Symmetric Functions And Combinatorial Operators On Polynomials PDF Download
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| Author | : Alain Lascoux |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 282 |
| Release | : 2003 |
| Genre | : Mathematics |
| ISBN | : 0821828711 |
Download Symmetric Functions and Combinatorial Operators on Polynomials Book in PDF, ePub and Kindle
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 | : Alain Lascoux |
| Publisher | : |
| Total Pages | : 268 |
| Release | : 2003 |
| Genre | : |
| ISBN | : 9780821828717 |
Download Symmetric functions and combinatorial operators on polynomials Book in PDF, ePub and Kindle
| Author | : Laurent Manivel |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 180 |
| Release | : 2001 |
| Genre | : Computers |
| ISBN | : 9780821821541 |
Download Symmetric Functions, Schubert Polynomials and Degeneracy Loci Book in PDF, ePub and Kindle
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 | : Ian Grant Macdonald |
| Publisher | : Oxford University Press |
| Total Pages | : 496 |
| Release | : 1998 |
| Genre | : Mathematics |
| ISBN | : 9780198504504 |
Download Symmetric Functions and Hall Polynomials Book in PDF, ePub and Kindle
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 | : Ian Grant Macdonald |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 71 |
| Release | : 1998 |
| Genre | : Mathematics |
| ISBN | : 0821807706 |
Download Symmetric Functions and Orthogonal Polynomials Book in PDF, ePub and Kindle
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. |
| Total Pages | : 342 |
| Release | : 2019-11-18 |
| Genre | : Education |
| ISBN | : 1470448998 |
Download An Introduction to Symmetric Functions and Their Combinatorics Book in PDF, ePub and Kindle
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 | : Douglas Bowman |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 73 |
| Release | : 2002 |
| Genre | : Mathematics |
| ISBN | : 082182774X |
Download $q$-Difference Operators, Orthogonal Polynomials, and Symmetric Expansions Book in PDF, ePub and Kindle
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
| Author | : Taekyun Kim |
| Publisher | : MDPI |
| Total Pages | : 206 |
| Release | : 2021-03-19 |
| Genre | : Mathematics |
| ISBN | : 3036503609 |
Download Current Trends in Symmetric Polynomials with Their Applications Ⅱ Book in PDF, ePub and Kindle
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.
| Author | : Evgeny Smirnov |
| Publisher | : Springer Nature |
| Total Pages | : 159 |
| Release | : 2024 |
| Genre | : Electronic books |
| ISBN | : 3031503414 |
Download Symmetric Functions Book in PDF, ePub and Kindle
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.
| Author | : James Haglund |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 178 |
| Release | : 2008 |
| Genre | : Mathematics |
| ISBN | : 0821844113 |
Download The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics Book in PDF, ePub and Kindle
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.
