Symmetric Functions and Hall Polynomials
Symmetric Functions and Hall Polynomials

Author: Ian Grant Macdonald

Publisher: Oxford University Press

Published: 1998

Total Pages: 496

ISBN-13: 9780198504504

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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.

An Introduction to Symmetric Functions and Their Combinatorics
An Introduction to Symmetric Functions and Their Combinatorics

Author: Eric S. Egge

Publisher: American Mathematical Soc.

Published: 2019-11-18

Total Pages: 359

ISBN-13: 1470448998

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This book is a reader-friendly introduction to the theory of symmetric functions, and it includes fundamental topics such as the monomial, elementary, homogeneous, and Schur function bases; the skew Schur functions; the Jacobi–Trudi identities; the involution ω ω; the Hall inner product; Cauchy's formula; the RSK correspondence and how to implement it with both insertion and growth diagrams; the Pieri rules; the Murnaghan–Nakayama rule; Knuth equivalence; jeu de taquin; and the Littlewood–Richardson rule. The book also includes glimpses of recent developments and active areas of research, including Grothendieck polynomials, dual stable Grothendieck polynomials, Stanley's chromatic symmetric function, and Stanley's chromatic tree conjecture. Written in a conversational style, the book contains many motivating and illustrative examples. Whenever possible it takes a combinatorial approach, using bijections, involutions, and combinatorial ideas to prove algebraic results. The prerequisites for this book are minimal—familiarity with linear algebra, partitions, and generating functions is all one needs to get started. This makes the book accessible to a wide array of undergraduates interested in combinatorics.

Symmetric Functions and Orthogonal Polynomials
Symmetric Functions and Orthogonal Polynomials

Author: Ian Grant Macdonald

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 71

ISBN-13: 0821807706

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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.

The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics
The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics

Author: James Haglund

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 178

ISBN-13: 0821844113

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This work contains detailed descriptions of developments in the combinatorics of the space of diagonal harmonics, a topic at the forefront of current research in algebraic combinatorics. These developments have led in turn to some surprising discoveries in the combinatorics of Macdonald polynomials.

Symmetric Functions and Combinatorial Operators on Polynomials
Symmetric Functions and Combinatorial Operators on Polynomials

Author: Alain Lascoux

Publisher: American Mathematical Soc.

Published: 2003

Total Pages: 282

ISBN-13: 0821828711

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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.

Algebraic Combinatorics and Coinvariant Spaces
Algebraic Combinatorics and Coinvariant Spaces

Author: Francois Bergeron

Publisher: CRC Press

Published: 2009-07-06

Total Pages: 227

ISBN-13: 1439865078

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Written for graduate students in mathematics or non-specialist mathematicians who wish to learn the basics about some of the most important current research in the field, this book provides an intensive, yet accessible, introduction to the subject of algebraic combinatorics. After recalling basic notions of combinatorics, representation theory, and

Vertex Operator Algebras and the Monster
Vertex Operator Algebras and the Monster

Author: Igor Frenkel

Publisher: Academic Press

Published: 1989-05-01

Total Pages: 563

ISBN-13: 0080874541

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This work is motivated by and develops connections between several branches of mathematics and physics--the theories of Lie algebras, finite groups and modular functions in mathematics, and string theory in physics. The first part of the book presents a new mathematical theory of vertex operator algebras, the algebraic counterpart of two-dimensional holomorphic conformal quantum field theory. The remaining part constructs the Monster finite simple group as the automorphism group of a very special vertex operator algebra, called the "moonshine module" because of its relevance to "monstrous moonshine."

Young Tableaux
Young Tableaux

Author: William Fulton

Publisher: Cambridge University Press

Published: 1997

Total Pages: 276

ISBN-13: 9780521567244

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Describes combinatorics involving Young tableaux and their uses in representation theory and algebraic geometry.

Affine Hecke Algebras and Orthogonal Polynomials
Affine Hecke Algebras and Orthogonal Polynomials

Author: I. G. Macdonald

Publisher: Cambridge University Press

Published: 2003-03-20

Total Pages: 200

ISBN-13: 9780521824729

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First account of a theory, created by Macdonald, of a class of orthogonal polynomial, which is related to mathematical physics.

The Symmetric Group
The Symmetric Group

Author: Bruce E. Sagan

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 254

ISBN-13: 1475768044

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This book brings together many of the important results in this field. From the reviews: ""A classic gets even better....The edition has new material including the Novelli-Pak-Stoyanovskii bijective proof of the hook formula, Stanley’s proof of the sum of squares formula using differential posets, Fomin’s bijective proof of the sum of squares formula, group acting on posets and their use in proving unimodality, and chromatic symmetric functions." --ZENTRALBLATT MATH