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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,...,''tk''), 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,...,''xn''), 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 orthogona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root System
In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory). Finally, root systems are important for their own sake, as in spectral graph theory. Definitions and examples As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right; call them roots. These vectors Li ... [...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. retrieved 2013-01-19. Selected publications *References External links * Haiman'[...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 ... [...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 as an end to 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 gr ... [...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 chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a .... 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 Mathem ... [...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 Wilson and Gunson. Andrews generalized it to the q-Dyson conjecture, proved by Zeilberger and Bressoud and sometimes called the Zeilberger–Bressoud theorem. Macdonald generalized it further to more general root systems with the Macdonald constant term conjecture, proved by 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 Dixon identity. and used a computer to find expressions for non-constant coefficients of Dyson's Laurent poly ... [...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-analog, ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is ... 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] |
Hall–Littlewood Polynomials
In mathematics, the Hall–Littlewood polynomials are symmetric functions depending on a parameter ''t'' and a partition λ. They are Schur functions when ''t'' is 0 and monomial symmetric functions when ''t'' is 1 and are special cases of Macdonald polynomials. They were first defined indirectly by Philip Hall using the Hall algebra, and later defined directly by Dudley E. Littlewood (1961). Definition The Hall–Littlewood polynomial ''P'' is defined by :P_\lambda(x_1,\ldots,x_n;t) = \left( \prod_ \prod_^ \frac \right) , where λ is a partition of at most ''n'' with elements λ''i'', and ''m''(''i'') elements equal to ''i'', and ''S''''n'' is the symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ... of order ''n''!. As an example, : P_(x_1,x_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zonal Spherical Function
In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group ''G'' with compact subgroup ''K'' (often a maximal compact subgroup) that arises as the matrix coefficient of a ''K''-invariant vector in an irreducible representation of ''G''. The key examples are the matrix coefficients of the ''Principal series representation, spherical principal series'', the irreducible representations appearing in the decomposition of the unitary representation of ''G'' on ''L''2(''G''/''K''). In this case the commutant of ''G'' is generated by the algebra of biinvariant functions on ''G'' with respect to ''K'' acting by right convolution. It is commutative if in addition ''G''/''K'' is a symmetric space, for example when ''G'' is a connected semisimple Lie group with finite centre and ''K'' is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Character
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra. In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation. Important consequences of the character formula are the Weyl dimension formula and the Kostant multiplicity formula. By definition, the character \chi of a representation \pi of ''G'' is the trace of \pi(g), as a function of a group element g\in G. The irreducible representations in this case are all finite-dimensional (this is part of the Peter–Weyl theorem); so the notion of trace is the usual one from linear algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
