# Symmetric Functions and Combinatorial Operators on Polynomials

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 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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