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A combinatorial model for the Macdonald polynomials. [PDF]
Proc Natl Acad Sci U S A, 2004 We prove a combinatorial formula for the Macdonald polynomial H_mu(x;q,t) which had been conjectured by the first author. Corollaries to our main theorem include the expansion of H_mu(x;q,t) in terms of LLT polynomials, a new proof of the charge formula ...
Haglund J.
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A probabilistic interpretation of the Macdonald polynomials [PDF]
The Annals of Probability, 2010 The two-parameter Macdonald polynomials are a central object of algebraic combinatorics and representation theory. We give a Markov chain on partitions of k with eigenfunctions the coefficients of the Macdonald polynomials when expanded in the power sum ...
Diaconis, Persi, Ram, Arun
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Staircase Macdonald polynomials and the $q$-Discriminant [PDF]
Discrete Mathematics & Theoretical Computer Science, 2008 We prove that a $q$-deformation $\mathfrak{D}_k(\mathbb{X};q)$ of the powers of the discriminant is equal, up to a normalization, to a specialization of a Macdonald polynomial indexed by a staircase partition. We investigate the expansion of $\mathfrak{D}
Adrien Boussicault, Jean-Gabriel Luque
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Highest weight Macdonald and Jack Polynomials [PDF]
Journal of Physics A: Mathematical and Theoretical, 2011 Fractional quantum Hall states of particles in the lowest Landau levels are described by multivariate polynomials. The incompressible liquid states when described on a sphere are fully invariant under the rotation group.
Bernevig B A +11 more
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Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials [PDF]
Discrete Mathematics & Theoretical Computer Science, 2009 A breakthrough in the theory of (type $A$) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams.
Cristian Lenart
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Type A partially-symmetric Macdonald polynomials [PDF]
Algebraic Combinatorics, 2023 We construct type A partially-symmetric Macdonald polynomials $P_{(\lambda \mid \gamma)}$, where $\lambda \in \mathbb{Z}_{\geq 0}^{n-k}$ is a partition and $\gamma \in \mathbb{Z}_{\geq 0}^k$ is a composition.
Ben Goodberry
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A raising operator formula for Macdonald polynomials
Forum of Mathematics, SigmaWe give an explicit raising operator formula for the modified Macdonald polynomials $\tilde {H}_{\mu }(X;q,t)$ , which follows from our recent formula for $\nabla $ on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing ...
J. Blasiak +4 more
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Macdonald polynomials at $t=q^k$ [PDF]
Discrete Mathematics & Theoretical Computer Science, 2009 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 ...
Jean-Gabriel Luque
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Spin-Ruijsenaars, q-Deformed Haldane–Shastry and Macdonald Polynomials [PDF]
Communications in Mathematical Physics, 2020 We study the q-analogue of the Haldane–Shastry model, a partially isotropic (xxz-like) long-range spin chain that by construction enjoys quantum-affine (really: quantum-loop) symmetries at finite system size.
Jules Lamers, V. Pasquier, D. Serban
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A bijective proof of a factorization formula for Macdonald polynomials at roots of unity [PDF]
Discrete Mathematics & Theoretical Computer Science, 2008 We give a combinatorial proof of the factorization formula of modified Macdonald polynomials $\widetilde{H}_{\lambda} (X;q,t)$ when $t$ is specialized at a primitive root of unity.
F. Descouens, H. Morita, Y. Numata
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