Clustering properties of rectangular Macdonald polynomials

Dunkl, Charles F.; Luque, Jean-Gabriel

doi:10.4171/aihpd/19

Cited by 9 publications

(14 citation statements)

References 27 publications

(59 reference statements)

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“…, λ N ] ⊂ µ then this means that µ N < λ N . Then, the factor (x N − q µ N ) in (14) vanishes for x N = q µ N . This proves that the two polynomials have the same vanishing properties and so, that they are equal.…”

Section: Saturated Partitionsmentioning

confidence: 99%

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Factorizations of Symmetric Macdonald Polynomials

2018

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Section: Saturated Partitionsmentioning

confidence: 99%

“…The aim of our paper is to show how the material described in [14] can help in this context. In particular, we focus on the second clustering property for HW polynomials.…”

Section: Fqht and Jack Polynomialsmentioning

confidence: 99%

Factorizations of Symmetric Macdonald Polynomials

2018

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“…In [6], there is described another set of Jucys-Murphy elements. The set described here is nicely linked to singularity and seems easier to manipulate in this setup.…”

Section: 2mentioning

confidence: 99%

Singular Nonsymmetric Macdonald Polynomials and Quasistaircases

Colmenarejo¹,

Dunkl

2020

SIGMA

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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 partitions and are associated to special parameter values (q, t). For N variables, there are singular polynomials for any pair of positive integers m and n, with 2 ≤ n ≤ N , and parameters values (q, t) satisfying q a t b = 1 exactly when a = rm and b = rn, for some integer r. The coefficients of nonsymmetric Macdonald polynomials with respect to the basis of monomials {x α } are rational functions of q and t.In this paper, we present the construction of subspaces of singular nonsymmetric Macdonald polynomials specialized to particular values of (q, t). The key part of this construction is to show the coefficients have no poles at the special values of (q, t). Moreover, this subspace of singular Macdonald polynomials for the special values of the parameters is an irreducible module for the Hecke algebra of type A N −1 .

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“…We propose to investigate the relationship between the notions of singularity and highest weight by using a more general class of polynomials, called vector-valued Macdonald and Jack polynomials [14,15], whose coefficients belong to a representation of the Hecke algebra and which project to the classical case while preserving singularity properties. This work is part of a larger study [9,16,20] on the conjectures of Bernevig and Haldane [3] on clustering properties of (symmetric) Jack polynomials. These clustering properties are closely related to the quasistaircase partition, which get our attention later on the paper.…”

Section: Introductionmentioning

confidence: 99%

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

Colmenarejo¹,

Dunkl²,

Luque³

2019

Preprint

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We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, specially commuting with the actions of the symmetric group or Hecke algebra, respectively, and with the Cherednik operators for which these polynomials are eigenfunctions. In the framework of representation theory of the symmetric group and the Hecke algebra, we study the relation between singular nonsymmetric Jack and Macdonald polynomials and highest weight symmetric Jack and Macdonald polynomials. Moreover, we study the quasistaircase partition as a continuation of our study on the conjectures of Bernevig and Haldane on clustering properties of symmetric Jack polynomials.Key words and phrases. Macdonald and Jack Polynomials, singular polynomials, highest weight polynomials, vector-valued polynomials, representation theory of symmetric group and Hecke algebra.This work is partially supported by the "European Regional Development Fund" (ERDF) via the regional (GRR) project MOUSTIC. 1 VECTOR-VALUED AND HIGHEST WEIGHT JACK AND MACDONALD POLYNOMIALS 2 5.3. The symmetric group case 20 5.3.1. Singularity of quasistaircase nonsymmetric Jack polynomials 20 5.3.2. From singular nonsymmetric Jack polynomials to highest weight symmetric Jack polynomials 23 5.4. The Hecke algebra case 23 5.4.1. Singularity of quasistaircase nonsymmetric Macdonald polynomials 24 5.4.2. From singular nonsymmetric Macdonald polynomials to highest weight symmetric Macdonald polynomials 25 5.4.3. A note on specializations 27 6. Factorizations at special points 27 6.1. The symmetric group case 28 6.2. The Hecke algebra case 29 6.2.1. The general polynomials of isotype τ 34 6.2.2. Application to projections and factorizations 35 7. Conclusion and perspective 38 References 39

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Clustering properties of rectangular Macdonald polynomials

Cited by 9 publications

References 27 publications

Factorizations of Symmetric Macdonald Polynomials

Factorizations of Symmetric Macdonald Polynomials

Singular Nonsymmetric Macdonald Polynomials and Quasistaircases

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

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