Symmetric Functions and Combinatorial Operators on Polynomials

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Brand new. We distribute directly for the publisher. 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, Pad???? 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.The last chapter sketches a non-commutative version of symmetric functions, using Young tableaux and the plactic monoid.The book contains numerous exercises clarifying and extending many points of the main text. It will make an excellent supplementary text for a graduate course in combinatorics. Read more Show Less

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Overview

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.

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Product Details

Table of Contents

Preface
Ch. 1 Symmetric functions 1
1.1 Alphabets 1
1.2 Partitions 1
1.3 Generating Functions of Symmetric Functions 5
1.4 Matrix Generating Functions 7
1.5 Cauchy Formula 13
1.6 Scalar Product 14
1.7 Differential Calculus 16
1.8 Operators on Isobaric Determinants 18
1.9 Pieri Formulas 22
Ch. 2 Symmetric Functions as Operators and [lambda]-Rings 31
2.1 Algebraic Operations on Alphabets 31
2.2 Lambda Operations 32
2.3 Interpreting Polynomials and q-series 33
2.4 Lagrange Inversion 35
2.5 Some relations between multiples of alphabets 36
Ch. 3 Euclidean Division 47
3.1 Euclid's Algorithm 47
3.2 Remainders as Schur functions 50
3.3 Companion Matrix 51
3.4 Bezout's Matrix 52
3.5 Sylvester's Summations 54
3.6 Sturm Sequence 55
3.7 Wronski's Algorithm 57
3.8 Division and Continued Fractions 57
Ch. 4 Reciprocal Differences and Continued Fractions 63
4.1 Euler's Recursions 63
4.2 Continued Fraction Expression of a Formal Series 64
4.3 Interpolation of a Function by a Continued Fraction 66
4.4 Relation between Stieltjes and Wronski Continued Fractions 69
4.5 Jacobi's Tridiagonal Matrix 70
4.6 Motzkin Paths 71
4.7 Dyck Paths 72
4.8 Link between Emmeration of Motzkiu and Dyck Paths 73
Ch. 5 Division, encore 79
5.1 Derived Alphabets 79
5.2 Normalized Differences 81
5.3 Associated Continued Fractions 82
5.4 Hankel Forms 83
Ch. 6 Pade Approximants 87
6.1 Recovering a Rational Function from a Taylor Series 87
6.2 Pade Table 88
6.3 Euclid and Pade 89
6.4 Rational Interpolation 90
Ch. 7 Symmetrizing Operators 95
7.1 Divided Differences 95
7.2 Compatibility with Complete Functions 97
7.3 Braid Relations 97
7.4 Decomposing in the Basis of Permutations 99
7.5 Generating Series by Symmetrization 100
7.6 Maximal Symmetrizers 101
7.7 Schur Functions and Bott's Theorem 102
7.8 Lagrange Interpolation 105
7.9 Finite Derivation 108
7.10 Calogero's Raising and Lowering Operators 110
Ch. 8 Orthogonal Polynomials 117
8.1 Orthogonal Polynomials as Symmetric Functions 117
8.2 Reproducing Kernels 118
8.3 Continued Fractions 119
8.4 Higher Order Kernels 120
8.5 Even Moments 123
8.6 Zeros 125
8.7 The Moment Generating Function 128
8.8 Jacobi's Matrix and Paths 129
8.9 Discrete Measures 131
Ch. 9 Schubert Polynomials 141
9.1 Newton Interpolation Formula 141
9.2 Newton and Euclid 142
9.3 Discrete Wronskian 143
9.4 Schubert Polynomials 145
9.5 Vanishing Properties 147
9.6 Newton Interpolation in Several Variables 148
9.7 Interpolation of Symmetric Functions 150
9.8 Key Polynomials 152
Ch. 10 The Ring of Polynomials as a Module over Symmetric Ones 157
10.1 Quadratic Form on [actual symbol not reproducible] 157
10.2 Kernel 158
10.3 Shifts 162
10.4 Generating Function in the NilCoxeter Algebra 163
10.5 NilPlactic Kernel 165
10.6 Basis of Elementary Symmetric Functions 168
10.7 Yang-Baxter Basis 170
10.8 Yang-Baxter Elements as Permutations 172
Ch. 11 The plactic algebra 175
11.1 Tableaux 175
11.2 Plactic Algebra 176
11.3 Littlewood-Richardson Rule 178
11.4 Action of the Symmetric Group on the Free Algebra 179
11.5 Free Key Polynomials 181
11.6 Plastic Schubert Polynomials 182
App. A: Complements 185
App. B: Solutions of Exercises 197
Bibliography 261
Index 267
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