Macdonald polynomials and the space of diagonal harmonics

2007 ◽  
pp. 27-39 ◽  
Author(s):  
James Haglund
Keyword(s):  
Macdonald Polynomials ◽  
Diagonal Harmonics

scholarly journals Evaluation of Nonsymmetric Macdonald Superpolynomials at Special Points

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 779
Author(s):  
Charles F. Dunkl
Keyword(s):  
Preceding Paper ◽  
Hecke Algebra ◽  
Type A ◽  
Macdonald Polynomials ◽  
Special Points ◽  
Two Parameters

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.

scholarly journals A combinatorial formula for nonsymmetric Macdonald polynomials

2008 ◽  
Vol 130 (2) ◽  
pp. 359-383 ◽  
Author(s):  
James. Haglund ◽  
Mark D. Haiman ◽  
N. Loehr
Keyword(s):  
Macdonald Polynomials ◽  
Combinatorial Formula

scholarly journals Macdonald polynomials at $t=q^k$

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Jean-Gabriel Luque
Keyword(s):  
Macdonald Polynomials ◽  
Nous Montrons ◽  
International Audience

International audience We investigate the homogeneous symmetric Macdonald polynomials $P_{\lambda} (\mathbb{X} ;q,t)$ for the specialization $t=q^k$. We show an identity relying the polynomials $P_{\lambda} (\mathbb{X} ;q,q^k)$ and $P_{\lambda} (\frac{1-q}{1-q^k}\mathbb{X} ;q,q^k)$. As a consequence, we describe an operator whose eigenvalues characterize the polynomials $P_{\lambda} (\mathbb{X} ;q,q^k)$. Nous nous intéressons aux propriétés des polynômes de Macdonald symétriques $P_{\lambda} (\mathbb{X} ;q,t)$ pour la spécialisation $t=q^k$. En particulier nous montrons une égalité reliant les polynômes $P_{\lambda} (\mathbb{X} ;q,q^k)$ et $P_{\lambda} (\frac{1-q}{1-q^k}\mathbb{X} ;q,q^k)$. Nous en déduisons la description d'un opérateur dont les valeurs propres caractérisent les polynômes $P_{\lambda} (\mathbb{X} ;q,q^k)$.

scholarly journals A Combinatorial Approach to the $q,t$-Symmetry Relation in Macdonald Polynomials

10.37236/5350 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Maria Monks Gillespie
Keyword(s):  
Macdonald Polynomials ◽  
Combinatorial Proof ◽  
Symmetry Relation ◽  
Combinatorial Approach ◽  
Combinatorial Formula ◽  
Recursive Structure ◽  
Littlewood Polynomials

Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation $\widetilde{H}_\mu(\mathbf{x};q,t)=\widetilde{H}_{\mu^\ast}(\mathbf{x};t,q)$. We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials ($q=0$) when $\mu$ is a partition with at most three rows, and for the coefficients of the square-free monomials in $\mathbf{x}$ for all shapes $\mu$. We also provide a proof for the full relation in the case when $\mu$ is a hook shape, and for all shapes at the specialization $t=1$. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.

scholarly journals A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

2012 ◽  
Vol 64 (4) ◽  
pp. 822-844 ◽  
Author(s):  
J. Haglund ◽  
J. Morse ◽  
M. Zabrocki
Keyword(s):  
Building Blocks ◽  
Main Diagonal ◽  
Combinatorial Theory ◽  
Dyck Path ◽  
Dyck Paths ◽  
Littlewood Polynomials ◽  
Diagonal Harmonics ◽  
Littlewood Polynomial ◽  
Theory Operator ◽  
Shuffle Conjecture

Abstract We introduce a q, t-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory ∇ operator applied to a Hall–Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the “shuffle conjecture” (Duke J. Math. 126 (2005), pp. 195 − 232) for ∇ en[X]. We bring to light that certain generalized Hall–Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of q, t-Catalan sequences, and we prove a number of identities involving these functions.

scholarly journals On form factors and Macdonald polynomials

2013 ◽  
Vol 2013 (9) ◽  
Author(s):  
Michael Lashkevich ◽  
Yaroslav Pugai
Keyword(s):  
Form Factors ◽  
Macdonald Polynomials
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