{2} + q t + 1$
⟅410⟆	⟅310⟆	⟅210⟆	⟅100⟆
$q^{4} t^{3} + q t + 1$	$q^{3} t^{2} + q t + 1$	$q^{2} t^{2} + q t + 1$	$q t^{2} + t + 1$
⟅400⟆	⟅300⟆	⟅200⟆	⟅000⟆
$q^{4} t^{2} + t + 1$	$q^{3} t^{2} + t + 1$	$q^{2} t^{2} + t + 1$	$t^{2} + t + 1$

Under the

(2, 2)

-admissible specialization

(t, q) = (u, u^{- 2})

, the table becomes:

⟅420⟆	⟅320⟆	⟅220⟆	⟅110⟆
$u^{- 6} + u^{- 3} + 1$	$u^{- 4} + u^{- 3} + 1$	$u^{- 3} + u^{- 2} + 1$	$u^{- 1} + 2$
⟅410⟆	⟅310⟆	⟅210⟆	⟅100⟆
$u^{- 6} + u^{- 1} + 1$	$u^{- 4} + u^{- 1} + 1$	$u^{- 2} + u^{- 1} + 1$	$2 + u$
⟅400⟆	⟅300⟆	⟅200⟆	⟅000⟆
$u^{- 6} + 1 + u$	$u^{- 4} + 1 + u$	$u^{- 2} + 1 + u$	$1 + u + u^{2}$

We observe that the only partition whose reciprocal sum equals

u^{- 6} + u^{- 3} + 1

[420]

Notice that Proposition 4 can be alternatively stated as follows.

Corollary 2.

Suppose that

μ \subseteq λ = Q S (ℓ, k; s, r; β)

is a partition such that

〈 μ 〉

is a permutation of

〈 λ 〉

. Then,

$μ = λ$ ,
The intersection of the eigenspace of ξ with eigenvalue ${⟅ λ ⟆}_{q^{- 1}, t^{- 1}}$ , and the space generated by $P_{μ}$ , with $μ \subseteq λ$ , 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 $μ \subseteq λ$ , 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:

〈 μ 〉

is a permutation of

〈 λ 〉

{〈 μ 〉}_{q^{- 1}, t^{- 1}}

is a permutation of

{〈 λ 〉}_{q^{- 1}, t^{- 1}}

⟅ μ ⟆ = ⟅ λ ⟆

, and

{⟅ μ ⟆}_{q^{- 1}, t^{- 1}} = {⟅ λ ⟆}_{q^{- 1}, t^{- 1}}

. Proposition 4 allows us to complete the proof. □

5.3. On the Reciprocal Vector $〈 Q S (ℓ, k; s, r; β) 〉$

In this section, we prove that, under a

(s, ℓ)

-admissible specialization, all the entries of

〈 Q S (ℓ, k; s, r; β) 〉

are distinct. For simplicity, we denote

Q S (ℓ, k; s, r; β)

λ

and the specialization

(t, q) = (u^{\frac{s - 1}{g}}, ω_{1} u^{- \frac{ℓ + 1}{g}})

(t, q) = (u^{b}, ω_{1} u^{- a})

Define the utility function

h (w, z) = b z - a w

, so that, for a partition

μ

〈μ〉 [i] = ω_{1}^{μ [i]} u^{h (μ [i], N - i)}

. Define also a decreasing sequence as

j_{1} = N - n_{0}

, with

n_{0} = (ℓ + 1) \frac{r}{s - 1} + ℓ

;

j_{m} = k + (β - m + 1) ℓ

, for

2 \leq m \leq β + 1

; and

j_{β + 2} = 0

. Notice this sequence splits the interval

[0, N]

into

β + 2

pieces.

j_{m + 1} < i \leq j_{m}

, for

1 \leq m \leq β + 1

, then

λ [i] = r + m s

and

〈λ〉 [i] = ω_{1}^{r + m s} u^{h (r + m s, N - i)}

. If

j_{1} < i \leq N

λ [i] = 0

and

〈λ〉 [i] = u^{h (0, N - i)}

Let

c \in N

such that

n_{0} = (ℓ + 1) c - 1

and

r = (s - 1) (c - 1)

. Then,

\begin{matrix} \begin{matrix} - \frac{m (s + ℓ)}{g} = h (r + m s, N - j_{m}) \leq h (r + m s, N - i) \leq h (r + m s, N - j_{m} + ℓ - 1) \\ = \frac{1}{g} [- m (s + ℓ) + (s - 1) (ℓ - 1)] . \end{matrix} \end{matrix}

(7)

Here is an example showing that the utility function h alone does not suffice to separate the

〈λ〉

values.

Example 6.

Let

ℓ = 5, s = 3, N = 15, n_{0} = 5

. Then,

g = 2

r = 0

, and the relation

15 = k + 10 β

implies that

k = 5

and

β = 1

. This corresponds to the quasistaircase partition

λ = [6^{5}, 3^{5}, 0^{5}]

, for which the values of the sequence are

j_{1} = 10

j_{2} = 5

and

j_{3} = 0

. Then, the vector of values of

h (λ [i], N - i)

\begin{matrix} [- 4, - 5, - 6, - 7, - 8, 0, - 1, - 2, - 3, - 4, 4, 3, 2, 1, 0] . \end{matrix}

There are two pairs of equal entries, however the corresponding entries of

〈λ〉

are different. For the value 0,

〈λ〉 [6] = ω^{3} \neq 1 = 〈λ〉 [15]

, where

ω = - 1

. In addition, for the value

- 4

〈λ〉 [1] = ω^{6} u^{- 4} \neq ω^{3} u^{- 4} = 〈λ〉 [10]

Proposition 5.

The entries of

〈λ〉

are pairwise distinct.

Proof.

Take two indices

i_{1}

and

i_{2}

and assume without lost of generality that

1 \leq i_{1} < i_{2} \leq N

. We check that the values of

〈λ〉 [i_{1}]

and

〈λ〉 [i_{2}]

are distinct. For that, it is enough to check that

h (λ [i_{1}], N - i_{1}) \neq h (λ [i_{2}], N - i_{2})

h (λ [i_{1}], i_{1}) \neq h (λ [i_{2}], i_{2})

. Let us analyze the differences by looking at the indices in the division of the interval

[1, N]

given by the sequence

{j_{m}}

j_{m + 1} < i_{1} < i_{2} \leq j_{m}

, then

h (r + m s, N - i_{1}) - h (r + m s, N - i_{2}) = b (i_{2} - i_{1}) > 0

. Thus,

〈λ〉 [i_{1}] \neq 〈λ〉 [i_{2}]

. The same happens if

j_{1} < i_{1} < i_{2} \leq N

since

h (0, N - i_{1}) - h (0, N - i_{2}) = b (i_{2} - i_{1}) > 0

Now, suppose

j_{m_{1} + 1} < i_{1} \leq j_{m_{1}} \leq j_{m_{2} + 1} < i_{2} \leq j_{m_{2}}

, then (7) shows that

\begin{matrix} h (r + m_{2} s, i_{2}) - h (r + m_{1} s, i_{1}) \geq \frac{1}{g} \{(m_{1} - m_{2}) (s + ℓ) - (s - 1) (ℓ - 1)\} . \end{matrix}

Thus, if

m_{1} - m_{2} > \frac{(s - 1) (ℓ - 1)}{s + ℓ}

, then

〈λ〉 [i_{1}] \neq 〈λ〉 [i_{2}]

Consider the case

1 \leq m_{1} - m_{2} \leq \frac{(s - 1) (ℓ - 1)}{s + ℓ}

; then,

〈λ〉 [i_{1}] = ω_{1}^{r + m_{1} s} u^{h (r + m_{1} s, N - i_{1})}

and

〈λ〉 [i_{2}] = ω_{1}^{r + m_{2} s} u^{h (r + m_{2} s, N - i_{2})}

. There are two different arguments depending on whether

ω_{1} \neq 1

ω_{1} \neq 1

, suppose that

1 = \frac{〈λ〉 [i_{1}]}{〈λ〉 [i_{2}]} = q^{(m_{1} - m_{2}) s} t^{i_{2} - i_{1}}

. By Lemma 1,

(m_{1} - m_{2}) s = p (s - 1)

, for some

p \in N

, but

1 \leq m_{1} - m_{2} \leq \frac{(s - 1) (ℓ - 1)}{s + ℓ} < s - 1

and

\gcd (s, s - 1) = 1

, which is a contradiction.

ω_{1} = 1

, or equivalently

\gcd (s - 1, ℓ + 1) = 1

, then

h (λ [i], N - i) \equiv - (ℓ + 1) λ [i] \mod (s - 1)

. If

j_{m + 1} < i \leq j_{m}

, then

h (λ [i], N - i) \equiv m (s + ℓ) \mod (s - 1)

and

\gcd (s + ℓ, s - 1) = 1

. In the case

j_{m_{1} + 1} < i_{1} \leq j_{m_{1}} \leq j_{m_{2} + 1} < i_{2} \leq j_{m_{2}}

, with

1 \leq m_{1} - m_{2} \leq \frac{(s - 1) (ℓ - 1)}{s + ℓ}

, it follows that

\begin{matrix} h (r + m_{2} s, i_{2}) - h (r + m_{1} s, i_{1}) \equiv (m_{1} - m_{2}) (s + ℓ) \mod (s - 1) . \end{matrix}

This implies

h (r + m_{2} s, i_{2}) - h (r + m_{1} s, i_{1}) ≢ 0 \mod (s - 1)

, since

1 \leq m_{1} - m_{2} < s - 1

. The similar argument applies when

j_{1} < i_{2} \leq N

Suppose

j_{m + 1} = j_{m} < i_{1} \leq j_{m}

and

j_{1} < i_{2} \leq N

. By (7), and the fact that

h (0, N - i_{2}) \geq 0

\begin{matrix} h (0, N - i_{2}) - h (r + m_{1} s, N - i_{1}) \geq - h (r + m_{1} s, N - j_{m_{1}} + ℓ - 1) = \frac{1}{g} \{m_{1} (s + ℓ) - (s - 1) (ℓ - 1)\} . \end{matrix}

m_{1} > \frac{(s - 1) (ℓ - 1)}{s + ℓ}

, then

h (0, N - i_{2}) - h (r + m_{1} s, N - i_{1}) > 0

and

〈λ〉 [i_{1}] \neq 〈λ〉 [i_{2}]

. Otherwise, suppose

1 = \frac{〈λ〉 [i_{1}]}{〈λ〉 [i_{2}]} = q^{m_{1} s + r} t^{i_{2} - i_{1}}

and

m_{1} s + r = p (s - 1)

. Then,

m_{1} s = (s - 1) (p - c + 1)

, which is impossible for

1 \leq m_{1} < s - 1 .

□

6. Factorizations and Wheel Condition

In this section, we investigate the case of the staircase partitions that are a particular case of quasistaircase partitions of the form

\begin{matrix} S t (ℓ, k; s; β) : = Q S (ℓ, k; s, 0; β) = [{((β + 1) s)}^{k}, {(β s)}^{ℓ}, \dots, s^{ℓ}, 0^{ℓ}], \end{matrix}

for any

ℓ \geq 1

0 \leq k < ℓ

s \geq 2

and

β \geq 1

. We also assume that the parameters q and t specialize as

\begin{matrix} (t, q) = (u^{\frac{s - 1}{g}}, ω_{1} u^{- \frac{ℓ + 1}{g}}), \end{matrix}

where

g = \gcd (ℓ + 1, s - 1)

and

ω_{1}^{\frac{s - 1}{g}}

is a primitive gth root of the unity.

6.1. Wheel Condition and Admissible Partitions

We start recalling the main results of [10].

Definition 7.

A symmetric polynomial

p (x_{1}, \dots, x_{N})

satisfies the

(s, ℓ)

-wheel condition if

\begin{matrix} \{\frac{x_{2}}{x_{1}}, \dots, \frac{x_{ℓ + 1}}{x_{ℓ}}, \frac{x_{1}}{x_{ℓ + 1}}\} \subset {t, t q, \dots, t q^{s - 1}} \end{matrix}

implies that

p (x_{1}, \dots, x_{N}) = 0

. It is easy to check that the set of the symmetric polynomials satisfying the wheel condition is an ideal, denoted by

J_{N}^{ℓ, s}

in [10]. A

(ℓ, s, N)

-admissible partition is a partition

λ = [λ_{1}, \dots, λ_{N}]

satisfying

λ_{i} - λ_{i + ℓ} \geq s

for any

i = 1, \dots, N - ℓ

The following theorem summarizes two results that relates both concepts.

Theorem 1

([10]). The ideal

J_{N}^{ℓ, s}

is generated by the Macdonald polynomials indexed by admissible partitions,

\begin{matrix} J_{N}^{ℓ, s} = span \{P_{λ} (X_{N}; q, t) : λ i s (ℓ, s, N) - a d m i s s i b l e p a r t i t i o n\} . \end{matrix}

Moreover,

J_{N}^{ℓ, s}

is stable under the action of

L_{q, t}^{+} = (1 - q) (ξ - Ξ)

Remarking that

P_{S t (ℓ, k; s; β)} (X_{(β + 1) ℓ + k}; q, t)

is a minimal degree polynomial belonging to

J_{(β + 1) ℓ + k}^{ℓ, s}

, by applying Theorem 1 we get the following result.

Proposition 6.

\begin{matrix} P_{S t (ℓ, k; s; β)} (X_{(β + 1) ℓ + k}; q, t) \overset{(*)}{=} M S_{S t (ℓ, k; s; β)} (X_{(β + 1) ℓ + k}; q, t) . \end{matrix}

Proof.

Since

M S_{S t (ℓ, k; s; β)} (X_{(β + 1) ℓ + k}; q, t)

is in the kernel of

L_{q, t}^{+}

P_{S t (ℓ, k; s; β)} (X_{(β + 1) ℓ + k}; q, t)

and

M S_{S t (ℓ, k; s; β)} (X_{(β + 1) ℓ + k}; q, t)

are in the same eigenspace of

ξ

with reciprocal sum

\begin{matrix} ⟅ S t (ℓ, k; s; β) ⟆ = q^{(β + 1) s} \cdot (\sum_{i = 1}^{k} t^{N - i + 1}) + \sum_{j = 0}^{β} q^{j s} \cdot (\sum_{i = 1}^{ℓ} t^{N - k - (β - j) ℓ - i + 1}) . \end{matrix}

Recall that, for some coefficients

c_{μ} (q, t)

\begin{matrix} M S_{S t (ℓ, k; s; β)} \overset{(*)}{=} P_{S t (ℓ, k; s; β)} + \sum_{μ \subset S t (ℓ, k; s; β)} c_{μ} (q, t) \cdot P_{μ} . \end{matrix}

Consider the spaces generated by the polynomials

P_{μ}

, with

μ \subseteq S t (ℓ, k; s; β)

. This space splits into several eigenspaces, and each of them is associated with a reciprocal sum

⟅ μ ⟆

. From Corollary 2, the subspace of the eigenspace associated with the reciprocal sum

⟅ S t (ℓ, k; s; β) ⟆

generated by the polynomials

M S_{μ}

, with

μ \subseteq S t (ℓ, k; s; β)

, has dimension 1. Therefore,

P_{S t (ℓ, k; s; β)} (X_{(β + 1) ℓ + k}; q, t)

and

M S_{S t (ℓ, k; s; β)} (X_{(β + 1) ℓ + k}; q, t)

are proportional. □

Example 7.

For instance, observe that the shifted Macdonald polynomial

\begin{matrix} M S_{[20]} (x_{1}, x_{2}; q = \frac{1}{u}, t = u^{2}) = \frac{(x_{2} x_{1} u + x_{1}^{2} u + x_{2}^{2} u - \frac{x_{2} x_{1}}{u} + x_{1} u^{2} x_{2} - x_{1}^{2} - x_{2}^{2} - x_{2} x_{1})}{(1 - u^{- 2}) {(1 - u^{- 1})}^{2}} \end{matrix}

is homogeneous and equal to

P_{[20]} (x_{1}, x_{2}; q = \frac{1}{u}, u^{2})

up to a multiplicative factor.

6.2. Factorizations

From Proposition 6, the polynomial

M S_{S t (ℓ, k; s; β)}

is homogeneous for

(ℓ, s)

-admissible specializations. Moreover,

\begin{matrix} M S_{S t (ℓ, k; s; β)} (X_{k + β ℓ} + \frac{1 - t^{ℓ}}{1 - t} y; q, t) = y^{\frac{ℓ s β (β - 1)}{2} + k s (β + 1)} M S_{S t (ℓ, k; s; β)} (\frac{X_{k + β ℓ}}{y} + \frac{1 - t^{ℓ}}{1 - t}; q, t) . \end{matrix}

Applying Proposition 3,

\begin{matrix} M S_{S t (ℓ, k; s; β)} (X_{k + β ℓ} + \frac{1 - t^{ℓ}}{1 - t}; q, t) \overset{(*)}{=} R (X_{β ℓ + k}; t^{ℓ} \frac{1 - q^{s}}{1 - q}) M S_{S t (ℓ, k; s; β - 1)} (q^{- s} t^{- ℓ} X_{k + β ℓ}; q, t) . \end{matrix}

Again, Proposition 6 shows that the polynomial

M S_{S t (ℓ, k; s; β - 1)} (X_{k + β ℓ}; q, t)

is homogeneous.

Proposition 7.

\begin{matrix} M S_{S t (ℓ, k; s; β)} (X_{k + β ℓ} + \frac{1 - t^{ℓ}}{1 - t} y; q, t) \overset{(*)}{=} R (X_{k + β ℓ}; y t^{ℓ} \frac{1 - q^{s}}{1 - q}) M S_{S t (ℓ, k; s; β - 1)} (q^{- s} t^{- ℓ} X_{k + β ℓ}; q, t) . \end{matrix}

Example 8.

Consider the partition

[42200] = S t (2, 1; 2; 1)

and the specialization

(t, q) = (u, u^{- 3})

. Then,

\begin{matrix} \frac{M S_{[42200]} (x_{1}, x_{2}, x_{3}, y t, y)}{M S_{[200]} (q^{- 2} t^{- 2} x_{1}, q^{- 2} t^{- 2} x_{2}, q^{- 2} t^{- 2} x_{3})} \overset{(*)}{=} \prod_{i = 1}^{3} (x_{i} - u^{2} y) (x_{1} - u^{- 1} y) . \end{matrix}

Set

Y_{i, β} = y_{i} + y_{i + 1} + \dots + y_{β}

, for any

0 \leq i \leq β

, and consider the polynomial which is symmetric in both alphabets

X_{k}

and

Y_{0, β}

\begin{matrix} P_{ℓ, k; s; β} (X_{k}; Y_{0, β}) : = M S_{S t (ℓ, k; s; β)} (X_{k} + \frac{1 - t^{ℓ}}{1 - t} Y_{0, β}; q, t) . \end{matrix}

By Proposition 7,

\begin{matrix} P_{S t (ℓ, k; s; β)} (X_{k}; Y_{0, β}) \overset{(*)}{=} R (X_{k} + \frac{1 - t^{ℓ}}{1 - t} Y_{1, β}; t^{ℓ} \frac{1 - q^{s}}{1 - q} y_{0}) P_{S t (ℓ, k; s; β - 1)} (q^{- s} t^{ℓ} X_{k}; q^{- s} t^{ℓ} Y_{1, β}) . \end{matrix}

Iterating, one finds

\begin{matrix} P_{S t (ℓ, k; s; β)} (X_{k}; Y_{0, β}) & \overset{(*)}{=} & \prod_{α = 0}^{β - 1} R (X_{k} + \frac{1 - t^{ℓ}}{1 - t} Y_{α + 1, β}; t^{ℓ} \frac{1 - q^{s}}{1 - q} y_{α}) P_{S t (ℓ, k; s; 0)} (q^{- β s} t^{- β ℓ} X_{k}; y_{β} q^{- β s} t^{- β ℓ}), \\ with P_{S t (ℓ, k; s; 0)} (X_{k}; y_{β}) & = & M S_{S t (ℓ, k; s; 0)} (X_{k} + y_{β} \frac{1 - t^{ℓ}}{1 - t}; q, t) . \end{matrix}

Once again, applying Propositions 6 and 3, one gets the following result.

Theorem 2.

\begin{matrix} P_{S t (ℓ, k; s; β)} (X_{k}; Y_{0, β}) \overset{(*)}{=} \prod_{α = 0}^{β} R (X_{k} + \frac{1 - t^{ℓ}}{1 - t} Y_{α + 1, β}; t^{ℓ} \frac{1 - q^{s}}{1 - q} y_{α}) \\ = \prod_{α = 0}^{β} \prod_{j = 0}^{s - 1} [\prod_{i = 1}^{k} (x_{i} - t^{ℓ} q^{j} y_{α}) \prod_{i = α + 1}^{β} \prod_{a = 0}^{ℓ - 1} (t^{a} y_{i} - t^{ℓ} q^{j} y_{α})] . \end{matrix}

Example 9.

For

(t, q) = (u, u^{- 3})

, we have

\begin{matrix} M S_{[42200]} (x_{1}, t y_{1}, y_{1}, t y_{0}, y_{0}) \overset{(*)}{=} (x_{1} - u^{2} y_{0}) (x_{1} - u^{- 1} y_{0}) (x_{1} - u^{2} y_{1}) (x_{1} - u^{- 1} y_{1}) \\ \times (y_{1} - u^{2} y_{0}) (y_{1} - u^{- 1} y_{0}) (u y_{1} - u^{2} y_{0}) (u y_{1} - u^{- 1} y_{0}) . \end{matrix}

7. Beyond the Wheel Condition

First, we recall that, for some specializations of

(q, t)

, the quasistaircase polynomials satisfy the HW condition, [11]. Using an argument of dimension of eigenspaces, we show that the shifted Macdonald are homogeneous.

Theorem 3

([11]). For β, s, r, k,

ℓ \in N

, with

k \leq ℓ

, consider

λ = [{((β + 1) s + r)}^{k}, {(β s + r)}^{ℓ}, \dots, {(s + r)}^{ℓ}]

and the specialization

\begin{matrix} (t, q) = (u^{\frac{s - 1}{g}}, u^{- \frac{ℓ + 1}{g}} ω_{1}), \end{matrix}

where

g = \gcd (ℓ + 1, s - 1)

and

ω_{1}

is a rth root of the unity such that

ω_{1}^{\frac{s - 1}{g}}

is a gth primitive root of the unity. The polynomial

P_{λ} (x_{1} + \dots + x_{n}; q, t)

is in the kernel of

L_{q, t}^{+}

when

n = \frac{ℓ + 1}{s - 1} r + ℓ (β + 1) + k

is an integer.

The shifted symmetric polynomial

M S_{Q S (ℓ, k; s, r; β)}

is an eigenfunction of

Ξ

having the same eigenvalue as

P_{Q S (ℓ, k; s, r; β)}

. However, from Corollary 2, the corresponding eigenspace has dimension 1, which proves the following result.

Corollary 3.

P_{Q S (ℓ, k; s, r; β)} (X; q, t) \overset{(*)}{=} M S_{Q S (ℓ, k; s, r; β)} (X; q, t) .

Moreover, this implies that

M S_{Q S (ℓ, k; s, r; β)} (X; q, t)

is homogeneous.

By Corollary 3, we have

\begin{matrix} M S_{Q S (ℓ, k; s, r; β)} (X_{k + β ℓ} + \frac{1 - t^{r \frac{ℓ + 1}{s - 1} + ℓ}}{1 - t} y_{β}; q, t) \\ = y_{β}^{\frac{1}{2} (β + 1) (ℓ β + 2 k) s + r (k + l β)} M S_{Q S (ℓ, k; s, r; β)} (\frac{X_{k + β ℓ}}{y_{β}} + \frac{1 - t^{r \frac{ℓ + 1}{s - 1} + ℓ}}{1 - t}; q, t) . \end{matrix}

Applying Proposition 3, we obtain

\begin{matrix} M S_{Q S (ℓ, k; s, r; β)} (X_{k + β ℓ} + \frac{1 - t^{r \frac{ℓ + 1}{s - 1} + ℓ}}{1 - t}; q, t) \overset{(*)}{=} \\ R (X_{k + β ℓ}; t^{\frac{ℓ + 1}{s - 1} r + ℓ} \frac{1 - q^{s + r}}{1 - q}) M S_{Q S (ℓ, k; s, 0; β)} (q^{- r - s} t^{- r \frac{ℓ + 1}{s - 1} - ℓ} X_{k + β ℓ}; q, t) . \end{matrix}

Again, by Corollary 3,

M S_{Q S (ℓ, k; s, 0; β)}

is an homogeneous polynomial, and then,

\begin{matrix} \begin{matrix} M S_{Q S (ℓ, k; s, r; β)} (X_{k + β ℓ} + \frac{1 - t^{r \frac{ℓ + 1}{s - 1} + ℓ}}{1 - t} y_{β}; q, t) \overset{(*)}{=} \\ R (X_{k + β ℓ}; t^{\frac{ℓ + 1}{s - 1} r + ℓ} \frac{1 - q^{s + r}}{1 - q} y_{β}) M S_{Q S (ℓ, k; s, 0; β - 1)} (q^{- r - s} t^{- r \frac{ℓ + 1}{s - 1} - ℓ} X_{k + β ℓ}; q, t) . \end{matrix} \end{matrix}

(8)

Theorem 2 completes the proof of the following theorem.

Theorem 4.

\begin{matrix} M S_{Q S (ℓ, k; s, r; β)} (X_{k} + \frac{1 - t^{ℓ}}{1 - t} Y_{0, β - 1} + \frac{1 - t^{r \frac{ℓ + 1}{s - 1} + ℓ}}{1 - t} y_{β}; q, t) \overset{(*)}{=} \\ R (X_{k} + \frac{1 - t^{ℓ}}{1 - t} Y_{0, β - 1}; t^{\frac{ℓ + 1}{s - 1} r + ℓ} \frac{1 - q^{s + r}}{1 - q} y_{β}) \prod_{α = 0}^{β - 1} R (X_{k} + \frac{1 - t^{s}}{1 - t} Y_{α + 1, β - 1}; t^{ℓ} \frac{1 - q^{s}}{1 - q} y_{α}) . \end{matrix}

Let us show this result with an example.

Example 10.

For

(t, q) = (u, u^{- 3})

\begin{matrix} M S_{[533]} (x_{1} + (1 + t) y_{0} + (1 + t + t^{2} + t^{3} + t^{4}) y_{1}; q = u^{- 3}, t = u) \\ \overset{(*)}{=} R (x_{1} + (1 + t) y_{0}; t^{5} (1 + q + q^{2}) y_{1}) M S_{2} (x_{1} + (1 + t) y_{0}; q = u^{- 3}, t = u) \overset{(*)}{=} \\ R (x_{1} + (1 + t) y_{0}; t^{5} (1 + q + q^{2}) y_{1}) R (x_{1}; t^{2} (1 + q) y_{0}) \\ = \prod_{i = 0}^{2} [(x_{1} - t^{5} q^{i} y_{1}) (y_{0} - t^{5} q^{i} y_{1}) (t y_{0} - t^{5} q^{i} y_{1})] (x_{1} - t^{2} y_{0}) (x_{1} - t^{2} q y_{0}) . \end{matrix}

8. Conclusions and Perspectives

In this last section, we explore the second clustering property conjectured by Bernevig and Haldane and we present some examples that illustrate future work that can be done in this direction.

8.1. The Second Clustering Property

Most of the different variants of Macdonald polynomials presented in this paper specialize to Jack polynomials by sending u to 1 when

ℓ + 1

and

s - 1

are coprime, which corresponds to setting

t = q^{α}

and sending

α

to 1. Moreover, this implies also that

s - 1

divides r. In that context, Proposition 7 gives that

\begin{matrix} J_{S t (ℓ, k; s; β)}^{(- \frac{ℓ + 1}{s - 1})} (X_{k + β ℓ} + ℓ y) \overset{(*)}{=} \prod_{i = 1}^{β ℓ} {(x_{i} - y)}^{s} J_{S t (ℓ, k; s; β - 1)}^{(- \frac{ℓ + 1}{s - 1})} (X_{k + β ℓ}) . \end{matrix}

By Equation (8),

\begin{matrix} J_{Q S (ℓ, k; s, r; β)}^{(- \frac{ℓ + 1}{s - 1})} (X_{k + β ℓ} + (r \frac{ℓ + 1}{s - 1} + ℓ) y) \overset{(*)}{=} \prod_{i = 1}^{β ℓ} {(x_{i} - y)}^{s + r} J_{S t (ℓ, k; s; β - 1)}^{(- \frac{ℓ + 1}{s - 1})} (X_{k + β ℓ}) . \end{matrix}

(9)

The fact that

\prod_{i = 1}^{β ℓ} {(x_{i} - y)}^{s + r}

divides the polynomial

J_{Q S (ℓ, k; s, r; β)}^{(- \frac{ℓ + 1}{s - 1})} (X_{k + β ℓ} + (r \frac{ℓ + 1}{s - 1} + ℓ) y)

is a special case of a result of [4] [Theorem 1.1]. The following table contrasts Bernevig and Haldane notation (B-H notation) with our notation in order to transcribe the second clustering property into our notation.

Our notation	B-H notation
ℓ	k
k	0
s	r
$\frac{r}{s - 1} + 1$	s
$s + r$	$s (r - 1) + 1$
$S t (ℓ, 0; s; β)$	$λ_{k, r, 1}^{β}$
$Q S (ℓ, 0; s, r; β)$	$λ_{k, r, s}^{β}$
$r \frac{ℓ + 1}{s - 1} + ℓ$	$n_{0}$
$X_{k + β ℓ}$	$z_{n_{0} + 1} + \dots + z_{N}$
y	$Z = z_{1} = \dots = z_{n_{0}}$

We recover the equality in Equation (1) by setting

k = 0

in Equation (9). This proves the second clustering property conjectured by Bernevig and Haldane [2]. Furthermore, Equation (8) is more general than those conjectured in [2] for two reasons. First, we consider quasistaircase partitions

Q S (ℓ, k; s, r; β)

, with

0 \leq k < ℓ

, while only the case

k = 0

was investigated in [1,2]. It should be interesting to know if some of these polynomials can be interpreted as wave functions in FQHT. In addition, when

ω_{1} \neq 1

, the Macdonald polynomial does not degenerate to a Jack polynomial when u tends to 1.

More generally, the following equality is obtained from Theorem 4:

\begin{matrix} J_{Q S (ℓ, k; s, r; β)}^{(- \frac{ℓ + 1}{s - 1})} (X_{k} + ℓ Y_{0, β - 1} + (r \frac{ℓ + 1}{s - 1} + ℓ) y_{β}) \overset{(*)}{=} \\ \prod_{i = 1}^{k} {(x_{i} - y_{β})}^{r + s} \prod_{α = 0}^{β} [{(y_{α} - y_{β})}^{(s + r) ℓ} \prod_{i = 1}^{k} {(x_{i} - y_{α})}^{s} \prod_{i = α + 1}^{β - 1} {(y_{i} - y_{α})}^{ℓ s}] . \end{matrix}

We finish this subsection by presenting more general identities involving partitions which are not quasistaircase for the considered specialization. For instance,

\begin{matrix} J_{[43210]}^{(- 3)} (x + 2 y_{1} + 2 y_{2}) \overset{(*)}{=} y_{1} y_{2} {(y_{1} - y_{2})}^{4} {(x - y_{1})}^{2} {(x - y_{2})}^{2} . \end{matrix}

This formula is in fact a specialization of

\begin{matrix} P_{[43210]} (x + (1 + t) y_{1} + (1 + t) y_{2}; q = t^{- 3}, t) \\ \overset{(*)}{=} y_{1} y_{2} (y_{1} - t y_{2}) (y_{1} - t^{2} y_{2}) (y_{2} - t y_{1}) (y_{2} - t^{2} y_{1}) (t x - y_{1}) (x - t^{2} y_{1}) (t x - y_{2}) (x - t^{2} y_{2}) . \end{matrix}

Notice that this polynomial does not satisfy the HW condition. This suggests that there exists a Macdonald version of the result of [4], Theorem 1.1.

8.2. Other Clustering and Factorizations Properties

The first and third clustering conjectures suggest that there exist many ways to factorize HW Macdonald polynomials by specializing the variables involved. Let us illustrate this remark by giving an example.

For a partition

λ = [λ_{1}, λ_{2}, λ_{3}]

, consider the alphabet

X_{λ} = \frac{1 - u^{λ_{1}}}{1 - u} x_{1} + \frac{1 - u^{λ_{2}}}{1 - u} x_{2} + \frac{1 - u^{λ_{3}}}{1 - u} x_{3}

. Let

ρ_{i}

be the operator that adds 2 to the ith entry of

λ

if, after this operation, the resulting vector is still a partition. We set also

R_{i; k} = \prod_{\binom{1 \leq j \leq N}{j \neq i}} (u^{k} x_{i} - x_{j})

Starting with

M S_{[420]} (X_{111}; q = u^{- 2}, t = u)

, and by

X_{111} ρ_{1} = X_{311}

, one obtains

\begin{matrix} M S_{[530^{4}]} (X_{311}; q = u^{- 2}, t = u) = R_{1; 3} M S_{[420]} (X_{111}; q = u^{- 2}, t = u) . \end{matrix}

The next step is more interesting because there are two kinds of specializations that provide nice factorizations:

\begin{matrix} M S_{[640^{6}]} (X_{511}; q = u^{- 2}, t = u) & \overset{(*)}{=} & R_{1; 5} M S_{[530^{4}]} (X_{311}; q = u^{- 2}, t = u), and \\ M S_{[640^{6}]} (X_{331}; q = u^{- 2}, t = u) & \overset{(*)}{=} & R_{2; 3} M S_{[530^{4}]} (X_{311}; q = u^{- 2}, t = u) . \end{matrix}

Continuing, we get other new nice factorizations:

\begin{matrix} M S_{[750^{8}]} (X_{711}; q = u^{- 2}, t = u) & \overset{(*)}{=} & R_{1; 7} M S_{[640^{4}]} (X_{511}; q = u^{- 2}, t = u), \\ M S_{[750^{8}]} (X_{333}; q = u^{- 2}, t = u) & \overset{(*)}{=} & R_{3; 3} M S_{[640^{4}]} (X_{331}; q = u^{- 2}, t = u), \\ M S_{[750^{8}]} (X_{531}; q = u^{- 2}, t = u) & \overset{(*)}{=} & R_{2; 3} M S_{[640^{4}]} (X_{511}; q = u^{- 2}, t = u), \\ M S_{[750^{8}]} (X_{531}; q = u^{- 2}, t = u) & \overset{(*)}{=} & R_{1; 5} M S_{[640^{4}]} (X_{331}; q = u^{- 2}, t = u) . \end{matrix}

The computation can be graphically represented as in Figure 5.

Using vanishing properties, it is not too difficult to show that, when

\frac{λ_{1} + λ_{2} + λ_{3} - 3}{2} = r

M S_{[4 + r, 2 + r, 0^{1 + 2 r}]} (X_{λ}; q = u^{- 2}, t = u)

factorizes nicely, and that, if

λ ρ_{i}

is well defined, one has

\begin{matrix} M S_{[4 + (r + 1), 2 + (r + 1), 0^{1 + 2 r}]} (X_{λ ρ_{i}}; q^{- 2}, t) \overset{(*)}{=} R_{i, λ ρ_{i} [i]} M S_{[4 + r, 2 + r, 0^{1 + 2 r}]} (X_{λ}; q = u^{- 2}, t = u) . \end{matrix}

This kind of formula takes place in a wider picture which will be investigated in a future paper.

8.3. More Factorizations of Nonsymmetric Macdonald Polynomials

In the preparation of this paper, the authors used experiments with symbolic computation with Maple [43] to develop conjectures, for which the proofs presented in this paper are constructed with mathematical rigor and do not depend on computers. The experiments also resulted in some fascinating examples which as yet have not led to any guesses at possible underlying structure.

We finish this paper presenting some specializations of nonsymmetric Macdonald polynomials that factorize nicely and that are not included in the results presented.

\begin{matrix} \begin{matrix} E_{[210]} (x_{1}, x_{2}, x_{3}; q = u^{- 2}, t = u) \overset{(*)}{=} (u x_{2} - x_{1}) (u x_{3} - x_{1}) (u x_{3} - x_{2}), \\ E_{[630]} (x_{1}, x_{2}, x_{3}; q = u^{- 2}, t = u^{3}) \overset{(*)}{=} (x_{2} u - x_{3}) (- u x_{3} + x_{2}) (x_{2} - u^{3} x_{3}) (x_{1} u - x_{3}) \\ \times (- u x_{3} + x_{1}) (x_{1} - u^{3} x_{3}) (x_{1} u - x_{2}) (x_{1} - x_{2} u) (x_{1} - x_{2} u^{3}), \\ E_{[420]} (x_{1}, x_{2}, x_{3}; q = - t, t) \overset{(*)}{=} t (x_{2} + x_{3}) (- t x_{3} + x_{2}) (x_{3} + x_{1}) (- t x_{3} + x_{1}) (x_{1} + x_{2}) (x_{1} - x_{2} t), \end{matrix} \end{matrix}

\begin{matrix} E_{[221100]} (x_{1}, x_{2}, y_{1}, t y_{1}, y_{2}, t y_{2}; q = t^{- 3}, t) \overset{(*)}{=} (y_{1} - y_{2} t^{2}) (y_{1} - y_{2} t) (x_{2} - y_{2} t^{2}) (x_{2} - t^{2} y_{1}) (x_{1} - y_{2} t^{2}) (x_{1} - t^{2} y_{1}) \end{matrix}

\begin{matrix} \begin{matrix} E_{[42200]} (x_{1}, y_{1}, y_{1} u^{2}, y_{2}, y_{2} u^{2}; q = u^{- 3}, t = u^{2}) \overset{(*)}{=} (y_{1} - y_{2} u^{4}) (y_{1} - y_{2} u) (y_{1} u - y_{2}) \\ \times (y_{1} - y_{2} u^{2}) (x_{1} - y_{2} u^{4}) (x_{1} - y_{2} u) (x_{1} - y_{1} u^{4}) (x_{1} - y_{1} u), \\ E_{[0022]} (t y, y, x_{3}, x_{4}; q = t^{- 3}, t) \overset{(*)}{=} (t x_{4} - y) (t x_{3} - y) x_{4} x_{3} . \end{matrix} \end{matrix}

Notice that the last example is not singular and we have

\begin{matrix} E_{[0022]} (t y, y, x_{3}, x_{4}; q = t^{- 3}, t) \overset{(*)}{\neq} M_{[0022]} (t y, y, x_{3}, x_{4}; q = t^{- 3}, t) \overset{(*)}{=} (t x_{4} - y) (t x_{3} - y) (x_{4} - 1) (x_{3} - 1) . \end{matrix}

However, it is deduced from the singular polynomial

E_{[1100]} (t y, y, x_{3}, x_{4}; q = t^{- 3}, t) \overset{(*)}{=} (t x_{4} - y) (t x_{3} - y)

, by applying the affine step twice.

Some other examples involving vectors which are not partitions are more interesting. For instance, the polynomial

E_{1010} (x_{1}, x_{2}, x_{3}, x_{4}; q = t^{- 3}, t)

is singular and we have

\begin{matrix} E_{[1010]} (t y, x_{2}, y, x_{4}; q = t^{- 3}, t) \overset{(*)}{=} (t x_{4} - y) (t x_{2} - y) . \end{matrix}

More general formulas for quasistaircases are also observed. For instance,

\begin{matrix} \begin{matrix} E_{[32000]} (x_{1}, x_{2}, y, t y, t^{2} y; q = t^{- 2}, t) \overset{(*)}{=} (x_{2} - y t^{3}) (x_{2} - y t) (x_{1} - y t^{3}) (x_{1} - y t) (x_{1} - t x_{2}), \\ E_{[4300000]} (x_{1}, x_{2}, y, t y, t^{2} y, t^{3} y, t^{4} y; q = t^{- 2}, t) \overset{(*)}{=} (x_{2} - t y) (x_{2} - y t^{3}) (x_{2} - y t^{5}) (x_{1} - y t) \\ \times (x_{1} - y t^{3}) (x_{1} - y t^{5}) (x_{1} - t x_{2}) . \end{matrix} \end{matrix}

Numerical evidence suggests that one has a formula very close to those of Theorem 4 but for a specialization under the form

\begin{matrix} (t, q) = (u^{\frac{s}{g}}, u^{- \frac{ℓ + 1}{g}} ω_{1}), \end{matrix}

where

g = \gcd (ℓ + 1, s)

and

ω_{1}^{\frac{s}{g}}

is a gth primitive root of the unity.

In addition, as in Section 8.2, we observe factorizations for other specializations of the variables

x_{i}

’s. For instance

\begin{matrix} E_{[4300000]} (x_{1}, t^{2} y_{1}, y_{1}, t y_{1}, y_{2}, t y_{2}, t^{2} y_{2}; q = t^{- 2}, t) \overset{(*)}{=} (y_{1} - t y_{2}) (y_{1} - t^{3} y_{2}) (t y_{1} - y_{2}) \\ \times (x_{1} - t y_{2}) (x_{1} - t^{3} y_{2}) (x_{1} - t y_{1}) (x_{1} - t^{3} y_{1}) . \end{matrix}

The precise statements, proofs, and connection with the factorizations of symmetric Macdonald polynomials remain to be investigated.

Author Contributions

All authors contributed equally to this paper. L.C. and C.F.D. prepared the final draft.

Funding

J.-G.L. was partially funded by the GRR PROJECT MOUSTIC.

Acknowledgments

We thank Stephen Griffeth for pointing out the relevance of Theorem 1.1 in [4] to this paper. Jean-Gabriel Luque acknowledges Thierry Jolicoeur for fruitful discussions.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:

FQH	Fractional Quantum Hall
HW	Highest weight
LW	Lowest weight

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Figure 1. Classical Hall effect.

Figure 2. Integral quantum Hall effect.

Figure 3. Fractional quantum Hall (FQH) effect.

Figure 4. Illustration of the proof of Lemma 2.

Figure 5. Computation of

M S_{[4 + r, 2 + r, 0^{1 + 2 r}]} (X_{λ}; q = u^{- 2}, t = u) .

Figure 5. Computation of

M S_{[4 + r, 2 + r, 0^{1 + 2 r}]} (X_{λ}; q = u^{- 2}, t = u) .

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Colmenarejo, L.; Dunkl, C.F.; Luque, J.-G. Factorizations of Symmetric Macdonald Polynomials. Symmetry 2018, 10, 541. https://doi.org/10.3390/sym10110541

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Colmenarejo L, Dunkl CF, Luque J-G. Factorizations of Symmetric Macdonald Polynomials. Symmetry. 2018; 10(11):541. https://doi.org/10.3390/sym10110541

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Colmenarejo, Laura, Charles F. Dunkl, and Jean-Gabriel Luque. 2018. "Factorizations of Symmetric Macdonald Polynomials" Symmetry 10, no. 11: 541. https://doi.org/10.3390/sym10110541

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Colmenarejo, L., Dunkl, C. F., & Luque, J. -G. (2018). Factorizations of Symmetric Macdonald Polynomials. Symmetry, 10(11), 541. https://doi.org/10.3390/sym10110541

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Factorizations of Symmetric Macdonald Polynomials

Abstract

1. Introduction

2. Background

2.1. Partitions and Vectors

2.2. Affine Hecke Algebra

2.3. Symmetric Functions and Virtual Alphabets

2.4. Macdonald Polynomials and Variants

2.5. Computing Macdonald Polynomials Using the Yang–Baxter Graph

2.6. Vanishing Properties

3. Clustering Properties of Jack Polynomials and the Quantum Hall Effect

3.1. A Gentle History of the Quantum Hall Effect

3.2. Quantum Hall Wave Functions

3.3. FQHT and Jack Polynomials

3.4. The Interest of Shifted Macdonald Polynomials

4. Factorizations for Generic ( q , t ) Parameters

4.1. Saturated Partitions

4.2. Standard Specializations for the Variables

5. Specializations of the Type t a q b = 1 and Quasistaircase Partitions

5.1. Admissible Specializations

5.2. On the Reciprocal Sum ⟅ Q S ( ℓ , k ; s , r ; β ) ⟆

5.3. On the Reciprocal Vector 〈 Q S ( ℓ , k ; s , r ; β ) 〉

6. Factorizations and Wheel Condition

6.1. Wheel Condition and Admissible Partitions

6.2. Factorizations

7. Beyond the Wheel Condition

8. Conclusions and Perspectives

8.1. The Second Clustering Property

8.2. Other Clustering and Factorizations Properties

8.3. More Factorizations of Nonsymmetric Macdonald Polynomials

Author Contributions

Funding

Acknowledgments

Conflicts of Interest

Abbreviations

References

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4. Factorizations for Generic $(q, t)$ Parameters

5. Specializations of the Type $t^{a} q^{b} = 1$ and Quasistaircase Partitions

5.2. On the Reciprocal Sum $⟅ Q S (ℓ, k; s, r; β) ⟆$

5.3. On the Reciprocal Vector $〈 Q S (ℓ, k; s, r; β) 〉$