Symmetric Functions and Polynomials (Mathematics Essentials) PDF Download
Are you looking for read ebook online? Search for your book and save it on your Kindle device, PC, phones or tablets. Download Symmetric Functions and Polynomials (Mathematics Essentials) PDF full book. Access full book title Symmetric Functions and Polynomials (Mathematics Essentials) by Alma Adams. Download full books in PDF and EPUB format.
Author: Alma Adams
Publisher: NY Research Press
ISBN: 9781647254629
Category : Mathematics
Languages : en
Pages : 0
Get Book
Book Description
A function containing several variables that remains unchanged for any permutation of the variables is called a symmetric function. Polynomials are a type of function. A symmetric polynomial refers to a type of polynomial P in n variables such that if any of the variables are swapped with each other, it remains the same polynomial. There are various types of symmetric polynomials including power-sum symmetric polynomials, elementary symmetric polynomials, complete homogeneous symmetric polynomials, monomial symmetric polynomials, and Schur polynomials. Symmetric polynomials have numerous applications in various areas of combinatorics, representation theory, mathematical physics, and mathematics. They are frequently found in Newton's identities and Vieta's formula. This book includes some of the vital pieces of works being conducted across the world, on various topics related to symmetric functions and polynomials, and their applications. It will serve as a valuable source of reference for graduate and postgraduate students.
Author: Alma Adams
Publisher: NY Research Press
ISBN: 9781647254629
Category : Mathematics
Languages : en
Pages : 0
Get Book
Book Description
A function containing several variables that remains unchanged for any permutation of the variables is called a symmetric function. Polynomials are a type of function. A symmetric polynomial refers to a type of polynomial P in n variables such that if any of the variables are swapped with each other, it remains the same polynomial. There are various types of symmetric polynomials including power-sum symmetric polynomials, elementary symmetric polynomials, complete homogeneous symmetric polynomials, monomial symmetric polynomials, and Schur polynomials. Symmetric polynomials have numerous applications in various areas of combinatorics, representation theory, mathematical physics, and mathematics. They are frequently found in Newton's identities and Vieta's formula. This book includes some of the vital pieces of works being conducted across the world, on various topics related to symmetric functions and polynomials, and their applications. It will serve as a valuable source of reference for graduate and postgraduate students.
Author: Ian Grant Macdonald
Publisher: Oxford University Press
ISBN: 9780198504504
Category : Mathematics
Languages : en
Pages : 496
Get Book
Book Description
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.
Author: Alain Lascoux
Publisher: American Mathematical Soc.
ISBN: 0821828711
Category : Polynomials
Languages : en
Pages : 282
Get Book
Book Description
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.
Author: Ian Grant Macdonald
Publisher: American Mathematical Soc.
ISBN: 0821807706
Category : Mathematics
Languages : en
Pages : 71
Get Book
Book Description
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.
Author: Eric S. Egge
Publisher: American Mathematical Soc.
ISBN: 1470448998
Category : Education
Languages : en
Pages : 342
Get Book
Book Description
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.
Author: Alma Adams
Publisher: Willford Press
ISBN: 9781647285296
Category : Mathematics
Languages : en
Pages : 0
Get Book
Book Description
Orthogonal polynomials are a family of polynomials, wherein any two different polynomials in the sequence are orthogonal to each other under some inner product. Classical orthogonal polynomials, Hermite polynomials, Laguerre polynomials, Jacobi polynomials, and Gegenbauer polynomials are a few examples of orthogonal polynomials. These polynomials are used for least square approximations of a function, difference equations, and Fourier series. Another major application of orthogonal polynomials is error-correcting code and sphere packing. Orthogonal polynomials and special functions are useful mathematical functions, which have applications in various fields such as mathematical physics, statistics and probability, and engineering. These can be used to explain many physical and chemical phenomena. This book traces the recent studies in orthogonal polynomials and special functions. A number of latest researches have been included to keep the readers updated with the latest concepts in this area of study. With state-of-the-art inputs by acclaimed experts of mathematics, this book targets students and professionals.
Author: Lynne M. Butler
Publisher: American Mathematical Soc.
ISBN: 082182600X
Category : Mathematics
Languages : en
Pages : 173
Get Book
Book Description
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.
Author: Jeffrey Remmel
Publisher: Birkhäuser
ISBN: 3319236180
Category : Mathematics
Languages : en
Pages : 292
Get Book
Book Description
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.
Author: Laurent Manivel
Publisher: American Mathematical Soc.
ISBN: 9780821821541
Category : Computers
Languages : en
Pages : 180
Get Book
Book Description
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.
Author: K. F. Riley
Publisher: Cambridge University Press
ISBN: 1139492942
Category : Science
Languages : en
Pages : 847
Get Book
Book Description
The mathematical methods that physical scientists need for solving substantial problems in their fields of study are set out clearly and simply in this tutorial-style textbook. Students will develop problem-solving skills through hundreds of worked examples, self-test questions and homework problems. Each chapter concludes with a summary of the main procedures and results and all assumed prior knowledge is summarized in one of the appendices. Over 300 worked examples show how to use the techniques and around 100 self-test questions in the footnotes act as checkpoints to build student confidence. Nearly 400 end-of-chapter problems combine ideas from the chapter to reinforce the concepts. Hints and outline answers to the odd-numbered problems are given at the end of each chapter, with fully-worked solutions to these problems given in the accompanying Student Solutions Manual. Fully-worked solutions to all problems, password-protected for instructors, are available at www.cambridge.org/essential.
