Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal polynomials on the unit sphere 5. Examples of orthogonal polynomials in several variables 6. Root systems and Coxeter groups 7. Spherical harmonics associated with reflection groups 8. Generalized classical orthogonal polynomials 9. Summability of orthogonal expansions 10. Orthogonal polynomials associated with symmetric groups 11. Orthogonal polynomials associated with octahedral groups and applications References Author index Symbol index Subject index.

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Orthogonal Polynomials of Several Variables

Groupes et algébres de lie

Contents §0. Introduction §1. The Yangian §2. The quantum determinant and the centre of §3. The twisted Yangian §4. The Sklyanin determinant and the centre of §5. The quantum contraction and the quantum Liouville formula for the Yangian §6. The quantum contraction and the quantum Liouville formula for the twisted Yangian §7. The quantum determinant and the Sklyanin determinant of block matrices Bibliography

Yangians and Classical Lie Algebras

Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We include the important mathematics which is a very modern development, as well as the computational software that is transforming the theory into useful practice.

https://stuff.mit.edu/people/raj/Acta05rmt.pdf

Random matrix theory

We study the isospectral Hilbert scheme X_n, defined as the reduced fiber product of C^2n with the Hilbert scheme H_n of points in the plane, over the symmetric power S^n C^2. We prove that X_n is normal, Cohen-Macaulay, and Gorenstein, and hence flat over H_n. We derive two important consequences. 
(1) We prove the strong form of the "n! conjecture" of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients K_{lambda,mu}(q,t). This establishes the Macdonald positivity conjecture, that K_{lambda,mu}(q,t) is always a polynomial with non-negative integer coefficients. 
(2) We show that the Hilbert scheme H_n is isomorphic to the Hilbert scheme of orbits C^2n//S_n, in such a way that X_n is identified with the universal family over C^2n//S_n.

/pdf/hilbert-schemes-polygraphs-and-the-macdonald-positivity-2cgtkgiw64.pdf

Hilbert schemes, polygraphs and the Macdonald positivity conjecture

The Jack polynomials J (x; ) form a remarkable class of symmetric polynomials. They are indexed by a partition and depend on a parameter . One of their properties is that several classical families of symmetric functions can be obtained by specializing , e.g., the monomial symmetric functions m ( =∞), the elementary functions e ′ ( = 0), the Schur functions s ( = 1) and nally the two classes of zonal polynomials ( = 2, = 1=2). The Jack polynomials can be de ned in various ways, e.g.: a) as an orthogonal family of functions which is compatible with the canonical ltration of the ring of symmetric functions or b) as simultaneous eigenfunctions of certain di erential operators (the Sekiguchi–Debiard operators). Recently Opdam, [O], constructed a similar family F (x; ) of nonsymmetric polynomials. The index runs now through all compositions ∈ Nn. They are de ned in a completely similar fashion, e.g., the Sekiguchi–Debiard operators are being replaced by the Cherednik di erential-re ection operators (see Sect. 3). It is becoming more and more clear that these polynomials are as important as their symmetric counterparts. The purpose of this paper is to add to the existing characterizations of Jack polynomials two further ones: c) a recursion formula among the F together with two formulas to obtain J from them. d) combinatorial formulas of both J and F in terms of certain generalized tableaux.

A recursion and a combinatorial formula for Jack polynomials

The work of Kramer, Luna, Vust, Brion and others [Kra,LV,BLV,BP] established the importance of a very distinguished class of homogeneous varieties G/H, those which are now called spherical. Such varieties are homogeneous for a connected reductive group G and are characterized by many equivalent properties, the most important being (see [BLV]): — Any Borel subgroup B of G has an open orbit in G/H. — Every equivariant completion of G/H contains only finitely many orbits. — For every irreducible G-module V and any character χ of H

The Luna-Vust Theory of Spherical Embeddings

Let G be a reductive algebraic group and X an algebraic G-variety which admits a quotient it: X → X//G In this article we describe several results concerning the Picard group Pic(X//G) of the quotient and the group Picc(X) of G-line bundles on X For some further development of the subject we refer to the survey articles [Kr89a], [Kr89b]

The Picard Group of a G-Variety

In this article we present a fundamental result due to Sumihiro. It states that every normal G-variety X, where G is a connected linear algebraic group, is locally isomorphic to a quasi-projective G-variety, i.e., to a G-stable subvariety of the projective space P n with a linear G-action (Theorem 1.1). The central tools for the proof are G-linearization of line bundles (§2) and some properties of the Picard group of a linear algebraic group (§4).

Local Properties of Algebraic Group Actions

Heckman and Opdam introduced a non-symmetric analogue of Jack polynomials using Cherednik operators. In this paper, we derive a simple recursion formula for these polynomials and formulas relating the symmetric Jack polynomials with the non-symmetric ones. These formulas are then implemented by a closed expression of symmetric and non-symmetric Jack polynomials in terms of certain tableaux. The main application is a proof of a conjecture of Macdonald stating certain integrality and positivity properties of Jack polynomials.

Friedrich Knop

Papers

A recursion and a combinatorial formula for Jack polynomials

The Luna-Vust Theory of Spherical Embeddings

The Picard Group of a G-Variety

Local Properties of Algebraic Group Actions

A recursion and a combinatorial formula for Jack polynomials