scholarly journals The quantum Knizhnik–Zamolodchikov equation and non-symmetric Macdonald polynomials

2019 ◽  
Author(s):  
Masahiro Kasatani ◽  
Yoshihiro Takeyama
Keyword(s):  
Macdonald Polynomials ◽  
Zamolodchikov Equation

scholarly journals The Quantum Knizhnik-Zamolodchikov Equation and Non-symmetric Macdonald Polynomials

2007 ◽  
Vol 50 (3) ◽  
pp. 491-509 ◽  
Author(s):  
Masahiro Kasatani ◽  
Yoshihiro Takeyama
Keyword(s):  
Macdonald Polynomials ◽  
Zamolodchikov Equation

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

On the “Drinfeld-Sokolov” Reduction of the Knizhnik-Zamolodchikov Equation

1993 ◽  
pp. 131-141 ◽  
Author(s):  
P. Furlan ◽  
A. Ch. Ganchev ◽  
R. Paunov ◽  
V. B. Petkova
Keyword(s):  
Zamolodchikov Equation

scholarly journals Kontsevich’s integral for the Kauffman polynomial

1996 ◽  
Vol 142 ◽  
pp. 39-65 ◽  
Author(s):  
Thang Tu Quoc Le ◽  
Jun Murakami
Keyword(s):  
Finite Type ◽  
Knot Invariants ◽  
Chord Diagrams ◽  
Knot Invariant ◽  
Kauffman Polynomial ◽  
Zamolodchikov Equation

Kontsevich’s integral is a knot invariant which contains in itself all knot invariants of finite type, or Vassiliev’s invariants. The value of this integral lies in an algebra A0, spanned by chord diagrams, subject to relations corresponding to the flatness of the Knizhnik-Zamolodchikov equation, or the so called infinitesimal pure braid relations [11].

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.

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