An introduction to symmetric functions and their combinatorics
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 \omega; the Hall inner product; Cauchy&#...
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2019]
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| Schriftenreihe: | Student mathematical library
Volume 91 |
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| Zusammenfassung: | 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 \omega; 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 Cover -- Title page -- Preface -- Chapter 1. Symmetric Polynomials, the Monomial Symmetric Polynomials, and Symmetric Functions -- 1.1. Symmetric Polynomials -- 1.2. The Monomial Symmetric Polynomials -- 1.3. Symmetric Functions -- 1.4. Problems -- 1.5. Notes -- Chapter 2. The Elementary, Complete Homogeneous, and Power Sum Symmetric Functions -- 2.1. The Elementary Symmetric Functions -- 2.2. The Complete Homogeneous Symmetric Functions -- 2.3. The Power Sum Symmetric Functions -- 2.4. Problems -- Chapter 3. Interlude: Evaluations of Symmetric Functions -- 3.1. Symmetric Function Identities -- 3.2. Binomial Coefficients -- 3.3. Stirling Numbers of the First and Second Kinds -- 3.4. -Binomial Coefficients -- 3.5. Problems -- 3.6. Notes -- Chapter 4. Schur Polynomials and Schur Functions -- 4.1. Schur Functions and Semistandard Tableaux -- 4.2. Schur Polynomials as Ratios of Determinants -- 4.3. Problems -- 4.4. Notes -- Chapter 5. Interlude: A Rogues' Gallery of Symmetric Functions -- 5.1. Skew Schur Functions -- 5.2. Stable Grothendieck Polynomials -- 5.3. Dual Stable Grothendieck Polynomials -- 5.4. The Chromatic Symmetric Function -- 5.5. Problems -- 5.6. Notes -- Chapter 6. The Jacobi-Trudi Identities and an Involution on Λ -- 6.1. The First Jacobi-Trudi Identity -- 6.2. The Second Jacobi-Trudi Identity -- 6.3. The Involution -- 6.4. Problems -- 6.5. Notes -- Chapter 7. The Hall Inner Product -- 7.1. Inner Products on Λ_{ } -- 7.2. The Hall Inner Product and Cauchy's Formula -- 7.3. The Hall Inner Product on the Power Sum Symmetric Functions -- 7.4. Problems -- 7.5. Notes -- Chapter 8. The Robinson-Schensted-Knuth Correspondence -- 8.1. RSK Insertion: Constructing ( ) -- 8.2. Constructing ( ) -- 8.3. Implementing RSK with Growth Diagrams -- 8.4. Problems -- 8.5. Notes -- Chapter 9. Special Products Involving Schur Functions |
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| Beschreibung: | xiii, 342 Seiten Illustrationen |
| ISBN: | 9781470448998 |