The norm and the Evaluation of the Macdonald polynomials in superspace

González, Camilo; Lapointe, Luc

doi:10.1016/j.ejc.2019.103018

Cited by 5 publications

(4 citation statements)

References 23 publications

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“…González and Lapointe [10] proved an evaluation formula for the version of supersymmetric Macdonald polynomials constructed in [3], with z = t N −1 q −m , t N −2 q 1−m , . .…”

Section: Special Valuesmentioning

confidence: 99%

“…By defining representations of the Hecke algebra on anti-commuting variables the theory of vector-valued nonsymmetric Macdonald polynomials is applied to define and analyze superpolynomials. There is a theory of symmetric Macdonald superpolynomials initiated by Blondeau-Fournier, Desrosiers, Lapointe, and Mathieu [3] with further developments on norm and special point values by González and Lapointe [10]. Their approach and definitions are based on differential operators and linear combinations of the classical nonsymmetric Macdonald polynomials, whose coefficients involve anti-commuting variables.…”

Section: Introductionmentioning

confidence: 99%

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Nonsymmetric Macdonald Superpolynomials

Dunkl

2021

SIGMA

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There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald polynomials taking values in arbitrary modules of the Hecke algebra. In this paper the two ideas are combined to define and study nonsymmetric Macdonald polynomials taking values in the aforementioned anti-commuting polynomials, in other words, superpolynomials. The modules, their orthogonal bases and their properties are first derived. In terms of the standard Young tableau approach to representations these modules correspond to hook tableaux. The details of the Dunkl-Luque theory and the particular application are presented. There is an inner product on the polynomials for which the Macdonald polynomials are mutually orthogonal. The squared norms for this product are determined. By using techniques of Baker and Forrester [Ann. Comb. 3 (1999), 159-170, arXiv:q-alg/9707001] symmetric Macdonald polynomials are built up from the nonsymmetric theory. Here ''symmetric'' means in the Hecke algebra sense, not in the classical group sense. There is a concise formula for the squared norm of the minimal symmetric polynomial, and some formulas for anti-symmetric polynomials. For both symmetric and anti-symmetric polynomials there is a factorization when the polynomials are evaluated at special points.

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“…González and Lapointe [10] proved an evaluation formula for the version of supersymmetric Macdonald polynomials constructed in [3], with z = t N −1 q −m , t N −2 q 1−m , . .…”

Section: Special Valuesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nonsymmetric Macdonald Superpolynomials

Dunkl

2021

SIGMA

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“…Next we consider the transition from α to β (see (9)) with the affine step M β,F (x; θ) = x N wM α,F (x; θ) and as before the calculation is based on the formula…”

Section: From α To δ For Type (1)mentioning

confidence: 99%

“…The operators used in their work to define Macdonald polynomials are significantly different from ours. There are results on evaluations for these polynomials found by González and Lapointe [9]. In this section, we consider symmetrization over a subset of the coordinates, and associated evaluations.…”

Section: Restricted Symmetrization and Antisymmetrizationmentioning

confidence: 99%

Evaluation of Nonsymmetric Macdonald Superpolynomials at Special Points

Dunkl

2021

Symmetry

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In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of the Hecke algebra of type A (Dunkl and Luque SLC 2012) was applied to such modules consisting of polynomials in anti-commuting variables, to define nonsymmetric Macdonald superpolynomials. These polynomials depend on two parameters q,t and are defined by means of a Yang–Baxter graph. The present paper determines the values of a subclass of the polynomials at the special points 1,t,t2,… or 1,t−1,t−2,…. The arguments use induction on the degree and computations with products of generators of the Hecke algebra. The resulting formulas involve q,t-hook products. Evaluations are also found for Macdonald superpolynomials having restricted symmetry and antisymmetry properties.

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Symmetry and Pieri rules for the bisymmetric Macdonald polynomials

Concha,

Lapointe

2024

European Journal of Combinatorics

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The norm and the Evaluation of the Macdonald polynomials in superspace

Cited by 5 publications

References 23 publications

Nonsymmetric Macdonald Superpolynomials

Nonsymmetric Macdonald Superpolynomials

Evaluation of Nonsymmetric Macdonald Superpolynomials at Special Points

Symmetry and Pieri rules for the bisymmetric Macdonald polynomials

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