Results 1 to 10 of about 77,378 (232)
Combinatorial formula for Macdonald polynomials, Bethe Ansatz, and generic Macdonald polynomials [PDF]
arXiv, 2000 We give a direct proof of the combinatorial formula for interpolation Macdonald polynomials by introducing certain polynomials, which we call generic Macdonald polynomials, which depend on $d$ additional parameters and specialize to all Macdonald polynomials of degree $d$.
Andreĭ Okounkov
arxiv +5 more sources
A probabilistic interpretation of the Macdonald polynomials [PDF]
arXiv, 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 polynomials.
Persi Diaconis, Arun Ram
arxiv +7 more sources
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
doaj +6 more sources
On generalized Macdonald polynomials [PDF]
Journal of High Energy Physics, 2020 Generalized Macdonald polynomials (GMP) are eigenfunctions of specificallydeformed Ruijsenaars Hamiltonians and are built as triangular polylinear combinations of Macdonald polynomials.
A. Mironov, A. Morozov
doaj +5 more sources
On combinatorial formulas for Macdonald polynomials [PDF]
Advances in Mathematics, 2008 A recent 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 a pair of statistics on fillings of Young diagrams. Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of so-called alcove walks; these ...
Cristian Lenart
+7 more sources
A combinatorial model for the Macdonald polynomials. [PDF]
Proc Natl Acad Sci U S A, 2004 We introduce a polynomialC̃μ[Z;q, t], depending on a set of variablesZ=z1,z2,..., a partition μ, and two extra parametersq, t. The definition ofC̃μinvolves a pair of statistics (maj(σ, μ), inv(σ, μ)) on words σ of positive integers, and the coefficients of theziare manifestly in\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym ...
Haglund J.
europepmc +5 more sources
Breakthroughs in the theory of Macdonald polynomials. [PDF]
Proc Natl Acad Sci U S A, 2005 In 1998, I. G. Macdonald (1) introduced a remarkable new basis for the space of symmetric functions. The elements of this basis are denoted , where λ is a partition and p, q are two free parameters. The 's, which are now called “Macdonald polynomials,” specialize to many of the well known bases for the symmetric functions, by suitable choices of the ...
Garsia A, Remmel JB.
europepmc +5 more sources
Macdonald polynomials and algebraic integrability
Advances in Mathematics, 2002 54 ...
Oleg Chalykh
openalex +5 more sources
Paths and Kostka–Macdonald Polynomials [PDF]
Moscow Mathematical Journal, 2009 We give several equivalent combinatorial descriptions of the space of states for the box-ball systems, and connect certain partition functions for these models with the q-weight multiplicities of the tensor product of the fundamental representations of the Lie algebra gl(n).
Alexander Kirillov, R. Sakamoto
openalex +4 more sources
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
doaj +3 more sources