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Symmetric Functions and Hall Polynomials

Author: Ian Grant Macdonald

Publisher: Oxford University Press

Published: 1998

Total Pages: 496

ISBN-13: 9780198504504

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.

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.

An Introduction to Symmetric Functions and Their Combinatorics

Author: Eric S. Egge

Publisher: American Mathematical Soc.

Published: 2019-11-18

Total Pages: 342

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.

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.

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

k-Schur Functions and Affine Schubert Calculus

Author: Thomas Lam

Publisher: Springer

Published: 2016-09-03

Total Pages: 0

ISBN-13: 9781493949724

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This book gives an introduction to the very active field of combinatorics of affine Schubert calculus, explains the current state of the art, and states the current open problems. Affine Schubert calculus lies at the crossroads of combinatorics, geometry, and representation theory. Its modern development is motivated by two seemingly unrelated directions. One is the introduction of k-Schur functions in the study of Macdonald polynomial positivity, a mostly combinatorial branch of symmetric function theory. The other direction is the study of the Schubert bases of the (co)homology of the affine Grassmannian, an algebro-topological formulation of a problem in enumerative geometry. This is the first introductory text on this subject. It contains many examples in Sage, a free open source general purpose mathematical software system, to entice the reader to investigate the open problems. This book is written for advanced undergraduate and graduate students, as well as researchers, who want to become familiar with this fascinating new field.

Bounded Littlewood Identities

Author: Eric M. Rains

Publisher: American Mathematical Soc.

Published: 2021-07-21

Total Pages: 115

ISBN-13: 1470446901

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We describe a method, based on the theory of Macdonald–Koornwinder polynomials, for proving bounded Littlewood identities. Our approach provides an alternative to Macdonald’s partial fraction technique and results in the first examples of bounded Littlewood identities for Macdonald polynomials. These identities, which take the form of decomposition formulas for Macdonald polynomials of type (R, S) in terms of ordinary Macdonald polynomials, are q, t-analogues of known branching formulas for characters of the symplectic, orthogonal and special orthogonal groups. In the classical limit, our method implies that MacMahon’s famous ex-conjecture for the generating function of symmetric plane partitions in a box follows from the identification of GL(n, R), O(n) as a Gelfand pair. As further applications, we obtain combinatorial formulas for characters of affine Lie algebras; Rogers–Ramanujan identities for affine Lie algebras, complementing recent results of Griffin et al.; and quadratic transformation formulas for Kaneko–Macdonald-type basic hypergeometric series.

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.

Subgroup Lattices and Symmetric Functions

Author: Lynne M. Butler

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 160

ISBN-13: 082182600X

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This work presents foundational research on two approaches to studying subgroup lattices of finite abelian $p$-groups. The first approach is linear algebraic in nature and generalizes Knuth's study of subspace lattices. This approach yields a combinatorial interpretation of the Betti polynomials of these Cohen-Macaulay posets. The second approach, which employs Hall-Littlewood symmetric functions, exploits properties of Kostka polynomials to obtain enumerative results such as rank-unimodality. Butler completes Lascoux and Schutzenberger's proof that Kostka polynomials are nonnegative, then discusses their monotonicity result and a conjecture on Macdonald's two-variable Kostka functions.

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.