# Clustering properties of rectangular Macdonald polynomials

@article{Dunkl2012ClusteringPO, title={Clustering properties of rectangular Macdonald polynomials}, author={Charles F. Dunkl and Jean-Gabriel Luque}, journal={arXiv: Mathematical Physics}, year={2012} }

The clustering properties of Jack polynomials are relevant in the theoretical study of the fractional Hall states. In this context, some factorization properties have been conjectured for the $(q,t)$-deformed problem involving Macdonald polynomials. The present paper is devoted to the proof of this formula. To this aim we use four families of Jack/Macdonald polynomials: symmetric homogeneous, nonsymmetric homogeneous, shifted symmetric and shifted nonsymmetric.

## 8 Citations

### Connections between vector-valued and highest weight Jack and Macdonald polynomials

- Mathematics
- 2019

We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, especially commuting with the actions of the symmetric…

### The norm and the Evaluation of the Macdonald polynomials in superspace

- MathematicsEur. J. Comb.
- 2020

### Evaluation of Nonsymmetric Macdonald Superpolynomials at Special Points

- MathematicsSymmetry
- 2021

The values of a subclass of the polynomials at the special points 1,t,t2,… or 1, t−1,t−2,….

### The Classification of All Singular Nonsymmetric Macdonald Polynomials

- MathematicsAxioms
- 2022

The affine Hecke algebra of type A has two parameters q,t and acts on polynomials in N variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators…

### Jack Polynomials as Fractional Quantum Hall States and the Betti Numbers of the (k + 1)-Equals Ideal

- Mathematics
- 2014

We show that for Jack parameter α = −(k + 1)/(r − 1), certain Jack polynomials studied by Feigin–Jimbo–Miwa–Mukhin vanish to order r when k + 1 of the coordinates coincide. This result was…

### Singular Nonsymmetric Macdonald Polynomials and Quasistaircases

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2020

Singular nonsymmetric Macdonald polynomials are constructed by use of the representation theory of the Hecke algebras of the symmetric groups. These polynomials are labeled by quasistaircase…

### Jack Polynomials with Prescribed Symmetry and Some of Their Clustering Properties

- Mathematics
- 2013

We study Jack polynomials in N variables, with parameter α, and having a prescribed symmetry with respect to two disjoint subsets of variables. For instance, these polynomials can exhibit a symmetry…

### Factorizations of Symmetric Macdonald Polynomials

- MathematicsSymmetry
- 2018

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…

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