Symmetric Functions and Combinatorial Operators on Polynomials Book

	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