|
Ian G. Macdonald
Ian Grant Macdonald (born 11 October 1928 in London, England) is a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebra, algebraic combinatorics, and combinatorics. He was educated at Winchester College and Trinity College, Cambridge, graduating in 1952. He then spent five years as a civil servant. He was offered a position at Manchester University in 1957 by Max Newman, on the basis of work he had done while outside academia. In 1960 he moved to the University of Exeter, and in 1963 became a Fellow of Magdalen College, Oxford. He became Fielden Professor at Manchester in 1972, and professor at Queen Mary College, University of London, in 1976. He worked on symmetric products of algebraic curves, Jordan algebras and the representation theory of groups over local fields. In 1972 he proved the Macdonald identities, after a pattern known to Freeman Dyson. His 1979 book ''Symmetric Functions a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ian Macdonald
Ian MacCormick (known by the pseudonym Ian MacDonald; 3 October 1948 – 20 August 2003) was a British music critic and author, best known for both ''Revolution in the Head'', his critical history of the Beatles which borrowed techniques from art historians, and ''The New Shostakovich'', a study of Russian composer Dmitri Shostakovich. Biography MacDonald briefly attended King's College, Cambridge, at first to study English, then archaeology and anthropology. He dropped out after a year. While at Cambridge, he was distantly acquainted with the singer/songwriter Nick Drake. From 1972 to 1975 he served as assistant editor at the ''NME''. MacDonald began a songwriting collaboration as lyricist with Quiet Sun, which included his brother Bill MacCormick and future Roxy Music guitarist Phil Manzanera. The collaboration resumed in the late 1970s, with MacDonald providing lyrics for the album ''Listen Now''. Later, Brian Eno assisted MacDonald in producing ''Sub Rosa'', an album of his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Products Of Algebraic Curves
In mathematics, the ''n''-fold symmetric product of an algebraic curve ''C'' is the quotient space of the ''n''-fold cartesian product :''C'' × ''C'' × ... × ''C'' or ''C''''n'' by the group action of the symmetric group ''S''''n'' on ''n'' letters permuting the factors. It exists as a smooth algebraic variety denoted by Σ''n''''C''. If ''C'' is a compact Riemann surface, Σ''n''''C'' is therefore a complex manifold. Its interest in relation to the classical geometry of curves is that its points correspond to effective divisors on ''C'' of degree ''n'', that is, formal sums of points with non-negative integer coefficients. For ''C'' the projective line (say the Riemann sphere \mathbb ∪ ≈ ''S''''2''), its nth symmetric product Σ''n''''C'' can be identified with complex projective space \mathbb^n of dimension ''n''. If ''G'' has genus ''g'' ≥ 1 then the Σ''n''''C'' are closely related to the Jacobian variety ''J'' of ''C''. More accurately for '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Macdonald Conjecture (other)
Macdonald conjecture may refer to one of several conjectures: *Macdonald's conjectures about Macdonald polynomials *Macdonald's generalization of the 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-Dy ... *Macdonald's generalization of the Mehta integral {{mathdab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Macdonald Polynomial
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] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Littlewood–Richardson Rule
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. Littlewood–Richardson coefficients depend on three partitions, say \lambda,\mu,\nu, of which \lambda and \mu describe the Schur functions being multiplied, and \nu gives the Schur function of which this is the coefficient in the linear combination; in other words they ... [...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] |
Hall Polynomial
In mathematics, the Hall algebra is an associative algebra with a basis corresponding to isomorphism classes of finite abelian ''p''-groups. It was first discussed by but forgotten until it was rediscovered by , both of whom published no more than brief summaries of their work. The Hall polynomials are the structure constants of the Hall algebra. The Hall algebra plays an important role in the theory of Masaki Kashiwara and George Lusztig regarding canonical bases in quantum groups. generalized Hall algebras to more general categories, such as the category of representations of a quiver. Construction A finite abelian ''p''-group ''M'' is a direct sum of cyclic ''p''-power components C_, where \lambda=(\lambda_1,\lambda_2,\ldots) is a partition of n called the ''type'' of ''M''. Let g^\lambda_(p) be the number of subgroups ''N'' of ''M'' such that ''N'' has type \nu and the quotient ''M/N'' has type \mu. Hall proved that the functions ''g'' are polynomial functions of ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices. K-theory involves the construction of families of ''K''-functors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the Grothendieck–Riemann–Roch theorem, Bott periodicity, the Atiyahâ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory Of Equations
In algebra, the theory of equations is the study of algebraic equations (also called "polynomial equations"), which are equations defined by a polynomial. The main problem of the theory of equations was to know when an algebraic equation has an algebraic solution. This problem was completely solved in 1830 by Évariste Galois, by introducing what is now called Galois theory. Before Galois, there was no clear distinction between the "theory of equations" and "algebra". Since then algebra has been dramatically enlarged to include many new subareas, and the theory of algebraic equations receives much less attention. Thus, the term "theory of equations" is mainly used in the context of the history of mathematics, to avoid confusion between old and new meanings of "algebra". History Until the end of the 19th century, "theory of equations" was almost synonymous with "algebra". For a long time, the main problem was to find the solutions of a single non-linear polynomial equation in a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Freeman Dyson
Freeman John Dyson (15 December 1923 – 28 February 2020) was an English-American theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrices, mathematical formulation of quantum mechanics, condensed matter physics, nuclear physics, and engineering. He was Professor Emeritus in the Institute for Advanced Study in Princeton and a member of the Board of Sponsors of the Bulletin of the Atomic Scientists. Dyson originated several concepts that bear his name, such as Dyson's transform, a fundamental technique in additive number theory, which he developed as part of his proof of Mann's theorem; the Dyson tree, a hypothetical genetically engineered plant capable of growing in a comet; the Dyson series, a perturbative series where each term is represented by Feynman diagrams; the Dyson sphere, a thought experiment that attempts to explain how a spacefaring, space-faring civilization would meet its energy requirements with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Macdonald Identities
In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by . They include as special cases the Jacobi triple product identity, Watson's quintuple product identity, several identities found by , and a 10-fold product identity found by . and pointed out that the Macdonald identities are the analogs of the Weyl denominator formula for affine Kac–Moody algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody a ...s and superalgebras. References * * * * * *{{Citation , last1=Winquist , first1=Lasse , title=An elementary proof of p(11m+6) ≡ 0 mod 11 , mr=0236136 , year=1969 , journal=Journal of Combinatorial Theory , volume=6 , pages=56–59 , doi=10.1016/s0021-9800(69)80105-5, doi-access=free Lie algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
