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Symmetric Functions
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Book Synopsis An Introduction to Symmetric Functions and Their Combinatorics by : Eric S. Egge
Download or read book An Introduction to Symmetric Functions and Their Combinatorics written by Eric S. Egge and published by American Mathematical Soc.. This book was released on 2019-11-18 with total page 359 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
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 and published by American Mathematical Soc.. This book was released on 2003 with total page 282 pages. 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.
Book Synopsis Symmetric Functions, Schubert Polynomials and Degeneracy Loci by : Laurent Manivel
Download or read book Symmetric Functions, Schubert Polynomials and Degeneracy Loci written by Laurent Manivel and published by American Mathematical Soc.. This book was released on 2001 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text grew out of an advanced course taught by the author at the Fourier Institute (Grenoble, France). It serves as an introduction to the combinatorics of symmetric functions, more precisely to Schur and Schubert polynomials. Also studied is the geometry of Grassmannians, flag varieties, and especially, their Schubert varieties. This book examines profound connections that unite these two subjects. The book is divided into three chapters. The first is devoted to symmetricfunctions and especially to Schur polynomials. These are polynomials with positive integer coefficients in which each of the monomials correspond to a Young tableau with the property of being ``semistandard''. The second chapter is devoted to Schubert polynomials, which were discovered by A. Lascoux andM.-P. Schutzenberger who deeply probed their combinatorial properties. It is shown, for example, that these polynomials support the subtle connections between problems of enumeration of reduced decompositions of permutations and the Littlewood-Richardson rule, a particularly efficacious version of which may be derived from these connections. The final chapter is geometric. It is devoted to Schubert varieties, subvarieties of Grassmannians, and flag varieties defined by certain incidenceconditions with fixed subspaces. This volume makes accessible a number of results, creating a solid stepping stone for scaling more ambitious heights in the area. The author's intent was to remain elementary: The first two chapters require no prior knowledge, the third chapter uses some rudimentary notionsof topology and algebraic geometry. For this reason, a comprehensive appendix on the topology of algebraic varieties is provided. This book is the English translation of a text previously published in French.
Book Synopsis The Symmetric Group by : Bruce E. Sagan
Download or read book The Symmetric Group written by Bruce E. Sagan and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis Harmonic Analysis and Special Functions on Symmetric Spaces by : Gerrit Heckman
Download or read book Harmonic Analysis and Special Functions on Symmetric Spaces written by Gerrit Heckman and published by Academic Press. This book was released on 1995-02-08 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: The two parts of this sharply focused book, Hypergeometric and Special Functions and Harmonic Analysis on Semisimple Symmetric Spaces, are derived from lecture notes for the European School of Group Theory, a forum providing high-level courses on recent developments in group theory. The authors provide students and researchers with a thorough and thoughtful overview, elaborating on the topic with clear statements of definitions and theorems and augmenting these withtime-saving examples. An extensive set of notes supplements the text.Heckman and Schlichtkrull extend the ideas of harmonic analysis on semisimple symmetric spaces to embrace the theory of hypergeometric and spherical functions and show that the K-variant Eisenstein integrals for G/H are hypergeometric functions under this theory. They lead readers from the fundamentals of semisimple symmetric spaces of G/H to the frontier, including generalization, to the Riemannian case. This volume will interest harmonic analysts, those working on or applying the theory of symmetric spaces; it will also appeal to those with an interest in special functions.Extends ideas of harmonic analysis on symmetric spacesFirst treatment of the theory to include hypergeometric and spherical functionsLinks algebraic, analytic, and geometric methods
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 and published by American Mathematical Soc.. This book was released on 1998 with total page 71 pages. 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.
Book Synopsis Symmetric Functions by : Evgeny Smirnov
Download or read book Symmetric Functions written by Evgeny Smirnov and published by Springer Nature. This book was released on 2024 with total page 159 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to combinatorial aspects of the theory of symmetric functions. This rich, interesting and highly nontrivial part of algebraic combinatorics has numerous applications to algebraic geometry, topology, representation theory and other areas of mathematics. Along with classical material, such as Schur polynomials and Young diagrams, less standard subjects are also covered, including Schubert polynomials and Danilov–Koshevoy arrays. Requiring only standard prerequisites in algebra and discrete mathematics, the book will be accessible to undergraduate students and can serve as a basis for a semester-long course. It contains more than a hundred exercises of various difficulty, with hints and solutions. Primarily aimed at undergraduate and graduate students, it will also be of interest to anyone who wishes to learn more about modern algebraic combinatorics and its usage in other areas of mathematics.
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 and published by Oxford University Press. This book was released on 1998 with total page 496 pages. 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.
Book Synopsis Symmetric Functions 2001: Surveys of Developments and Perspectives by : Sergey Fomin
Download or read book Symmetric Functions 2001: Surveys of Developments and Perspectives written by Sergey Fomin and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: Proceedings of the NATO Advanced Study Institute, held in Cambridge, UK, from 25th June to 6th July, 2001
Book Synopsis Symmetric Properties of Real Functions by : Brian thomson
Download or read book Symmetric Properties of Real Functions written by Brian thomson and published by CRC Press. This book was released on 2020-08-26 with total page 474 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work offers detailed coverage of every important aspect of symmetric structures in function of a single real variable, providing a historical perspective, proofs and useful methods for addressing problems. It provides assistance for real analysis problems involving symmetric derivatives, symmetric continuity and local symmetric structure of sets or functions.
Book Synopsis The Symmetric Function Tables of the Fifteenthic by : Floyd Fiske Decker
Download or read book The Symmetric Function Tables of the Fifteenthic written by Floyd Fiske Decker and published by . This book was released on 1910 with total page 68 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Current Trends in Symmetric Polynomials with Their Applications Ⅱ by : Taekyun Kim
Download or read book Current Trends in Symmetric Polynomials with Their Applications Ⅱ written by Taekyun Kim and published by MDPI. This book was released on 2021-03-19 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: The special issue contains research papers with various topics in many different branches of mathematics, applied mathematics, and mathematical physics. Each paper presents mathematical theory, methods, and their application based on current and recent developing symmetric polynomials. Also, each one aims to provide the full understanding of current research problems, theories, and applications on the chosen topics and contains the most recent advances made in the area of symmetric functions and polynomials.
Book Synopsis The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics by : James Haglund
Download or read book The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics written by James Haglund and published by American Mathematical Soc.. This book was released on 2008 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group by : Valery V. Volchkov
Download or read book Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group written by Valery V. Volchkov and published by Springer. This book was released on 2011-11-30 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of mean periodic functions is a subject which goes back to works of Littlewood, Delsarte, John and that has undergone a vigorous development in recent years. There has been much progress in a number of problems concerning local - pects of spectral analysis and spectral synthesis on homogeneous spaces. The study oftheseproblemsturnsouttobecloselyrelatedtoavarietyofquestionsinharmonic analysis, complex analysis, partial differential equations, integral geometry, appr- imation theory, and other branches of contemporary mathematics. The present book describes recent advances in this direction of research. Symmetric spaces and the Heisenberg group are an active ?eld of investigation at 2 the moment. The simplest examples of symmetric spaces, the classical 2-sphere S 2 and the hyperbolic plane H , play familiar roles in many areas in mathematics. The n Heisenberg groupH is a principal model for nilpotent groups, and results obtained n forH may suggest results that hold more generally for this important class of Lie groups. The purpose of this book is to develop harmonic analysis of mean periodic functions on the above spaces.
Book Synopsis Counting with Symmetric Functions by : Jeffrey Remmel
Download or read book Counting with Symmetric Functions written by Jeffrey Remmel and published by Birkhäuser. This book was released on 2015-11-28 with total page 297 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph provides a self-contained introduction to symmetric functions and their use in enumerative combinatorics. It is the first book to explore many of the methods and results that the authors present. Numerous exercises are included throughout, along with full solutions, to illustrate concepts and also highlight many interesting mathematical ideas. The text begins by introducing fundamental combinatorial objects such as permutations and integer partitions, as well as generating functions. Symmetric functions are considered in the next chapter, with a unique emphasis on the combinatorics of the transition matrices between bases of symmetric functions. Chapter 3 uses this introductory material to describe how to find an assortment of generating functions for permutation statistics, and then these techniques are extended to find generating functions for a variety of objects in Chapter 4. The next two chapters present the Robinson-Schensted-Knuth algorithm and a method for proving Pólya’s enumeration theorem using symmetric functions. Chapters 7 and 8 are more specialized than the preceding ones, covering consecutive pattern matches in permutations, words, cycles, and alternating permutations and introducing the reciprocity method as a way to define ring homomorphisms with desirable properties. Counting with Symmetric Functions will appeal to graduate students and researchers in mathematics or related subjects who are interested in counting methods, generating functions, or symmetric functions. The unique approach taken and results and exercises explored by the authors make it an important contribution to the mathematical literature.
Book Synopsis Subgroup Lattices and Symmetric Functions by : Lynne M. Butler
Download or read book Subgroup Lattices and Symmetric Functions written by Lynne M. Butler and published by American Mathematical Soc.. This book was released on 1994 with total page 173 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 Schützenberger's proof that Kostka polynomials are nonnegative, then discusses their monotonicity result and a conjecture on Macdonald's two-variable Kostka functions.
Book Synopsis Symmetric Functions in Statistics by : Derrick S. Tracy
Download or read book Symmetric Functions in Statistics written by Derrick S. Tracy and published by . This book was released on 1972 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt:
