Macdonald Polynomials: Commuting Family of q-Difference Operators and Their Joint Eigenfunctions (SpringerBriefs in Mathematical Physics)

Preface
Contents
1 Overview of Macdonald Polynomials
	1.1 Macdonald Polynomials
	1.2 Fundamental Properties of Macdonald Polynomials
	1.3 Outlook
2 Preliminaries on Symmetric Functions
	2.1 Symmetric Functions ek(x), hk(x) and pk(x)
	2.2 Fundamental Theorem of Symmetric Polynomials
	2.3 Wronski Relations and Newton Relations
	2.4 Monomial Symmetric Functions
	2.5 Comments on Fundamental Theorems
3 Schur Functions
	3.1 Definitions of the Schur Functions
		3.1.1 Two Definitions
		3.1.2 Combinatorial Definition
		3.1.3 Determinantal Definition
	3.2 Principal Specialization and Self-duality
		3.2.1 Principal Specialization: Evaluation at x=tδ
		3.2.2 Hook Length Formula
		3.2.3 Self-duality
	3.3 Cauchy Formula
		3.3.1 Cauchy Determinant
		3.3.2 Cauchy Formula for Schur Functions
	3.4 Recurrence on the Number of Variables
	3.5 Equivalence of the Two Definitions
	3.6 Dual Cauchy Formula
	3.7 Jacobi–Trudi Formula
	3.8 q-Difference and Differential Equations
	3.9 Link to the Representation Theory of GLn (Overview)
		3.9.1 Polynomial Representations of GLn
		3.9.2 Weyl Character Formula and Branching Rules
		3.9.3 (GLm,GLn) Duality
4 Macdonald Polynomials: Definition and Examples
	4.1 Macdonald–Ruijsenaars q-Difference Operator
		4.1.1 Macdonald–Ruijsenaars Operator Dx
		4.1.2 Fundamental Properties of Dx
		4.1.3 Diagonalization of Dx
	4.2 Some Examples
	4.3 Eigenfunctions in Two Variables
		4.3.1 Eigenfunctions in Power Series
		4.3.2 Macdonald Polynomials in Two Variables
	4.4 q-Binomial Theorem and q-Hypergeometric Series
		4.4.1 q-Binomial Theorem
		4.4.2 q-Hypergeometric Series
		4.4.3 Generating Function of Two Variable Macdonald Polynomials
	4.5 Macdonald Polynomials Attached to Single Rows
		4.5.1 Macdonald Polynomials P(l)(x) and gl(x)
		4.5.2 Expression in Terms of φD
		4.5.3 Wronski Relations
5 Orthogonality and Higher-Order q-Difference Operators
	5.1 Scalar Product and Orthogonality
		5.1.1 Weight Function and Scalar Product
		5.1.2 Constant Term and Scalar Products
	5.2 Proof of Orthogonality
		5.2.1 Cauchy\'s Theorem as a Basis of q-Difference de Rham Theory
		5.2.2 Formal Adjoint of a q-Difference Operator
		5.2.3 Dx Is Self-Adjoint with Respect to w(x)
		5.2.4 Orthogonality
	5.3 Commuting Family of q-Difference Operators
		5.3.1 Macdonald–Ruijsenaars Operator of rth Order
		5.3.2 Fundamental Properties of D(r)x
		5.3.3 Macdonald Polynomials as Joint Eigenfunctions
	5.4 Commutativity of the Operators D(r)x
		5.4.1 Orthogonality Implies Commutativity
		5.4.2 A Direct Proof of Commutativity
	5.5 Refinement of the Existence Theorem
	5.6 Some Remarks Related to Dx(u)
		5.6.1 Macdonald Polynomials in x-1
		5.6.2 Determinant Representation of Dx(u)
		5.6.3 Limit to the Differential (Jack) Case
6 Self-duality, Pieri Formula and Cauchy Formulas
	6.1 Self-duality and Pieri Formula
		6.1.1 Principal Specialization
		6.1.2 Self-duality
		6.1.3 Pieri Formula
	6.2 Self-duality Implies the Pieri Formula
	6.3 Principal Specialization: Evaluation at x=tδ
	6.4 Koornwinder\'s Proof of Self-duality
	6.5 Cauchy Formula and Dual Cauchy Formula
	6.6 Kernel Identities
		6.6.1 Kernel Identity for the Cauchy Formula
		6.6.2 Kernel Identity for the Dual Cauchy Formula
7 Littlewood–Richardson Coefficients and Branching Coefficients
	7.1 Littlewood–Richardson Coefficients and Branching Coefficients
	7.2 Relation Between cλµ,ν and bλµ,ν
	7.3 Explicit Formula for bλ
	7.4 Tableau Representation of Pλ(x)
	7.5 Macdonald–Ruijsenaars Operators of Row Type (Overview)
		7.5.1 q-Difference Operators Hx(l) of Row Type
		7.5.2 Wronski Relations
		7.5.3 Pieri Formula of Row Type
8 Affine Hecke Algebra and q-Dunkl Operators (Overview)
	8.1 Affine Weyl Groups and Affine Hecke Algebras
	8.2 q-Dunkl Operators
	8.3 From q-Dunkl Operators to Macdonald–Ruijsenaars Operators
	8.4 Nonsymmetric Macdonald Polynomials
	8.5 Double Affine Hecke Algebra and Cherednik Involution
Appendix  Notation
References