# Factorizations of Symmetric Macdonald Polynomials

^{1}

^{2}

^{3}

^{*}

^{†}

## 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

**Definition**

**1.**

**Definition**

**2.**

#### 2.5. Computing Macdonald Polynomials Using the Yang–Baxter Graph

**Proposition**

**1.**

- If $v\left[i\right]<v[i+1]$, ${M}_{v.{s}_{i}}={M}_{v}\left({T}_{i}+\frac{1-t}{1-\frac{\langle v\rangle [i+1]}{\langle v\rangle \left[i\right]}}\right)$, where $v.{s}_{i}$ is the vector obtained from v by exchanging the values $v\left[i\right]$ and $v[i+1]$.
- ${M}_{v\mathsf{\Phi}}={M}_{v}\tau ({x}_{N}-1)$, where $v\mathsf{\Phi}=\left[v\left[2\right],\dots ,v\left[N\right],v\left[1\right]+1\right]$. We refer to this step as affine step.

#### 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

**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 ${Z}_{1}={z}_{1}=\cdots ={z}_{k+1}$, ${Z}_{2}={z}_{k+1}=\cdots ={z}_{2(k+1)}$,⋯,${Z}_{s-1}={z}_{(s-2)(k+1)+1}=\cdots ={z}_{(s-1)(k+1)}$, and ${Z}_{F}={z}_{(s-1)(k+1)}=\cdots ={z}_{s(k+1)-1}$. Then, the Jack polynomial ${J}_{{\lambda}_{k,r,s}^{\beta}}^{-\frac{k+1}{r-1}}((k+1)({Z}_{1}+\cdots +{Z}_{s-1})+k{Z}_{F}+{z}_{s(k+1)}+\cdots +{z}_{N})$ behaves as $\prod _{i=s(k+1)}^{N}{\left({Z}_{F}-{z}_{i}\right)}^{r}$ when each ${z}_{i}$, with $i=s(k+1),\dots ,N$, tends to ${Z}_{F}$. For instance, we have$$\begin{array}{c}{J}_{53}^{(-2)}(2{Z}_{1}+{Z}_{F}+{z}_{3}+{z}_{4})={\left({\mathit{Z}}_{\mathit{F}}-{\mathit{z}}_{\mathit{4}}\right)}^{2}{\left({\mathit{Z}}_{\mathit{F}}-{\mathit{z}}_{\mathit{3}}\right)}^{2}P({Z}_{1},{Z}_{F},{z}_{3},{z}_{4}),\phantom{\rule{4.pt}{0ex}}\mathrm{with}\hfill \\ \hfill P({Z}_{1},{Z}_{F},{z}_{3},{z}_{4})=144\phantom{\rule{0.166667em}{0ex}}{\left({\mathit{z}}_{\mathit{3}}-{\mathit{z}}_{\mathit{4}}\right)}^{2}\left({\mathit{z}}_{\mathit{3}}\phantom{\rule{0.166667em}{0ex}}{\mathit{z}}_{\mathit{4}}+{\mathit{Z}}_{\mathit{F}}\phantom{\rule{0.166667em}{0ex}}{\mathit{z}}_{\mathit{4}}+{\mathit{Z}}_{\mathit{F}}\phantom{\rule{0.166667em}{0ex}}{\mathit{z}}_{\mathit{3}}-2\phantom{\rule{0.166667em}{0ex}}{\mathit{Z}}_{\mathit{1}}\phantom{\rule{0.166667em}{0ex}}{\mathit{z}}_{\mathit{4}}-2\phantom{\rule{0.166667em}{0ex}}{\mathit{Z}}_{\mathit{1}}\phantom{\rule{0.166667em}{0ex}}{\mathit{z}}_{\mathit{3}}-2\phantom{\rule{0.166667em}{0ex}}{\mathit{Z}}_{\mathit{1}}\phantom{\rule{0.166667em}{0ex}}{\mathit{Z}}_{\mathit{F}}+3\phantom{\rule{0.166667em}{0ex}}{{\mathit{Z}}_{\mathit{1}}}^{2}\right).\end{array}$$**Second clustering property:**A cluster of ${n}_{0}=(k+1)s-1$ particles.To formalize this situation, we set ${z}_{1}=\cdots ={z}_{(k+1)s-1}=Z$. Then, the Jack polynomial ${J}_{{\lambda}_{k,r,s}}^{\left(-\frac{k+1}{r-1}\right)}({n}_{0}Z+{z}_{{n}_{0}+1}+\dots +{z}_{N})$ 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,$$\begin{array}{c}\hfill {\displaystyle {J}_{{\lambda}_{k,r,s}^{\beta}}^{\left(-\frac{k+1}{r-1}\right)}({n}_{0}Z+{z}_{{n}_{0}+1}+\cdots +{z}_{N})\stackrel{(\ast )}{=}\prod _{i=s(k+1)}^{N}{(Z-{z}_{i})}^{(r-1)s+1}{J}_{{\lambda}_{k,r,1}^{\beta -1}}^{\left(-\frac{k+1}{r-1}\right)}({z}_{{n}_{0}+1}+\cdots +{z}_{N}).}\end{array}$$For instance,$$\begin{array}{c}\hfill {J}_{53}^{(-2)}(3Z+{z}_{4}+{z}_{5})=-144{(Z-{z}_{4})}^{3}{(Z-{z}_{5})}^{3}{J}_{2}^{(-2)}({z}_{4}+{z}_{5}).\end{array}$$**Third clustering property:**$s-1$ clusters of $2k+1$ particles.By setting $s-1$ sets of variables as follows: ${Z}_{1}={z}_{1}=\cdots ={z}_{2k+1}$,…, ${Z}_{s-1}={z}_{(s-2)(2k+1)+1}=\cdots ={z}_{(s-1)(2k+1)}$, the HW Jack ${J}_{{\lambda}_{k,r,s}^{\beta}}$ satisfies$$\begin{array}{c}{\displaystyle {J}_{{\lambda}_{k,r,s}^{\beta}}^{\left(-\frac{k+1}{r-1}\right)}((2k+1)({Z}_{1}+\dots +{Z}_{s-1})+{z}_{(s-1)(2k+1)+1}+\cdots +{z}_{N})\stackrel{(\ast )}{=}}\hfill \\ \hfill {\displaystyle \prod _{1\le i<j\le s-1}{({Z}_{i}-{Z}_{j})}^{k(3r-2)}\prod _{i=1}^{s-1}\prod _{\ell =(s-1)(2k+1)+1}^{N}{({Z}_{i}-{z}_{\ell})}^{2r-1}\xb7{J}_{{\lambda}_{k,r}^{\beta -s+1}}({z}_{(s-1)(2k+1)+1}+\cdots +{z}_{N}).}\end{array}$$For instance,$$\begin{array}{c}\hfill {J}_{64}^{\left(-2\right)}(3({Z}_{1}+{Z}_{2})+{z}_{7})=-3456{({Z}_{1}-{Z}_{2})}^{4}{({Z}_{1}-{z}_{7})}^{3}{({Z}_{2}-{z}_{7})}^{3}.\end{array}$$

#### 3.4. The Interest of Shifted Macdonald Polynomials

- 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 ${L}^{+}$).
- 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 $t={q}^{\alpha}$ and q to 1.

## 4. Factorizations for Generic $(\mathit{q},\mathit{t})$ Parameters

#### 4.1. Saturated Partitions

**Definition**

**3.**

**Proposition**

**2.**

- ${P}_{\lambda}({x}_{1},\dots ,{x}_{N};q,t)={({x}_{1}\dots {x}_{N})}^{{\lambda}_{N}}{P}_{[{\lambda}_{1}-{\lambda}_{N},\dots ,{\lambda}_{N-1}-{\lambda}_{N},0]}({q}^{-{\lambda}_{N}}{x}_{1},\dots ,{q}^{-{\lambda}_{N}}{x}_{N};q,t)$.
- $M{S}_{\lambda}({x}_{1},\dots ,{x}_{N};q,t)\stackrel{(\ast )}{=}\prod _{k=0}^{{\lambda}_{N}-1}\prod _{i=1}^{N}({x}_{i}-{q}^{k})M{S}_{[{\lambda}_{1}-{\lambda}_{N},\dots ,{\lambda}_{N-1}-{\lambda}_{N},0]}({q}^{-{\lambda}_{N}}{x}_{1},\dots ,{q}^{-{\lambda}_{N}}{x}_{N};q,t)$.

**Proof.**

#### 4.2. Standard Specializations for the Variables

**Definition**

**4.**

**Proposition**

**3.**

**Proof.**

**Example**

**1.**

$\langle 50\rangle $ | $\langle 41\rangle $ | $\langle 32\rangle $ |

$[{q}^{5}t,1]$ | $[{q}^{4}t,q]$ | × |

$\langle 40\rangle $ | $\langle 31\rangle $ | $\langle 22\rangle $ |

$[{q}^{4}t,1]$ | $[{q}^{3}t,q]$ | $[{q}^{2}t,{q}^{2}]$ |

$\langle 30\rangle $ | $\langle 21\rangle $ | |

$[{q}^{3}t,1]$ | $[{q}^{2}t,q]$ | |

$\langle 20\rangle $ | $\langle 11\rangle $ | |

$[{q}^{2}t,1]$ | $[qt,q]$ | |

$\langle 10\rangle $ | ||

$[qt,1]$ | ||

$\langle 00\rangle $ | ||

$[t,1]$ |

**Corollary**

**1.**

**Proof.**

**Example**

**2.**

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

**Definition**

**5.**

#### 5.1. Admissible Specializations

**Definition**

**6.**

**Example**

**3.**

- $(t,q)=(u,{u}^{-3})$ 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.
- $(t,q)=(u,{e}^{\frac{2i\pi}{3}}{u}^{-2})$ and $(t,q)=(u,{e}^{\frac{2i\pi}{3}}{u}^{-2})$ 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.

**Lemma**

**1.**

**Proof.**

#### 5.2. On the Reciprocal Sum $\u27c5QS(\ell ,k;s,r;\beta )\u27c6$

**Lemma**

**2.**

**Proof.**

**Example**

**4.**

**Proposition**

**4.**

**Proof.**

**Example**

**5.**

⟅420⟆ | ⟅320⟆ | ⟅220⟆ | ⟅110⟆ |

${q}^{4}{t}^{2}+{q}^{2}t+1$ | ${q}^{3}{t}^{2}+{q}^{2}t+1$ | ${q}^{2}{t}^{2}+{q}^{2}t+1$ | $q{t}^{2}+qt+1$ |

⟅410⟆ | ⟅310⟆ | ⟅210⟆ | ⟅100⟆ |

${q}^{4}{t}^{3}+qt+1$ | ${q}^{3}{t}^{2}+qt+1$ | ${q}^{2}{t}^{2}+qt+1$ | $q{t}^{2}+t+1$ |

⟅400⟆ | ⟅300⟆ | ⟅200⟆ | ⟅000⟆ |

${q}^{4}{t}^{2}+t+1$ | ${q}^{3}{t}^{2}+t+1$ | ${q}^{2}{t}^{2}+t+1$ | ${t}^{2}+t+1$ |

⟅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⟆ |

${u}^{-6}+1+u$ | ${u}^{-4}+1+u$ | ${u}^{-2}+1+u$ | $1+u+{u}^{2}$ |

**Corollary**

**2.**

- $\mu =\lambda $,
- The intersection of the eigenspace of ξ with eigenvalue ${\u27c5\lambda \u27c6}_{{q}^{-1},{t}^{-1}}$, and the space generated by ${P}_{\mu}$, with $\mu \subseteq \lambda $, has dimension 1.
- The intersection of the eigenspace of Ξ with eigenvalue ${\u27c5\lambda \u27c6}_{{q}^{-1},{t}^{-1}}$, and the space generated by $M{S}_{\mu}$, with $\mu \subseteq \lambda $, has dimension 1.

**Proof.**

#### 5.3. On the Reciprocal Vector $\langle QS(\ell ,k;s,r;\beta )\rangle $

**Example**

**6.**

**Proposition**

**5.**

**Proof.**

## 6. Factorizations and Wheel Condition

#### 6.1. Wheel Condition and Admissible Partitions

**Definition**

**7.**

**Theorem**

**1**

**.**The ideal ${J}_{N}^{\ell ,s}$ is generated by the Macdonald polynomials indexed by admissible partitions,

**Proposition**

**6.**

**Proof.**

**Example**

**7.**

#### 6.2. Factorizations

**Proposition**

**7.**

**Example**

**8.**

**Theorem**

**2.**

**Example**

**9.**

## 7. Beyond the Wheel Condition

**Theorem**

**3**

**.**For β, s, r, k, $\ell \in \mathbb{N}$, with $k\le \ell $, consider $\lambda =[{((\beta +1)s+r)}^{k},{(\beta s+r)}^{\ell},\dots ,{(s+r)}^{\ell}]$ and the specialization

**Corollary**

**3.**

**Theorem**

**4.**

**Example**

**10.**

## 8. Conclusions and Perspectives

#### 8.1. The Second Clustering Property

Our notation | B-H notation |

ℓ | k |

k | 0 |

s | r |

$\frac{r}{s-1}+1$ | s |

$s+r$ | $s(r-1)+1$ |

$St(\ell ,0;s;\beta )$ | ${\lambda}_{k,r,1}^{\beta}$ |

$QS(\ell ,0;s,r;\beta )$ | ${\lambda}_{k,r,s}^{\beta}$ |

$r\frac{\ell +1}{s-1}+\ell $ | ${n}_{0}$ |

${\mathbb{X}}_{k+\beta \ell}$ | ${z}_{{n}_{0}+1}+\cdots +{z}_{N}$ |

y | $Z={z}_{1}=\cdots ={z}_{{n}_{0}}$ |

#### 8.2. Other Clustering and Factorizations Properties

#### 8.3. More Factorizations of Nonsymmetric Macdonald Polynomials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FQH | Fractional Quantum Hall |

HW | Highest weight |

LW | Lowest weight |

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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

**AMA Style**

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

**Chicago/Turabian Style**

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