Singular Nonsymmetric Macdonald Polynomials and Quasistaircases

Colmenarejo, Laura; Dunkl, Charles F.

doi:10.3842/sigma.2020.010

Cited by 4 publications

(8 citation statements)

References 11 publications

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“…We proceed to analyze the Hecke algebra case. To do so, for the rest of the discussion we assume the following result, whose proof is included in [8]. This theorem shows a very natural phenomenon.…”

Section: The Hecke Algebra Casementioning

confidence: 99%

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

Colmenarejo

Dunkl

Luque

2022

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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Section: The Hecke Algebra Casementioning

confidence: 99%

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

Colmenarejo

Dunkl

Luque

2022

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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“…The following is a restatement of the development in [3] with significant differences in notation. First there is an informal discussion of the beginning of the process of building Y by placing N, N −1, N −2, .…”

Section: Necessary Conditions For Singular Polynomialsmentioning

confidence: 99%

“…These are called singular polynomials with singular parameter q m = t −n . In a previous paper [3] Colmenarejo and the author found a large class of such polynomials associated with tableaux of quasi-staircase shape. In this paper we will show that there are no other occurrences.…”

Section: Introductionmentioning

confidence: 97%

The Classification of All Singular Nonsymmetric Macdonald Polynomials

Dunkl¹

2021

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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 and the Jucys-Murphy elements whose simultaneous eigenfunctions are the nonsymmetric Macdonald polynomials, and basis vectors of irreducible modules of the Hecke algebra, respectively. For certain parameter values it is possible for special polynomials to be simultaneous eigenfunctions with equal corresponding eigenvalues of both sets of operators. These are called singular polynomials. The possible parameter values are of the form q m = t −n with 2 ≤ n ≤ N. For a fixed parameter the singular polynomials span an irreducible module of the Hecke algebra. Colmenarejo and the author (SIGMA 16 (2020), 010) showed that there exist singular polynomials for each of these parameter values, they coincide with specializations of nonsymmetric Macdonald polynomials, and the isotype (a partition of N ) of the Hecke algebra module is (dn − 1, n − 1, . . . , n − 1, r) for some d ≥ 1. In the present paper it is shown that there are no other singular polynomials.

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“…We proceed to analyze the Hecke algebra case. To do so, for the rest of the discussion we assume the following result, whose proof is included in a forthcoming paper [8].…”

Section: 32mentioning

confidence: 99%

“…Taking the Jack polynomials case as a guide, we replicate the study for Macdonald polynomials. Our study is based on Theorem 5.9, which is already included in [9] as a conjecture as part of our conclusions and that will be proved in a forthcoming paper of the authors, [8].…”

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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Singular Nonsymmetric Macdonald Polynomials and Quasistaircases

Cited by 4 publications

References 11 publications

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

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

The Classification of All Singular Nonsymmetric Macdonald Polynomials

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

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