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Macdonald Polynomials
In mathematics, Macdonald polynomials ''P''λ(''x''; ''t'',''q'') are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald originally associated his polynomials with weights λ of finite root systems and used just one variable ''t'', but later realized that it is more natural to associate them with affine root systems rather than finite root systems, in which case the variable ''t'' can be replaced by several different variables ''t''=(''t''1,...,''t''''k''), one for each of the ''k'' orbits of roots in the affine root system. The Macdonald polynomials are polynomials in ''n'' variables ''x''=(''x''1,...,''x''''n''), where ''n'' is the rank of the affine root system. They generalize many other families of orthogonal polynomials, such as Jack polynomials and Hall–Littlewood polynomials and Askey–Wilson polynomials, which in turn include most of the named 1-va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Pafnuty Chebyshev, P. L. Chebyshev and was pursued by Andrey Markov, A. A. Markov and Thomas Joannes Stieltjes, T. J. Stieltjes. They appear in a wide variety of fields: numerical analysis (Gaussian quadrature, quadrature rules), probability theory, representation theory (of Lie group, Lie groups, quantum group, quantum groups, and re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Pochhammer Symbol
In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symbol (x)_n = x(x+1)\dots(x+n-1), in the sense that \lim_ \frac = (x)_n. The ''q''-Pochhammer symbol is a major building block in the construction of ''q''-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series. Unlike the ordinary Pochhammer symbol, the ''q''-Pochhammer symbol can be extended to an infinite product: (a;q)_\infty = \prod_^ (1-aq^k). This is an analytic function of ''q'' in the interior of the unit disk, and can also be considered as a formal power series in ''q''. The special case \phi(q) = (q;q)_\infty=\prod_^\infty (1-q^k) is known as Euler's function, and is important in combinatorics, number theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isospectral Hilbert Scheme
In mathematics, two linear operators are called isospectral or cospectral if they have the same spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues, when those are counted with multiplicity. The theory of isospectral operators is markedly different depending on whether the space is finite or infinite dimensional. In finite-dimensions, one essentially deals with square matrices. In infinite dimensions, the spectrum need not consist solely of isolated eigenvalues. However, the case of a compact operator on a Hilbert space (or Banach space) is still tractable, since the eigenvalues are at most countable with at most a single limit point λ = 0. The most studied isospectral problem in infinite dimensions is that of the Laplace operator on a domain in R2. Two such domains are called isospectral if their Laplacians are isospectral. The problem of inferring the geometrical properties of a domain from the spectrum of its Laplacian is often kno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adriano Garsia
Adriano Mario Garsia (born 20 August 1928) is a Tunisian-born Italian American mathematician who works in analysis, combinatorics, representation theory, and algebraic geometry. He is a student of Charles Loewner and has published work on representation theory, symmetric functions, and algebraic combinatorics. He and Mark Haiman made the N!_conjecture. He is also the namesake of the Garsia–Wachs algorithm for optimal binary search trees, which he published with his student Michelle L. Wachs in 1977. Born to Italian Tunisians in Tunis on 20 August 1928, Garsia moved to Rome in 1946. , he had 36 students and at least 200 descendants, according to the data at the Mathematics Genealogy Project. He was on the faculty of the University of California, San Diego. He retired in 2013 after 57 years at UCSD as a founding member of the Mathematics Department. At his 90 Birthday Conference in 2019, it was notable that he was the oldest principal investigator of a grant from the Nati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N! Conjecture
In mathematics, the ''n''! conjecture is the conjecture that the dimension of a certain bi-graded module of diagonal harmonics is ''n''!. It was made by A. M. Garsia and M. Haiman and later proved by M. Haiman. It implies Macdonald's positivity conjecture about the Macdonald polynomials. Formulation and background The Macdonald polynomials P_\lambda are a two-parameter family of orthogonal polynomials indexed by a positive weight λ of a root system, introduced by Ian G. Macdonald (1987). They generalize several other families of orthogonal polynomials, such as Jack polynomials and Hall–Littlewood polynomials. They are known to have deep relationships with affine Hecke algebras and Hilbert schemes, which were used to prove several conjectures made by Macdonald about them. introduced a new basis for the space of symmetric functions, which specializes to many of the well-known bases for the symmetric functions, by suitable substitutions for the parameters ''q'' and ''t'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mark Haiman
Mark David Haiman is a mathematician at the University of California at Berkeley who proved the Macdonald positivity conjecture for Macdonald polynomials. He received his Ph.D in 1984 in the Massachusetts Institute of Technology under the direction of Gian-Carlo Rota. Previous to his appointment at Berkeley, he held positions at the University of California, San Diego and the Massachusetts Institute of Technology. In 2004 he received the inaugural AMS Moore Prize. In 2012 he became a fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, .... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Polynomial
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in ''n'' variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible representations of the general linear groups. The Schur polynomials form a linear basis for the space of all symmetric polynomials. Any product of Schur polynomials can be written as a linear combination of Schur polynomials with non-negative integral coefficients; the values of these coefficients is given combinatorially by the Littlewood–Richardson rule. More generally, skew Schur polynomials are associated with pairs of partitions and have similar properties to Schur polynomials. Definition (Jacobi's bialternant formula) Schur polynomials are indexed by integer partitions. Given a partition , where , and each is a non-negative integer, the functions a_ (x_1, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Formula For The Macdonald Polynomials
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Affine Hecke Algebra
In mathematics, a double affine Hecke algebra, or Cherednik algebra, is an algebra containing the Hecke algebra of an affine Weyl group, given as the quotient of the group ring of a double affine braid group. They were introduced by Cherednik, who used them to prove Macdonald's constant term conjecture for Macdonald polynomials. Infinitesimal Cherednik algebras have significant implications in representation theory, and therefore have important applications in particle physics and in chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions .... References * * *A. A. KirilloLectures on affine Hecke algebras and Macdonald's conjectures Bull. Amer. Math. Soc. 34 (1997), 251–292. * Macdonald, I. G. ''Affine Hecke algebras and orthogonal polynomials.'' Cambridge Tracts in Mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dyson Conjecture
In mathematics, the Dyson conjecture is a conjecture about the constant term of certain Laurent polynomials, proved independently in 1962 by Kenneth G. Wilson, Wilson and Gunson. George Andrews (mathematician), Andrews generalized it to the q-Dyson conjecture, proved by Doron Zeilberger, Zeilberger and David Bressoud, Bressoud and sometimes called the Zeilberger–Bressoud theorem. Ian G. Macdonald, Macdonald generalized it further to more general root systems with the Macdonald constant term conjecture, proved by Ivan Cherednik, Cherednik. Dyson conjecture The Dyson conjecture states that the Laurent polynomial :\prod _(1-t_i/t_j)^ has constant term :\frac. The conjecture was first proved independently by and . later found a short proof, by observing that the Laurent polynomials, and therefore their constant terms, satisfy the recursion relations :F(a_1,\dots,a_n) = \sum_^nF(a_1,\dots,a_i-1,\dots,a_n). The case ''n'' = 3 of Dyson's conjecture follows from the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rogers Polynomials
In mathematics, the Rogers polynomials, also called Rogers–Askey–Ismail polynomials and continuous q-ultraspherical polynomials, are a family of orthogonal polynomials introduced by in the course of his work on the Rogers–Ramanujan identities. They are ''q''-analogs of ultraspherical polynomials, and are the Macdonald polynomials for the special case of the ''A''1 affine root system . and discuss the properties of Rogers polynomials in detail. Definition The Rogers polynomials can be defined in terms of the ''q''-Pochhammer symbol and the basic hypergeometric series In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called ... by : C_n(x;\beta, q) = \frace^ _2\phi_1(q^,\beta;\beta^q^;q,q\beta^e^) where ''x'' = cos(''θ''). References * * * * * *{{Citation , last1=Rogers , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heckman–Opdam Polynomials
In mathematics, Heckman–Opdam polynomials (sometimes called Jacobi polynomials) ''P''λ(''k'') are orthogonal polynomials in several variables associated to root systems. They were introduced by . They generalize Jack polynomials when the roots system is of type ''A'', and are limits of Macdonald polynomials In mathematics, Macdonald polynomials ''P''λ(''x''; ''t'',''q'') are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald origi ... ''P''λ(''q'', ''t'') as ''q'' tends to 1 and (1 − ''t'')/(1 − ''q'') tends to ''k''. Main properties of the Heckman–Opdam polynomials have been detailed by Siddhartha Sahi A new formula for weight multiplicities and characters, Theorem 1.3. about Heckman–Opdam polynomials, Siddhartha Sahi References * * * * {{DEFAULTSORT:Heckman-Opdam polynomials Orthogonal polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
