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In mathematics, Macdonald polynomialsPλ(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=(t1,...,tk), one for each of the k orbits of roots in the affine root system. The Macdonald polynomials are polynomials in n variables x=(x1,...,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 1-variable orthogonal polynomials as special cases. Koornwinder polynomials are Macdonald polynomials of certain non-reduced root systems. They have deep relationships with affine Hecke algebras and Hilbert schemes, which were used to prove several conjectures made by Macdonald about them.
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mathematics, Macdonaldpolynomials Pλ(x; t,q) are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He...
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affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonaldpolynomials. Let V {\displaystyle V} be a Euclidean space...
theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane. These polynomials are named after...
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elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed...
affine An−1 root system. Macdonald reformulated these conjectures as conjectures about the norms of Macdonaldpolynomials. Macdonald's conjectures were proved...
Rogers polynomials, also called Rogers–Askey–Ismail polynomials and continuous q-ultraspherical polynomials, are a family of orthogonal polynomials introduced...
to expand Macdonaldpolynomials in terms of LLT polynomials. Ian Grojnowski and Mark Haiman proved a positivity conjecture for LLT polynomials that combined...
a polynomial. In this context other collections of specific symmetric polynomials, such as complete homogeneous, power sum, and Schur polynomials play...
introduced by Cherednik, who used them to prove Macdonald's constant term conjecture for Macdonaldpolynomials. Infinitesimal Cherednik algebras have significant...
developed a new characterization of both symmetric and nonsymmetric Macdonaldpolynomials using the combinatorial exclusion process. Bergeron, Anne; Corteel...
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semisimple p-adic algebraic groups, and correspond to families of Macdonaldpolynomials. The reduced affine root systems were used by Kac and Moody in their...
affine braid groups. Haiman, Mark (2006), "Cherednik algebras, Macdonaldpolynomials and combinatorics", International Congress of Mathematicians, Vol...
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