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Symmetric Functions and Orthogonal Polynomials

1998 Mathematics
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Release : 1998
Genre : Mathematics
Kind : eBook
Book Rating : 706/5 ( reviews)

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Book Synopsis Symmetric Functions and Orthogonal Polynomials by : Ian Grant Macdonald

Download or read book Symmetric Functions and Orthogonal Polynomials written by Ian Grant Macdonald. This book was released on 1998. Available in PDF, EPUB and Kindle. Book excerpt: One of the most classical areas of algebra, the theory of symmetric functions and orthogonal polynomials, has long been known to be connected to combinatorics, representation theory and other branches of mathematics. Written by perhaps the most famous author on the topic, this volume explains some of the current developments regarding these connections. It is based on lectures presented by the author at Rutgers University. Specifically, he gives recent results on orthogonal polynomials associated with affine Hecke algebras, surveying the proofs of certain famous combinatorial conjectures.

Symmetric Functions and Combinatorial Operators on Polynomials

2003 Mathematics
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Release : 2003
Genre : Mathematics
Kind : eBook
Book Rating : 711/5 ( reviews)

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Book Synopsis Symmetric Functions and Combinatorial Operators on Polynomials by : Alain Lascoux

Download or read book Symmetric Functions and Combinatorial Operators on Polynomials written by Alain Lascoux. This book was released on 2003. Available in PDF, EPUB and Kindle. Book excerpt: The theory of symmetric functions is an old topic in mathematics, which is used as an algebraic tool in many classical fields. With $\lambda$-rings, one can regard symmetric functions as operators on polynomials and reduce the theory to just a handful of fundamental formulas. One of the main goals of the book is to describe the technique of $\lambda$-rings. The main applications of this technique to the theory of symmetric functions are related to the Euclid algorithm and its occurrence in division, continued fractions, Pade approximants, and orthogonal polynomials. Putting the emphasis on the symmetric group instead of symmetric functions, one can extend the theory to non-symmetric polynomials, with Schur functions being replaced by Schubert polynomials. In two independent chapters, the author describes the main properties of these polynomials, following either the approach of Newton and interpolation methods, or the method of Cauchy and the diagonalization of a kernel generalizing the resultant. The last chapter sketches a non-commutative version of symmetric functions, with the help of Young tableaux and the plactic monoid. The book also contains numerous exercises clarifying and extending many points of the main text.

Symmetric Functions and Hall Polynomials

1998 Mathematics
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Release : 1998
Genre : Mathematics
Kind : eBook
Book Rating : 504/5 ( reviews)

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Book Synopsis Symmetric Functions and Hall Polynomials by : Ian Grant Macdonald

Download or read book Symmetric Functions and Hall Polynomials written by Ian Grant Macdonald. This book was released on 1998. Available in PDF, EPUB and Kindle. Book excerpt: This reissued classic text is the acclaimed second edition of Professor Ian Macdonald's groundbreaking monograph on symmetric functions and Hall polynomials. The first edition was published in 1979, before being significantly expanded into the present edition in 1995. This text is widely regarded as the best source of information on Hall polynomials and what have come to be known as Macdonald polynomials, central to a number of key developments in mathematics and mathematical physics in the 21st century Macdonald polynomials gave rise to the subject of double affine Hecke algebras (or Cherednik algebras) important in representation theory. String theorists use Macdonald polynomials to attack the so-called AGT conjectures. Macdonald polynomials have been recently used to construct knot invariants. They are also a central tool for a theory of integrable stochastic models that have found a number of applications in probability, such as random matrices, directed polymers in random media, driven lattice gases, and so on. Macdonald polynomials have become a part of basic material that a researcher simply must know if (s)he wants to work in one of the above domains, ensuring this new edition will appeal to a very broad mathematical audience. Featuring a new foreword by Professor Richard Stanley of MIT.

Laredo Lectures on Orthogonal Polynomials and Special Functions

2004 Mathematics
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Release : 2004
Genre : Mathematics
Kind : eBook
Book Rating : 097/5 ( reviews)

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Book Synopsis Laredo Lectures on Orthogonal Polynomials and Special Functions by : Renato Alvarez-Nodarse

Download or read book Laredo Lectures on Orthogonal Polynomials and Special Functions written by Renato Alvarez-Nodarse. This book was released on 2004. Available in PDF, EPUB and Kindle. Book excerpt: This new book presents research in orthogonal polynomials and special functions. Recent developments in the theory and accomplishments of the last decade are pointed out and directions for research in the future are identified. The topics covered include matrix orthogonal polynomials, spectral theory and special functions, Asymptotics for orthogonal polynomials via Riemann-Hilbert methods, Polynomial wavelets and Koornwinder polynomials.

$q$-Difference Operators, Orthogonal Polynomials, and Symmetric Expansions

2002 Mathematics
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Release : 2002
Genre : Mathematics
Kind : eBook
Book Rating : 74X/5 ( reviews)

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Book Synopsis $q$-Difference Operators, Orthogonal Polynomials, and Symmetric Expansions by : Douglas Bowman

Download or read book $q$-Difference Operators, Orthogonal Polynomials, and Symmetric Expansions written by Douglas Bowman. This book was released on 2002. Available in PDF, EPUB and Kindle. Book excerpt: The author explores ramifications and extensions of a $q$-difference operator method first used by L.J. Rogers for deriving relationships between special functions involving certain fundamental $q$-symmetric polynomials. In special cases these symmetric polynomials reduce to well-known classes of orthogonal polynomials. A number of basic properties of these polynomials follow from this approach. This leads naturally to the evaluation of the Askey-Wilson integral and generalizations. Expansions of certain generalized basic hypergeometric functions in terms of the symmetric polynomials are also found. This provides a quick route to understanding the group structure generated by iterating the two-term transformations of these functions. Some infrastructure is also laid for more general investigations in the future

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