I. G. Macdonald

Search for another author

Author Description

I. G. Macdonald (Ian Grant Macdonald) is a British mathematician renowned for his foundational contributions to algebraic combinatorics. He is particularly famous for introducing Macdonald polynomials, which have deep connections to representation theory, symmetric functions, and other areas of mathematics. Macdonald served as a professor at Queen Mary University of London, and his work has significantly influenced modern developments in combinatorics and related fields.

Ask a Question About This Mathematician

Introduction to Commutative Algebra.

1970-08-01

D. J. Lewis , B. R. McDonald , M. F. Atiyah +1 more

SIMPLE GROUPS OF LIE TYPE

1975-03-01

I. G. Macdonald

By W. Carter Roger: pp. viii, 331. £7.50. John Wiley & Sons, December 1972.) By W. Carter Roger: pp. viii, 331. £7.50. John Wiley & Sons, December 1972.)

Affine Hecke Algebras and Orthogonal Polynomials

2003-03-20

I. G. Macdonald

In recent years there has developed a satisfactory and coherent theory of orthogonal polynomials in several variables, attached to root systems, and depending on two or more parameters. These polynomials … In recent years there has developed a satisfactory and coherent theory of orthogonal polynomials in several variables, attached to root systems, and depending on two or more parameters. These polynomials include as special cases: symmetric functions; zonal spherical functions on real and p-adic reductive Lie groups; the Jacobi polynomials of Heckman and Opdam; and the Askey–Wilson polynomials, which themselves include as special or limiting cases all the classical families of orthogonal polynomials in one variable. This book, first published in 2003, is a comprehensive and organised account of the subject aims to provide a unified foundation for this theory, to which the author has been a principal contributor. It is an essentially self-contained treatment, accessible to graduate students familiar with root systems and Weyl groups. The first four chapters are preparatory to Chapter V, which is the heart of the book and contains all the main results in full generality.

View PDF Chat

Affine root systems and Dedekind's?-function

1971-06-01

I. G. Macdonald

A new class of symmetric functions.

1988-01-01

I. G. Macdonald

I will begin by reviewing briefly some aspects of the theory of symmetric functions. This will serve to fix notation and to provide some motivation for the subject of these … I will begin by reviewing briefly some aspects of the theory of symmetric functions. This will serve to fix notation and to provide some motivation for the subject of these lectures. Let x1, . . . , xn be independent indeterminates. The symmetric group Sn acts on the polynomial ring Z[x1, . . . , xn] by permuting the x’s, and we shall write Λn = Z[x1, . . . , xn] for the subring of symmetric polynomials in x1, . . . , xn. If f ∈ Λn, we may write f = ∑

Symmetric products of an algebraic curve

1962-10-01

I. G. Macdonald

Orthogonal Polynomials Associated with Root Systems

1990-01-01

I. G. Macdonald

Some Conjectures for Root Systems

1982-11-01

I. G. Macdonald

We present a collection of conjectures relating to root systems (or, equivalently, to Lie groups) together with the evidence we possess in support of them. They may be regarded as … We present a collection of conjectures relating to root systems (or, equivalently, to Lie groups) together with the evidence we possess in support of them. They may be regarded as generalizations of Dyson's conjecture [J. Math. Phys. 3 (1962), pp. 140–156] and Mehta's conjecture [Random Matrices, Academic Press, New York, 1967].

The Poincar� series of a Coxeter group

1972-09-01

I. G. Macdonald

Schur functions: Theme and variations.

1992-01-01

I. G. Macdonald

Polynomials Associated with Finite Gell-Complexes

1971-07-01

I. G. Macdonald

The Poincare Polynomial of a Symmetric Product

1962-10-01

I. G. Macdonald

Let X be a compact polyhedron, X n the topological product of n factors equal to X . The symmetric group S n operates on X n by permuting the … Let X be a compact polyhedron, X n the topological product of n factors equal to X . The symmetric group S n operates on X n by permuting the factors, and hence if G is any subgroup of S n we have an orbit space X n /G obtained by identifying each point of X n with its images under G . In particular X n /S n is the n th symmetric product of X , and if G is a cyclic subgroup of order n then X n /G is the n th cyclic product of X .

The volume of a compact Lie group

1980-02-01

I. G. Macdonald

Orthogonal polynomials associated with root systems

2000-01-01

I. G. Macdonald

Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. … Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R,S) we construct a family of W-invariant orthogonal polynomials in several variables, whose coefficients are rational functions of parameters $q,t_1,t_2,...,t_r$, where r (=1,2 or 3) is the number of W-orbits in R. For particular values of these parameters, these polynomials give the values of zonal spherical functions on real and p-adic symmetric spaces. Also when R=S is of type $A_n$, they conincide with the symmetric polynomials described in I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press (1995), Chapter VI.

The volume of a lattice polyhedron

1963-10-01

I. G. Macdonald

Let L be the lattice of all points with integer coordinates in the real affine plane R 2 (with respect to some fixed coordinate system). Let X be a finite … Let L be the lattice of all points with integer coordinates in the real affine plane R 2 (with respect to some fixed coordinate system). Let X be a finite rectilinear simplicial complex in R 2 whose 0-simplexes are points of L . Suppose X is pure and the frontier Ẋ of X is a Jordan curve; then there is a well-known formula for the area of X in terms of the number of points of L which lie in X and Ẋ respectively, namely where L(X) (resp. L (Ẋ)) is the number of points of L which lie in X (resp. Ẋ), and μ(X) is the area of X, normalized so that a fundamental parallelogram of L has unit area.

Polynomial functors and wreath products

1980-09-01

I. G. Macdonald

Commuting differential operators and zonal spherical functions

1987-01-01

I. G. Macdonald

On the Degrees of the Irreducible Representations of Symmetric Groups

1971-07-01

I. G. Macdonald

Numbers of conjugacy classes in some finite classical groups

1981-02-01

I. G. Macdonald

In this paper we calculate the number of congugacy classes in the following finite classical groups: GL n (F q ); PGL n (F q ), SL n (F q … In this paper we calculate the number of congugacy classes in the following finite classical groups: GL n (F q ); PGL n (F q ), SL n (F q ), and more generally G (F q ), where G is any algebraic group isogenous to SL n ; PSL n (F q ); ; , , and more generally where G is any group isogenous to SU n over F q ; and .

View PDF Chat

Duality over complete local rings

1962-07-01

I. G. Macdonald

Common Tangents to Four Unit Balls in R 3

2001-01-01

I. G. Macdonald , János Pach , Thorsten Theobald

View PDF Chat

Some Irreducible Representations of Weyl Groups

1972-07-01

I. G. Macdonald

Introduction to Commutative Algebra

2018-04-27

M. F. Atiyah , I. G. Macdonald

This book is designed to be read by students who have had a first elementary course in general algebra. It provides a common generalization of the primes of arithmetic and … This book is designed to be read by students who have had a first elementary course in general algebra. It provides a common generalization of the primes of arithmetic and the points of geometry. The book explains the various elementary operations which can be performed on ideals.

Jordan Algebras with Three Generators

1960-01-01

I. G. Macdonald

Spherical functions on a 𝔭-adic Chevalley group

1968-01-01

I. G. Macdonald

View PDF Chat

Orthogonal polynomials

2003-03-20

I. G. Macdonald

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Zeta functions attached to finite general linear groups

1980-02-01

I. G. Macdonald

Some computations for m-dimensional partitions

1967-10-01

A. O. L. Atkin , Paul Bratley , I. G. Macdonald +1 more

1. It was known to Euler that p ( n ), the number of unrestricted partitions of n into non-increasing integral parts, is generated by with the usual convention that … 1. It was known to Euler that p ( n ), the number of unrestricted partitions of n into non-increasing integral parts, is generated by with the usual convention that p (0) = 1.

TOPICS IN M-ADIC TOPOLOGIES

1972-07-01

I. G. Macdonald

By S. Greco and P. Salmon: pp. vii, 74; DM24, US$$6.60. (Springer-Verlag, Berlin, 1971.) By S. Greco and P. Salmon: pp. vii, 74; DM24, US$$6.60. (Springer-Verlag, Berlin, 1971.)

An Elementary Proof of a q-Binomial Identity

1989-01-01

I. G. Macdonald

In the previous paper [Z] D. Zeilberger asks for an elementary, non-combinatorial proof of the identity (KOH). We shall give such a proof. In the previous paper [Z] D. Zeilberger asks for an elementary, non-combinatorial proof of the identity (KOH). We shall give such a proof.

Affine lie algebras and modular forms

2006-11-24

I. G. Macdonald

Hypergeometric Functions I

2013-01-01

I. G. Macdonald

This is the typewritten version of a handwritten manuscript which was completed by Ian G. Macdonald in 1987 or 1988. The manuscript is a very informal working paper, never intended … This is the typewritten version of a handwritten manuscript which was completed by Ian G. Macdonald in 1987 or 1988. The manuscript is a very informal working paper, never intended for formal publication. Nevertheless, copies of the manuscript have circulated widely, giving rise to quite a few citations in the subsequent 25 years. Therefore it seems justified to make the manuscript available for the whole mathematical community. The author kindly gave his permission that a typewritten version be posted on arXiv.

Introduction to Lie algebras

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

View PDF Chat

Algebraic structure of Lie groups

1980-01-31

I. G. Macdonald

This survey of the algebraic structure of Lie groups and Lie algebras (mainly semi simple) is a considerably expanded version of the oral lectures at the symposium. It is limited … This survey of the algebraic structure of Lie groups and Lie algebras (mainly semi simple) is a considerably expanded version of the oral lectures at the symposium. It is limited to what is necessary for representation theory, which is another way of saying that very little has been left out. In spite of its length, it contains few proofs or even indications of proofs, nor have I given chapter and verse for each of the multitude of unproved assertions throughout the text. Instead, I have appended references to each section, from which the diligent reader should have no difficulty in tracking down the proofs.

On the Degrees of the Irreducible Representations of Finite Coxeter Groups

1973-02-01

I. G. Macdonald

Some enumerative formulae for algebraic curves

1958-10-01

I. G. Macdonald

This paper is in two parts. In Part I we are concerned with one or more linear series on an algebraic curve; we consider a set of points on the … This paper is in two parts. In Part I we are concerned with one or more linear series on an algebraic curve; we consider a set of points on the curve which are contained with assigned multiplicities in a set of each of the linear series and, by persistent use of Severi's equivalence relation for the united points of an algebraic correspondence with valency, we derive formulae for the number of such sets of points when the constants involved are such as to make this number finite. All this is essentially a generalization of the formula for the number of points in the Jacobian set of a linear series of freedom 1, and the main result is Theorem 3.

Constant term identities, orthogonal polynomials and affine Hecke algebras

1998-01-01

I. G. Macdonald

Hall Polynomials

1995-03-30

I. G. Macdonald

Abstract Let o be a (commutative) discrete valuation ring, p its maximal ideal, k — o/p the residue field. Later we shall require k to be a finite field, but … Abstract Let o be a (commutative) discrete valuation ring, p its maximal ideal, k — o/p the residue field. Later we shall require k to be a finite field, but for the present this restriction is unnecessary. We shall be concerned with finite o-modules M, that is to say, modules M which possess a finite composition series, or equivalently finitely-generated o-modules M such that p’M=Q for some r&gt;0. If k is finite, the finite o-modules are precisely those which have a finite number of elements.

Hypergeometric Functions II (q-analogues)

2013-01-01

I. G. Macdonald

This is the typewritten version of a handwritten manuscript which was completed by Ian G. Macdonald in 1987 or 1988. It is the sequel to the manuscript "Hypergeometric functions I." … This is the typewritten version of a handwritten manuscript which was completed by Ian G. Macdonald in 1987 or 1988. It is the sequel to the manuscript "Hypergeometric functions I." The two manuscripts are very informal working papers, never intended for formal publication. Nevertheless, copies of the manuscripts have circulated widely, giving rise to quite a few citations in the subsequent 25 years. Therefore it seems justified to make the manuscripts available for the whole mathematical community. The author kindly gave his permission that typewritten versions be posted on arXiv.

Simple groups of Lie type

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

The affine Hecke algebra

2003-03-20

I. G. Macdonald

Some conjectures for root systems and finite coxeter groups

1981-01-01

I. G. Macdonald

Some sign properties of symmetric functions

1989-03-01

D. B. Hunter , I. G. Macdonald

Abstract This paper is concerned with the sign properties of the S -functions s λ for real arguments. We show first that s λ is indefinite if any part of … Abstract This paper is concerned with the sign properties of the S -functions s λ for real arguments. We show first that s λ is indefinite if any part of the partition λ is odd. Thus it is only if all parts of λ are even that s λ can possibly be positive definite or semi-definite. In this case we show that s λ ( x ) is positive provided that at least l (λ) of the components of x are non-zero, where l (λ) is the number of parts of the partition λ.

Symmetric Functions

1995-03-30

I. G. Macdonald

Abstract Many of the objects we shall consider in this book will turn out to be parametrized by partitions. The purpose of this section is to lay down some notation … Abstract Many of the objects we shall consider in this book will turn out to be parametrized by partitions. The purpose of this section is to lay down some notation and terminology which will be used throughout, and to collect together some elementary results on orderings of partitions which will be used later.

A Note on Local Cohomology

1975-07-01

I. G. Macdonald

Maximal compact subgroups

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

ASSOCIATED WITH ROOT SYSTEMS

2000-01-01

I. G. Macdonald

Let R and S be two irreducible root systems spanning the same vec- tor space and having the same Weyl group W, such that S (but not necessarily R) is … Let R and S be two irreducible root systems spanning the same vec- tor space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R,S) we construct a family of W-invariant or- thogonal polynomials in several variables, whose coefficientsare rational functions of parameters q,t1,t2,...,tr, where r (= 1,2 or 3) is the number of W-orbits in R. For particular values of these parameters, these polynomials give the values of zonal spherical functions on real and p-adic symmetric spaces. Also when R = S is of type An, they conincide with the symmetric polynomials described in I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press (1995), Chapter VI.

Borel subgroups and maximal tori

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

CONVEX POLYTOPES AND THE UPPER BOUND CONJECTURE

1973-03-01

I. G. Macdonald

By P. McMullen and G. C. Shephard: pp. iv, 184, £2. (Cambridge University Press, London, 1971.) By P. McMullen and G. C. Shephard: pp. iv, 184, £2. (Cambridge University Press, London, 1971.)

The extended affine Weyl group

2003-03-20

I. G. Macdonald

Rings and Modules of Fractions

2018-04-27

Michael Atiyah , I. G. Macdonald

Rings and Ideals

2018-04-27

M. F. Atiyah , I. G. Macdonald

Introduction to Commutative Algebra

2018-04-27

M. F. Atiyah , I. G. Macdonald

Hypergeometric Functions II (q-analogues)

2013-01-01

I. G. Macdonald

Hypergeometric Functions I

2013-01-01

I. G. Macdonald

Affine lie algebras and modular forms

2006-11-24

I. G. Macdonald

Commentary on the previous paper: “A class of polynomials in search of a definition, or the symmetric group parametrized” [in Jack, Hall-Littlewood and Macdonald polynomials, 75–106, Contemp. Math., 417, Amer. Math. Soc., Providence, RI, 2006; MR2284122] by H. Jack

2006-01-01

I. G. Macdonald

The extended affine Weyl group

2003-03-20

I. G. Macdonald

Affine Hecke Algebras and Orthogonal Polynomials

2003-03-20

I. G. Macdonald

View PDF Chat

Affine root systems

2003-03-20

I. G. Macdonald

Orthogonal polynomials

2003-03-20

I. G. Macdonald

The rank 1 case

2003-03-20

I. G. Macdonald

Introduction

2003-03-20

I. G. Macdonald

Bibliography

2003-03-20

I. G. Macdonald

The affine Hecke algebra

2003-03-20

I. G. Macdonald

Common Tangents to Four Unit Balls in R 3

2001-01-01

I. G. Macdonald , János Pach , Thorsten Theobald

View PDF Chat

ASSOCIATED WITH ROOT SYSTEMS

2000-01-01

I. G. Macdonald

Orthogonal polynomials associated with root systems

2000-01-01

I. G. Macdonald

Constant term identities, orthogonal polynomials and affine Hecke algebras

1998-01-01

I. G. Macdonald

The Borel-Weil theorem

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Maximal compact subgroups

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Homogeneous spaces

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Lie theory

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Linear algebraic groups: definition and elementary properties

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

The root structure of a linear algebraic group

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

The complexification of a compact group

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Projective algebraic varieties

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Borel subgroups and maximal tori

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Introduction to Lie algebras

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

View PDF Chat

Homogeneous spaces and quotients

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Let G be a linear algebraic group, X an algebraic variety. An action of G on X is a morphism G × X → X, written (g, x) ↦ gx, … Let G be a linear algebraic group, X an algebraic variety. An action of G on X is a morphism G × X → X, written (g, x) ↦ gx, such that

The Peter-Weyl theorem

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

The Lie algebra of a linear algebraic group

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Representations of non-compact groups

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Affine algebraic varieties

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Representations of SL2ℝ

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

The unitary and symmetric groups

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Representations of simple Lie algebras

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Functions on ℝ n and Sn-1

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Simple Lie algebras over ℂ

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Simple groups of Lie type

1995-08-17

Roger W. Carter , I. G. Macdonald , Graeme Segal

Symmetric Functions

1995-03-30

I. G. Macdonald

The Hecke Ring Of GlnOver A Local Field

1995-03-30

I. G. Macdonald

Abstract Let F be a locally compact topological field. We shall assume that the topology of F is not the discrete topology, since any field whatsoever is locally compact when … Abstract Let F be a locally compact topological field. We shall assume that the topology of F is not the discrete topology, since any field whatsoever is locally compact when given the discrete topology. A non-discrete locally compact field is called a local field.

Hall-Littlewood Symmetric Functions

1995-03-30

I. G. Macdonald

Abstract Let x,...,x and t be independent indeterminates over Z, and let A be a partition of length &lt; n. Abstract Let x,...,x and t be independent indeterminates over Z, and let A be a partition of length &lt; n.

Zonal Polynomials

1995-03-30

I. G. Macdonald

Abstract For a subgroup K of G, the following conditions are equivalent: (a) the induced representation 1G K is multiplicity-free: (b) the algebra C(G, K) is commutative If these equivalent … Abstract For a subgroup K of G, the following conditions are equivalent: (a) the induced representation 1G K is multiplicity-free: (b) the algebra C(G, K) is commutative If these equivalent conditions are satisfied, the pair (G, K) is called a Gelfand pair.

Hall Polynomials

1995-03-30

I. G. Macdonald

Schur functions: Theme and variations.

1992-01-01

I. G. Macdonald

Orthogonal Polynomials Associated with Root Systems

1990-01-01

I. G. Macdonald

Some sign properties of symmetric functions

1989-03-01

D. B. Hunter , I. G. Macdonald

An Elementary Proof of a q-Binomial Identity

1989-01-01

I. G. Macdonald

A new class of symmetric functions.

1988-01-01

I. G. Macdonald

Coauthor	Papers Together
Graeme Segal	21
Roger W. Carter	21
M. F. Atiyah	3
Michael Atiyah	1
John McKay	1
B. R. McDonald	1
Paul Bratley	1
D. B. Hunter	1
Thorsten Theobald	1
A. O. L. Atkin	1
János Pach	1
D. J. Lewis	1

Affine root systems and Dedekind's?-function

1971-06-01

I. G. Macdonald

Some Conjectures for Root Systems

1982-11-01

I. G. Macdonald

The Poincar� series of a Coxeter group

1972-09-01

I. G. Macdonald

Statistical Theory of the Energy Levels of Complex Systems. I

1962-01-01

Freeman J. Dyson

New kinds of statistical ensemble are defined, representing a mathematical idealization of the notion of ``all physical systems with equal probability.'' Three such ensembles are studied in detail, based mathematically … New kinds of statistical ensemble are defined, representing a mathematical idealization of the notion of ``all physical systems with equal probability.'' Three such ensembles are studied in detail, based mathematically upon the orthogonal, unitary, and symplectic groups. The orthogonal ensemble is relevant in most practical circumstances, the unitary ensemble applies only when time-reversal invariance is violated, and the symplectic ensemble applies only to odd-spin systems without rotational symmetry. The probability-distributions for the energy levels are calculated in the three cases. Repulsion between neighboring levels is strongest in the symplectic ensemble and weakest in the orthogonal ensemble. An exact mathematical correspondence is found between these eigenvalue distributions and the statistical mechanics of a one-dimensional classical Coulomb gas at three different temperatures. An unproved conjecture is put forward, expressing the thermodynamic variables of the Coulomb gas in closed analytic form as functions of temperature. By means of general group-theoretical arguments, the conjecture is proved for the three temperatures which are directly relevant to the eigenvalue distribution problem. The electrostatic analog is exploited in order to deduce precise statements concerning the entropy, or degree of irregularity, of the eigenvalue distributions. Comparison of the theory with experimental data will be made in a subsequent paper.

Root systems and hypergeometric functions. I

1988-01-01

Gerrit Heckman , Eric Opdam

On considere les systemes dynamiques d'hamiltoniens: H=∑ i=1 n p i 2 /2+g∑ i<j (q i −q j ) −2 et H=∑ i=1 n p i 2 /2+g∑ i<j sin … On considere les systemes dynamiques d'hamiltoniens: H=∑ i=1 n p i 2 /2+g∑ i<j (q i −q j ) −2 et H=∑ i=1 n p i 2 /2+g∑ i<j sin −2 (q i −q j ). On etudie la resolution spectrale simultanee de l'algebre commutante d'operateurs differentiels associee a l'operateur de Schrodinger S=−∑ i=1 n ∂(q i ) 2 /2+g∑ i<j sin −2 (q i −q j )

Some applications of hypergeometric shift operators

1989-02-01

Eric Opdam

View PDF Chat

Euclidean Lie Algebras

1969-01-01

Robert V. Moody

Our aim in this paper is to study a certain class of Lie algebras which arose naturally in ( 4 ). In ( 4 ), we showed that beginning with … Our aim in this paper is to study a certain class of Lie algebras which arose naturally in ( 4 ). In ( 4 ), we showed that beginning with an indecomposable symmetrizable generalized Cartan matrix ( A ij ) and a field Φ of characteristic zero, we could construct a Lie algebra E(( A ij )) over Φ patterned on the finite-dimensional split simple Lie algebras. We were able to show that E(( A ij )) is simple providing that ( A ij ) does not fall in the list given in ( 4 , Table). We did not prove the converse, however. The diagrams of the table of ( 4 ) appear in Table 2. Call the matrices that they represent Euclidean matrices and their corresponding algebras Euclidean Lie algebras. Our first objective is to show that Euclidean Lie algebras are not simple.

View PDF Chat

Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials

1985-01-01

Richard Askey , James Wilson

A new class of Lie algebras

1968-10-01

Robert V. Moody

Problems and Prospects for Basic Hypergeometric Functions

1975-01-01

George E. Andrews

Some combinatorial properties of Jack symmetric functions

1989-09-01

Richard P. Stanley

Groupes Réductifs Sur Un Corps Local

1972-12-01

François Bruhat , Jacques Tits

Harmonic analysis for certain representations of graded Hecke algebras

1995-01-01

Eric Opdam

View PDF Chat

Askey-Wilson polynomials for root systems of type 𝐵𝐶

1992-01-01

Tom H. Koornwinder

Some properties of Koornwinder polynomials

2000-01-01

Siddhartha Sahi

View PDF Chat

Nonsymmetric Koornwinder Polynomials and Duality

1999-07-01

Siddhartha Sahi

In the fundamental work of Lusztig [L] on affine Hecke algebras, a special role is played by the root system of type Cn. The affine Hecke algebra is a deformation … In the fundamental work of Lusztig [L] on affine Hecke algebras, a special role is played by the root system of type Cn. The affine Hecke algebra is a deformation of the group algebra of an affine Weyl group which usually depends on as many parameters as there are distinct root lengths, i.e. one or two for an irreducible root system. However in the Cn case, the Hecke algebra H has three parameters, corresponding to the fact that there is a simple coroot which is divisible by 2. Recently, Cherednik [C1]-[C3] has introduced the notion of a double affine Hecke algebra, and has used it to prove several conjectures on Macdonald polynomials. These polynomials, and Cherednik's double affine Hecke algebra, involve two or three parameters, i.e., one more than the number of root lengths.

View PDF Chat

Double Affine Hecke Algebras and Macdonald's Conjectures

1995-01-01

Ivan Cherednik

In this paper we prove the main results about the structure of double affine Hecke algebras announced in [C1], [C2]. The technique is based on the realization of these algebras … In this paper we prove the main results about the structure of double affine Hecke algebras announced in [C1], [C2]. The technique is based on the realization of these algebras in terms of Demazure-Lusztig operators [BGG], [D], [L2], [LS], [C3] and rather standard facts from the theory of affine Weyl groups. In particular, it completes the proof (partially published in [C2]) of the Macdonald scalar product conjecture (see [Ml], (12,6')), including the famous Macdonald constant-term conjecture (the q, t-case). We mainly follow the Opdam paper [0] where the Macdonald-Mehta conjectures in the degenerate (differential) case were deduced from certain properties of the Heckman-Opdam operators [HO] and the existence of the shift operators. Heckman's interpretation of these operators via the so-called Dunkl operators (see [He] and also [C5]) was important to our approach. We note that the HO operators are closely related to the so-called quantum many-body problem (Calogero, Sutherland, Moser, Olshanetsky, Perelomov), the conformal field theory (Knizhnik-Zamolodchikov equations), the harmonic analysis on symmetric spaces (Harish-Chandra, Helgason etc.), and (last but not the least) the classic theory of the hypergeometric functions. Establishing the connection between the difference counterparts of Heckman-Opdam operators introduced in [C4] and the Macdonald theory [Ml], [M2] including the construction of the difference shift operators is the main thrust of this paper. Once the connection is established it is not very difficult to calculate the scalar squares of the Macdonald polynomials and to prove the constant-term conjecture from his fundamental paper [M3]. To simplify the exposition, we discuss the reduced root systems only and impose the relation q = tk for k E Z+ (to avoid infinite products in the definition of Macdonald's pairing). The purpose of this work is to present a concrete application of the new technique. Arbitrary q, t can be handled in a

Symmetric Functions and Orthogonal Polynomials

1997-11-26

I. Macdonald

Macdonald's evaluation conjectures and difference Fourier transform

1995-12-01

Ivan Cherednik

View PDF Chat

A generalization of Selberg’s beta integral

1990-01-01

Robert A. Gustafson

We evaluate several infinite families of multidimensional integrals which are generalizations or analogs of Euler's classical beta integral.We first evaluate a ^-analog of Selberg's beta integral.This integral is then used … We evaluate several infinite families of multidimensional integrals which are generalizations or analogs of Euler's classical beta integral.We first evaluate a ^-analog of Selberg's beta integral.This integral is then used to prove the Macdonald-Morris conjectures for the affine root systems of types S{Ci) and S{Cj) v and to give a new proof of these conjectures for SiBC;), S{B ( ), 5 , (B / ) V and S*(£/) •

View PDF Chat

Self-dual Koornwinder-Macdonald polynomials

1996-10-02

J. F. van Diejen

View PDF Chat

Lectures on affine Hecke algebras and Macdonald’s conjectures

1997-01-01

Alexander Kirillov

This paper gives a review of Cherednik’s results on the representation-theoretic approach to Macdonald polynomials and related special functions. Macdonald polynomials are a remarkable 2-parameter family of polynomials which can … This paper gives a review of Cherednik’s results on the representation-theoretic approach to Macdonald polynomials and related special functions. Macdonald polynomials are a remarkable 2-parameter family of polynomials which can be associated to every root system. As special cases, they include the Schur functions, the<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"><mml:semantics><mml:mi>q</mml:mi><mml:annotation encoding="application/x-tex">q</mml:annotation></mml:semantics></mml:math></inline-formula>-Jacobi polynomials, and certain spherical functions on real and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding="application/x-tex">p</mml:annotation></mml:semantics></mml:math></inline-formula>-adic symmetric spaces. They have a number of elegant combinatorial properties, which, however, are extremely difficult to prove. In this paper we show that a natural setup for studying these polynomials is provided by the representation theory of Hecke algebras and show how this can be used to prove some of the combinatorial identities for Macdonald polynomials.

View PDF Chat

Affine Hecke algebras and their graded version

1989-01-01

G. Lusztig

0.1. Let H,o be an affine Hecke algebra with parameter v0 E C* assumed to be of infinite order. (The basis elements Ts E H,o corresponding to simple reflections s … 0.1. Let H,o be an affine Hecke algebra with parameter v0 E C* assumed to be of infinite order. (The basis elements Ts E H,o corresponding to simple reflections s satisfy (Ts + l)(Ts v2c(s)) = 0, where C(S) E N depend on s and are subject only to c(s) = c(s') whenever s, s are conjugate in the affine Weyl group.) Such Hecke algebras appear naturally in the representation theory of semisimple p-adic groups, and understanding their representation theory is a question of considerable interest. Consider the special where c(s) is independent of s and the coroots generate a direct summand. In this special case, the question above has been studied in [1] and a classification of the simple modules was obtained. The approach of [1] was based on equivariant K-theory. This approach can be attempted in the general case (some indications are given in [5, 0.3]), but there appear to be some serious difficulties in carrying it out.

View PDF Chat

Affine Hecke Algebras and Orthogonal Polynomials

2003-03-20

I. G. Macdonald

View PDF Chat

Intertwining operators of double affine Hecke algebras

1997-12-01

Ivan Cherednik

Infinite Dimensional Lie Algebras

1983-01-01

Victor G. Kač

View PDF Chat

Infinite-Dimensional Lie Algebras

1990-09-20

Victor G. Kač

This is the third, substantially revised edition of this important monograph. The book is concerned with Kac–Moody algebras, a particular class of infinite-dimensional Lie algebras, and their representations. It is … This is the third, substantially revised edition of this important monograph. The book is concerned with Kac–Moody algebras, a particular class of infinite-dimensional Lie algebras, and their representations. It is based on courses given over a number of years at MIT and in Paris, and is sufficiently self-contained and detailed to be used for graduate courses. Each chapter begins with a motivating discussion and ends with a collection of exercises, with hints to the more challenging problems.

View PDF Chat

Involutions of double affine Hecke algebras

2001-01-01

Bogdan Ion

The main aim of the paper is to formulate and prove a result about the structure of double affine Hecke algebras which allows its two commutative subalgebras to play a … The main aim of the paper is to formulate and prove a result about the structure of double affine Hecke algebras which allows its two commutative subalgebras to play a symmetric role. This result is essential for the theory of intertwiners of double affine Hecke algebras.

Groupes réductifs sur un corps local

1984-12-01

François Bruhat , Jacques Tits

Sur quelques points d'algèbre homologique, I

1957-01-01

Alexander Grothendieck

View PDF Chat

Groupes et algèbres de Lie

1971-01-01

Nicolás Bourbaki

Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes … Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes et algebre de Lie, neuvieme Livre du traite, est consacre aux concepts fondamentaux pour les algebres de Lie. Il comprend les paragraphes: - 1 Definition des algebres de Lie; 2 Algebre enveloppante d une algebre de Lie; 3 Representations; 4 Algebres de Lie nilpotentes; 5 Algebres de Lie resolubles; 6 Algebres de Lie semi-simples; 7 Le theoreme d Ado. Ce volume est une reimpression de l edition de 1971.

The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group

1959-10-01

Bertram Kostant

Groups and Geometric Analysis

2000-10-03

Sigurđur Helgason

Geometric Fourier analysis on spaces of constant curvature Integral geometry and Radon transforms Invariant differential operators Invariants and harmonic polynomials Spherical functions and spherical transforms Analysis on compact symmetric spaces … Geometric Fourier analysis on spaces of constant curvature Integral geometry and Radon transforms Invariant differential operators Invariants and harmonic polynomials Spherical functions and spherical transforms Analysis on compact symmetric spaces Appendix Some details Bibliography Symbols frequently used Index Errata.

Proof of a Conjecture by Dyson in the Statistical Theory of Energy Levels

1962-07-01

J. Gunson

A conjectured identity relating the statistical properties of two types of ensembles occurring in a statistical theory of the distribution of energy levels in nuclei and other complex systems is … A conjectured identity relating the statistical properties of two types of ensembles occurring in a statistical theory of the distribution of energy levels in nuclei and other complex systems is proved.

View PDF Chat

The volume of a compact Lie group

1980-02-01

I. G. Macdonald

The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group

2009-01-01

Bertram Kostant

Basic sets of invariant polynomials for finite reflection groups

1988-01-01

M. L. Mehta

Abstract A basic net of invariant polynomials is explicitly given for each non-factorizable finite reflection group. Abstract A basic net of invariant polynomials is explicitly given for each non-factorizable finite reflection group.

Jacobi functions as limit cases of q-ultraspherical polynomials

1990-05-01

Tom H. Koornwinder

View PDF Chat

Proof of a Conjecture by Dyson

1962-09-01

Kenneth G. Wilson

We prove a mathematical conjecture by Dyson which he used in a study of the statistical distribution of energy levels in complex nuclei. We prove a mathematical conjecture by Dyson which he used in a study of the statistical distribution of energy levels in complex nuclei.

Short Proof of a Conjecture by Dyson

1970-06-01

I. J. Good

Dyson made a mathematical conjecture in his work on the distribution of energy levels in complex systems. A proof is given, which is much shorter than two that have been … Dyson made a mathematical conjecture in his work on the distribution of energy levels in complex systems. A proof is given, which is much shorter than two that have been published before.

A generalization of ultraspherical polynomials

1983-01-01

Richard Askey , Mourad E. H. Ismail

Some old polynomials of L. J. Rogers are orthogonal. Their weight function is given. The connection coefficient problem, which Rogers solved by guessing the formula and proving it by induction, … Some old polynomials of L. J. Rogers are orthogonal. Their weight function is given. The connection coefficient problem, which Rogers solved by guessing the formula and proving it by induction, is derived in a natural way and some other formulas are obtained. These polynomials generalize zonal spherical harmonics on spheres and include as special cases polynomials that are spherical functions on rank one spaces over reductive p-adic groups. A limiting case contains some Jacobi polynomials studied by Hylleraas that arose in work on the Yukawa potential.

The Linearization of the Product of Continuous q-Jacobi Polynomials

1981-08-01

Mizan Rahman

The problem of linearizing the product of two Jacobi polynomials, P m ( α, β ) ( x ) P n ( α, β ) ( x ), and to … The problem of linearizing the product of two Jacobi polynomials, P m ( α, β ) ( x ) P n ( α, β ) ( x ), and to establish the conditions for the non-negativity of the coefficients has been of considerable interest for many years. Explicit non-negative representations were sought and found by many authors [ 7, 8, 13, 14 ], but only in the special case α = β , although Hylleraas [ 14 ] succeeded in finding a formula in another case α = β + 1. Gasper [ 9, 10 ] found the necessary and sufficient conditions for the non-negativity of the linearization coefficients by exploiting a recurrence relation obtained by Hylleraas for the above-mentioned product. Koornwinder [ 16 ] approached the same problem from a different point of view and managed to find a non-negative integral expression to these coefficients when . However, an exact formula in a hypergeometric series form for general α, β has been very elusive so far, in spite of the fact that all computation of special cases seemed to indicate that such a formula should exist.

View PDF Chat

Partial differential equations for hypergeometric functions 3F2 of matrix argument

1975-01-01

Yasunori Fujikoshi

Abstract Many multivariate non‐null distributions and moment formulas can be expressed in terms of hypergeometric functions p F q of matrix arqument. Muirhead [6] and Constantine and Muirhead [2] gave … Abstract Many multivariate non‐null distributions and moment formulas can be expressed in terms of hypergeometric functions p F q of matrix arqument. Muirhead [6] and Constantine and Muirhead [2] gave partial differential equations for the functions of 2 F 1 of one argument matrix and two argument matrices, respectively. Such differential equations have been used to obtain asymptotic expansions of the functions (Muirhead [7], [8], [9], Sugiura [10]). The purpose of this paper is to derive partial differential equations for the functions 3 F 2 (a 1 a 2 , a 3 ; b 1 , b 2 , R) and 3 F 2 (a 1 , a 2 , a 3 ; b 1 , b 2 ; R, S). Differential equations for 2 F 2 are also obtained.

Macdonald-type identities

1978-03-01

James Lepowsky

Characteristic Classes and Homogeneous Spaces, I

1958-04-01

A. Borel , F. Hirzebruch

The Twenty-Seven Lines upon the Cubic Surface

1913-01-01

Archibald Henderson

Harmonic analysis on real reductive groups I the theory of the constant term

1975-06-01

Harish-chandra Harish-Chandra

SIMPLE IRREDUCIBLE GRADED LIE ALGEBRAS OF FINITE GROWTH

1968-12-31

Victor G. Kač

We classify the simple graded Lie algebras , for which the dimension of the space grows as some power of , under the additional assumption that the adjoint representation of … We classify the simple graded Lie algebras , for which the dimension of the space grows as some power of , under the additional assumption that the adjoint representation of on is irreducible. From these results we obtain a classification of the primitive infinite-dimensional Cartan pseudogroups of transformations and a classification of symmetric spaces.

Some enumerative formulae for algebraic curves

1958-10-01

I. G. Macdonald

On some bruhat decomposition and the structure of the hecke rings of p-Adic chevalley groups

1965-12-01

Nagayoshi Iwahori , Hisatake Matsumoto

View PDF Chat