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In mathematics, a Young tableau is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900. They were then applied to the study of the symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Schützenberger and Richard P. Stanley.
Ian Grant Macdonald was a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebra, algebraic combinatorics, and combinatorics.
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.
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.
In mathematics, an affine Hecke algebra is the algebra associated to an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials.
In mathematics, the Littelmann path model is a combinatorial device due to Peter Littelmann for computing multiplicities without overcounting in the representation theory of symmetrisable Kac–Moody algebras. Its most important application is to complex semisimple Lie algebras or equivalently compact semisimple Lie groups, the case described in this article. Multiplicities in irreducible representations, tensor products and branching rules can be calculated using a coloured directed graph, with labels given by the simple roots of the Lie algebra.
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural numbers, which the Littlewood–Richardson rule describes as counting certain skew tableaux. They occur in many other mathematical contexts, for instance as multiplicity in the decomposition of tensor products of finite-dimensional representations of general linear groups, or in the decomposition of certain induced representations in the representation theory of the symmetric group, or in the area of algebraic combinatorics dealing with Young tableaux and symmetric 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).
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.
In mathematics, Macdonald-Koornwinder polynomials (also called Koornwinder polynomials) are a family of orthogonal polynomials in several variables, introduced by Koornwinder and I. G. Macdonald, that generalize the Askey–Wilson polynomials. They are the Macdonald polynomials attached to the non-reduced affine root system of type (C∨
n, Cn), and in particular satisfy analogues of Macdonald's conjectures. In addition Jan Felipe van Diejen showed that the Macdonald polynomials associated to any classical root system can be expressed as limits or special cases of Macdonald-Koornwinder polynomials and found complete sets of concrete commuting difference operators diagonalized by them. Furthermore, there is a large class of interesting families of multivariable orthogonal polynomials associated with classical root systems which are degenerate cases of the Macdonald-Koornwinder polynomials. The Macdonald-Koornwinder polynomials have also been studied with the aid of affine Hecke algebras.
In mathematics, the plactic monoid is the monoid of all words in the alphabet of positive integers modulo Knuth equivalence. Its elements can be identified with semistandard Young tableaux. It was discovered by Donald Knuth, using an operation given by Craige Schensted in his study of the longest increasing subsequence of a permutation.
In mathematics, Kostka polynomials, named after the mathematician Carl Kostka, are families of polynomials that generalize the Kostka numbers. They are studied primarily in algebraic combinatorics and representation theory.
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them.
In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring with a countable number of variables. This ring generalizes the ring of symmetric functions. This ring can be realized as a specific limit of the rings of quasisymmetric polynomials in n variables, as n goes to infinity. This ring serves as universal structure in which relations between quasisymmetric polynomials can be expressed in a way independent of the number n of variables.
In mathematics, Schubert polynomials are generalizations of Schur polynomials that represent cohomology classes of Schubert cycles in flag varieties. They were introduced by Lascoux & Schützenberger (1982) and are named after Hermann Schubert.
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.
Alain Lascoux was a French mathematician at Université de Paris VII, University of Marne la Vallée and Nankai University. His research was primarily in algebraic combinatorics, particularly Hecke algebras and Young tableaux.
Julius Bogdan Borcea was a Romanian Swedish mathematician. His scientific work included vertex operator algebra and zero distribution of polynomials and entire functions, via correlation inequalities and statistical mechanics.
The E. H. Moore Research Article Prize, also called the Moore Prize, is one of twenty-two prizes given out by the American Mathematical Society (AMS). It recognizes an outstanding research article to have appeared in one of the AMS primary research journals during the previous six years. The prize was funded in 2002 in memory of the former AMS president E. H. Moore. Beginning in 2004, it is awarded every three years at the Joint Mathematics Meetings and carries a cash reward of $5,000.
References
- ↑ Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and Unipotent Varieties MR 1434225 J. Math. Phys. 38 (1997), no. 2, 1041–1068.
- ↑ J. Haglund, M. Haiman, N. Loehr A Combinatorial Formula for Macdonald Polynomials MR 2138143 J. Amer. Math. Soc. 18 (2005), no. 3, 735–761
- ↑ I. Grojnowski, M. Haiman, Affine algebras and positivity (preprint available here)
- I. Grojnowski, M. Haiman, Affine algebras and positivity (preprint available here)
- J. Haglund, M. Haiman, N. Loehr A Combinatorial Formula for Macdonald Polynomials MR 2138143 J. Amer. Math. Soc. 18 (2005), no. 3, 735–761
- Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and Unipotent Varieties MR 1434225 J. Math. Phys. 38 (1997), no. 2, 1041–1068.
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