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Charles F Dunkl - One of the best experts on this subject based on the ideXlab platform.
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some singular vector valued jack and Macdonald polynomials
Symmetry, 2019Co-Authors: Charles F DunklAbstract:For each partition τ of N, there are irreducible modules of the symmetric groups S N and of the corresponding Hecke algebra H N t whose bases consist of the reverse standard Young tableaux of shape τ . There are associated spaces of nonsymmetric Jack and Macdonald polynomials taking values in these modules. The Jack polynomials form a special case of the polynomials constructed by Griffeth for the infinite family G n , p , N of complex reflection groups. The Macdonald polynomials were constructed by Luque and the author. For each of the groups S N and the Hecke algebra H N t , there is a commutative set of Dunkl operators. The Jack and the Macdonald polynomials are parametrized by κ and q , t , respectively. For certain values of these parameters (called singular values), there are polynomials annihilated by each Dunkl operator; these are called singular polynomials. This paper analyzes the singular polynomials whose leading term is x 1 m ⊗ S , where S is an arbitrary reverse standard Young tableau of shape τ . The singular values depend on the properties of the edge of the Ferrers diagram of τ .
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some singular vector valued jack and Macdonald polynomials
arXiv: Representation Theory, 2019Co-Authors: Charles F DunklAbstract:For each partition $\tau$ of $N$ there are irreducible modules of the symmetric groups $\mathcal{S}_{N}$ or the corresponding Hecke algebra $\mathcal{H}_{N}\left( t\right) $ whose bases consist of reverse standard Young tableaux of shape $\tau$. There are associated spaces of nonsymmetric Jack and Macdonald polynomials taking values in these modules, respectively.The Jack polynomials are a special case of those constructed by Griffeth for the infinite family $G\left( n,p,N\right) $ of complex reflection groups. The Macdonald polynomials were constructed by Luque and the author. For both the group $\mathcal{S}_{N}$ and the Hecke algebra $\mathcal{H}_{N}\left( t\right) $ there is a commutative set of Dunkl operators. The Jack and the Macdonald polynomials are parametrized by $\kappa$ and $\left( q,t\right) $ respectively. For certain values of the parameters (called singular values) there are polynomials annihilated by each Dunkl operator; these are called singular polynomials. This paper analyzes the singular polynomials whose leading term is $x_{1}^{m}\otimes S$, where $S$ is an arbitrary reverse standard Young tableau of shape $\tau$. The singular values depend on properties of the edge of the Ferrers diagram of $\tau$.
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Some Singular Vector-Valued Jack and Macdonald Polynomials
MDPI AG, 2019Co-Authors: Charles F DunklAbstract:For each partition τ of N, there are irreducible modules of the symmetric groups S N and of the corresponding Hecke algebra H N t whose bases consist of the reverse standard Young tableaux of shape τ . There are associated spaces of nonsymmetric Jack and Macdonald polynomials taking values in these modules. The Jack polynomials form a special case of the polynomials constructed by Griffeth for the infinite family G n , p , N of complex reflection groups. The Macdonald polynomials were constructed by Luque and the author. For each of the groups S N and the Hecke algebra H N t , there is a commutative set of Dunkl operators. The Jack and the Macdonald polynomials are parametrized by κ and q , t , respectively. For certain values of these parameters (called singular values), there are polynomials annihilated by each Dunkl operator; these are called singular polynomials. This paper analyzes the singular polynomials whose leading term is x 1 m ⊗ S , where S is an arbitrary reverse standard Young tableau of shape τ . The singular values depend on the properties of the edge of the Ferrers diagram of τ
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Séminaire Lotharingien de Combinatoire 66 (2012), Article B66b VECTOR VALUED Macdonald POLYNOMIALS
2013Co-Authors: Charles F Dunkl, -g. J. LuqueAbstract:Abstract. This paper defines and investigates nonsymmetric Macdonald polynomials with values in an irreducible module of the Hecke algebra of type AN−1. These polynomials appear as simultaneous eigenfunctions of Cherednik operators. Several objects and properties are analyzed, such as the canonical bilinear form which pairs polynomials with those arising from reciprocals of the original parameters, and the symmetrization of the Macdonald polynomials. The main tool of the study is the Yang–Baxter graph. We show that these Macdonald polynomials can be easily computed following this graph. We give also an interpretation of the symmetrization and the bilinear forms applied to the Macdonald polynomials in terms of the Yang–Baxter graph. 1
Koppers A. A. P. - One of the best experts on this subject based on the ideXlab platform.
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"Missing links" for the long-lived Macdonald and Arago hotspots, South Pacific Ocean
'Geological Society of America', 2021Co-Authors: Buff L, Jackson M, Konrad K, Konter J, Bizimis M., Price A, Rose-koga Estelle, Blusztajn J., Koppers A. A. P.Abstract:Co-auteur étrangerInternational audienceThe Cook-Austral volcanic lineament extends from Macdonald Seamount (east) 18 to Aitutaki Island (west) in the South Pacific Ocean and consists of hotspot-related volcanic islands, seamounts, and atolls. The Cook-Australs [[lineament, or chain? Or Cook and Austal Islands?]] have been characterized as multiple overlapping, ageprogressive hotspot tracks generated by at least two mantle plumes, including the and Macdonald plumes, which have fed volcano construction for ~20 m.y. The Arago and Macdonald hotspot tracks are argued to have been active for at least 70 m.y. and toextend northwest of the Cook-Australs into the Cretaceous-aged Tuvalu-Gilbert and Tokelau Island chains, respectively. Large gaps in sampling exist along the predicted hotspot tracks, complicating efforts seeking to show that the Arago and Macdonald hotspots have been continuous, long-lived sources of hotspot volcanism back into the Cretaceous. We present new major- and trace-element concentrations and radiogenic isotopes for three seamounts (Moki, Malulu, Dino) and one atoll (Rose), and new clinopyroxene 40Ar/39 Ar ages for Rose (24.81 ± 1.02 Ma) and Moki (44.53 ± 10.05 Ma).All volcanoes are located in the poorly sampled region between the younger Cook Austral and the older, Cretaceous portions of the Arago and Macdonald hotspot tracks. Absolute plate motion modeling indicates that the Rose and Moki volcanoes lie on or near the reconstructed traces of the Arago and Macdonald hotspots, respectively, and the 40Ar/39 Ar ages for Rose and Moki align with the predicted age progression for the Arago (Rose) and Macdonald (Moki) hotspots, thereby linking the younger Cook-Austral and older Cretaceous portions of the long-lived (>70 m.y.) Arago and Macdonald hotspot tracks
Michael Wheeler - One of the best experts on this subject based on the ideXlab platform.
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a summation formula for Macdonald polynomials
Letters in Mathematical Physics, 2016Co-Authors: Jan De Gier, Michael WheelerAbstract:We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases \({t = 1}\) and \({q = 0}\), we recover known expressions for the monomial symmetric and Hall–Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and q–Whittaker polynomials.
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matrix product formula for Macdonald polynomials
arXiv: Mathematical Physics, 2015Co-Authors: Luigi Cantini, Jan De Gier, Michael WheelerAbstract:We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik--Zamolodchikov equations, which arise by considering representations of the Zamolodchikov--Faddeev and Yang--Baxter algebras in terms of $t$-deformed bosonic operators. These solutions form a basis of the ring of polynomials in $n$ variables, whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalisations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at $q=1$.
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matrix product formula for Macdonald polynomials
Journal of Physics A, 2015Co-Authors: Luigi Cantini, Jan De Gier, Michael WheelerAbstract:We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik-Zamolodchikov equations, which arise by considering representations of the Zamolodchikov-Faddeev and Yang-Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalised probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalisations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1.
Alexei Borodin - One of the best experts on this subject based on the ideXlab platform.
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nearest neighbor markov dynamics on Macdonald processes
Advances in Mathematics, 2016Co-Authors: Alexei Borodin, Leonid PetrovAbstract:Abstract Macdonald processes are certain probability measures on two-dimensional arrays of interlacing particles introduced by Borodin and Corwin in [7] . They are defined in terms of nonnegative specializations of the Macdonald symmetric functions and depend on two parameters q , t ∈ [ 0 ; 1 ) . Our main result is a classification of continuous time, nearest neighbor Markov dynamics on the space of interlacing arrays that act nicely on Macdonald processes. The classification unites known examples of such dynamics and also yields many new ones. When t = 0 , one dynamics leads to a new integrable interacting particle system on the one-dimensional lattice, which is a q-deformation of the PushTASEP (= long-range TASEP). When q = t , the Macdonald processes become the Schur processes of Okounkov and Reshetikhin [41] . In this degeneration, we discover new Robinson–Schensted-type correspondences between words and pairs of Young tableaux that govern some of our dynamics.
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general β jacobi corners process and the gaussian free field
Communications on Pure and Applied Mathematics, 2015Co-Authors: Alexei Borodin, Vadim GorinAbstract:We prove that the two-dimensional Gaussian free field describes the asymptotics of global fluctuations of a multilevel extension of the general β-Jacobi random matrix ensembles. Our approach is based on the connection of the Jacobi ensembles to a degeneration of the Macdonald processes that parallels the degeneration of the Macdonald polynomials to the Heckman-Opdam hypergeometric functions (of type A). We also discuss the β ∞ limit. © 2015 Wiley Periodicals, Inc.
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integrable probability from representation theory to Macdonald processes
arXiv: Probability, 2013Co-Authors: Alexei Borodin, Leonid PetrovAbstract:These are lecture notes for a mini-course given at the Cornell Probability Summer School in July 2013. Topics include lozenge tilings of polygons and their representation theoretic interpretation, the (q,t)-deformation of those leading to the Macdonald processes, nearest neighbor dynamics on Macdonald processes, their limit to semi-discrete Brownian polymers, and large time asymptotic analysis of polymer's partition function.
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Observables of Macdonald processes
arXiv: Probability, 2013Co-Authors: Alexei Borodin, Ivan Corwin, Vadim Gorin, Shamil ShakirovAbstract:We present a framework for computing averages of various observables of Macdonald processes. This leads to new contour--integral formulas for averages of a large class of multilevel observables, as well as Fredholm determinants for averages of two different single level observables.
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nearest neighbor markov dynamics on Macdonald processes
arXiv: Probability, 2013Co-Authors: Alexei Borodin, Leonid PetrovAbstract:Macdonald processes are certain probability measures on two-dimensional arrays of interlacing particles introduced by Borodin and Corwin (arXiv:1111.4408 [math.PR]). They are defined in terms of nonnegative specializations of the Macdonald symmetric functions and depend on two parameters (q,t), where 0<= q, t < 1. Our main result is a classification of continuous time, nearest neighbor Markov dynamics on the space of interlacing arrays that act nicely on Macdonald processes. The classification unites known examples of such dynamics and also yields many new ones. When t = 0, one dynamics leads to a new integrable interacting particle system on the one-dimensional lattice, which is a q-deformation of the PushTASEP (= long-range TASEP). When q = t, the Macdonald processes become the Schur processes of Okounkov and Reshetikhin (arXiv:math/0107056 [math.CO]). In this degeneration, we discover new Robinson--Schensted-type correspondences between words and pairs of Young tableaux that govern some of our dynamics.
Leonid Petrov - One of the best experts on this subject based on the ideXlab platform.
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nearest neighbor markov dynamics on Macdonald processes
Advances in Mathematics, 2016Co-Authors: Alexei Borodin, Leonid PetrovAbstract:Abstract Macdonald processes are certain probability measures on two-dimensional arrays of interlacing particles introduced by Borodin and Corwin in [7] . They are defined in terms of nonnegative specializations of the Macdonald symmetric functions and depend on two parameters q , t ∈ [ 0 ; 1 ) . Our main result is a classification of continuous time, nearest neighbor Markov dynamics on the space of interlacing arrays that act nicely on Macdonald processes. The classification unites known examples of such dynamics and also yields many new ones. When t = 0 , one dynamics leads to a new integrable interacting particle system on the one-dimensional lattice, which is a q-deformation of the PushTASEP (= long-range TASEP). When q = t , the Macdonald processes become the Schur processes of Okounkov and Reshetikhin [41] . In this degeneration, we discover new Robinson–Schensted-type correspondences between words and pairs of Young tableaux that govern some of our dynamics.
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integrable probability from representation theory to Macdonald processes
arXiv: Probability, 2013Co-Authors: Alexei Borodin, Leonid PetrovAbstract:These are lecture notes for a mini-course given at the Cornell Probability Summer School in July 2013. Topics include lozenge tilings of polygons and their representation theoretic interpretation, the (q,t)-deformation of those leading to the Macdonald processes, nearest neighbor dynamics on Macdonald processes, their limit to semi-discrete Brownian polymers, and large time asymptotic analysis of polymer's partition function.
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nearest neighbor markov dynamics on Macdonald processes
arXiv: Probability, 2013Co-Authors: Alexei Borodin, Leonid PetrovAbstract:Macdonald processes are certain probability measures on two-dimensional arrays of interlacing particles introduced by Borodin and Corwin (arXiv:1111.4408 [math.PR]). They are defined in terms of nonnegative specializations of the Macdonald symmetric functions and depend on two parameters (q,t), where 0<= q, t < 1. Our main result is a classification of continuous time, nearest neighbor Markov dynamics on the space of interlacing arrays that act nicely on Macdonald processes. The classification unites known examples of such dynamics and also yields many new ones. When t = 0, one dynamics leads to a new integrable interacting particle system on the one-dimensional lattice, which is a q-deformation of the PushTASEP (= long-range TASEP). When q = t, the Macdonald processes become the Schur processes of Okounkov and Reshetikhin (arXiv:math/0107056 [math.CO]). In this degeneration, we discover new Robinson--Schensted-type correspondences between words and pairs of Young tableaux that govern some of our dynamics.