Affine Hecke Algebras and Orthogonal Polynomials by I. G. MacDonald – A Comprehensive Study of Root Systems and Multivar

Introduction

For students and researchers delving into the intricate world of advanced algebra and special functions, AFFINE HECKE ALGEBRAS AND ORTHOGONAL POLYNOMIALS by I. G. MacDonald stands as a definitive resource. This hardcover volume from Cambridge University Press offers a rigorous yet accessible journey into the heart of modern mathematics, bridging the gap between representation theory and classical analysis. Whether you are a graduate student in India or a seasoned mathematician, this book provides the essential framework for understanding orthogonal polynomials in several variables, attached to root systems, and depending on multiple parameters.

Book Overview

This comprehensive monograph systematically develops the theory of orthogonal polynomials that arise from root systems, a field to which I. G. MacDonald has made groundbreaking contributions. The book is structured to guide readers from foundational concepts to advanced results, culminating in Chapter 5 which contains all the main theorems. The first four chapters lay the necessary groundwork—covering Hecke algebras, affine root systems, and basic hypergeometric series—before unlocking the powerful Macdonald polynomials. This is not merely a reference; it is a teaching text designed to build deep understanding through clear exposition and logical progression.

Key Highlights

• Unified Foundation: Presents a coherent theory of orthogonal polynomials in several variables attached to root systems, integrating two or more parameters seamlessly.
• Authoritative Author: Written by I. G. MacDonald, a principal contributor to the field, ensuring depth and accuracy.
• Progressive Structure: Chapters 1–4 build essential tools before Chapter 5 delivers the main results, making complex ideas digestible.
• Hardcover Durability: A premium Cambridge University Press edition designed for long-term academic use.
• Indian Market Relevance: Ideal for advanced mathematics curricula in Indian universities, particularly in algebra and special functions.

Inside the Book

The book opens with a thorough introduction to affine Hecke algebras, including the Iwahori–Hecke algebra and its representations. It then explores root systems and affine Weyl groups, providing the geometric backbone for the polynomial theory. Subsequent chapters introduce basic hypergeometric series and the Macdonald–Koornwinder polynomials, leading to the celebrated Macdonald conjectures and their proofs. The final chapter presents the core results: orthogonality, duality, and evaluation formulas for Macdonald polynomials, along with applications to constant term identities and combinatorial interpretations.

Key Topics

• Affine Hecke algebras and their representations
• Root systems, affine Weyl groups, and weight lattices
• Macdonald polynomials: definitions, properties, and specializations
• Orthogonality and duality for multivariate orthogonal polynomials
• Basic hypergeometric series and q-analogues
• Constant term identities and combinatorial applications

Reader Benefits

By studying this book, readers gain a solid command of advanced algebraic structures and their connections to special functions. The clear, step-by-step exposition helps demystify complex topics, making them accessible for self-study or classroom use. The book also serves as a gateway to current research areas such as representation theory of p-adic groups, integrable systems, and algebraic combinatorics. Indian students preparing for competitive exams like NET, GATE, or pursuing PhDs will find this an invaluable addition to their library.

Learning Outcomes

• Understand the construction and representation theory of affine Hecke algebras.
• Derive and manipulate Macdonald polynomials for any root system.
• Prove orthogonality and duality relations for multivariate orthogonal polynomials.
• Apply the theory to solve constant term identities and combinatorial problems.
• Connect affine Hecke algebras to other areas like quantum groups and statistical mechanics.

Who Should Read

This book is intended for graduate students and researchers in mathematics, particularly those specializing in algebra, representation theory, special functions, or combinatorics. It is also suitable for advanced undergraduate students with a strong background in abstract algebra and Lie theory. Physicists working in integrable models or conformal field theory will also benefit from the mathematical foundations presented here.

About the Author

I. G. MacDonald is a distinguished British mathematician and a leading figure in algebraic combinatorics and representation theory. He is best known for his foundational contributions to symmetric functions, Hecke algebras, and orthogonal polynomials. His books, including Symmetric Functions and Hall Polynomials, are considered classics in the field. A Fellow of the Royal Society, MacDonald’s work continues to inspire generations of mathematicians worldwide.

About the Publisher

Cambridge University Press is a world-renowned academic publisher with a legacy spanning over four centuries. Known for its rigorous editorial standards and commitment to scholarly excellence, CUP publishes seminal works across all disciplines. This hardcover edition reflects the high-quality production values that Indian readers and libraries have come to expect from this prestigious press.

Conclusion

AFFINE HECKE ALGEBRAS AND ORTHOGONAL POLYNOMIALS is more than a textbook—it is a masterclass in advanced mathematics. For anyone serious about understanding the deep interplay between algebra and analysis, this book is an indispensable companion. Order your copy from Bookshops.in today and add a cornerstone of modern mathematics to your collection.