Macdonald Polynomials

This book is a volume of the Springer Briefs in Mathematical Physics and serves as an introductory textbook on the theory of Macdonald polynomials. It is based on a series of online lectures given by the author at the Royal Institute of Technology (KTH), Stockholm, in February and March 2021. Macdonald polynomials are a class of symmetric orthogonal polynomials in many variables. They include important classes of special functions such as Schur functions and HallLittlewood polynomials and play important roles in various fields of mathematics and mathematical physics. After an overview of Schur functions, the author introduces Macdonald polynomials (of type A, in the GL n version) as eigenfunctions of a q -difference operator, called the MacdonaldRuijsenaars operator, in the ring of symmetric polynomials. Starting from this definition, various remarkable properties of Macdonald polynomials are explained, such as orthogonality, evaluation formulas, and self-duality, with emphasis on the roles of commuting q -difference operators. The author also explains how Macdonald polynomials are formulated in the framework of affine Hecke algebras and q -Dunkl operators.

Provides an introduction to Macdonald polynomials requiring only an undergraduate knowledge of algebra and analysis Presents selected topics that are easily accessible to readers with a background in mathematical physics Gives direct proofs to important theorems and formulas whose proofs are missing or hard to find in the literature

Autorentext

The author is currently Professor Emeritus at Kobe University and Professor at Rikkyo University. He previously held positions at Sophia University and the University of Tokyo. He was Invited Speaker at the ICM 2002 and also Plenary Speaker at the ICMP 2018.

Inhalt
Overview of Macdonald polynomials.- Preliminaries on symmetric functions.- Schur functions.- Macdonald polynomials: Definition and examples.- Orthogonality and higher order q -dierence operators.- Self-duality, Pieri formula and Cauchy formulas.- LittlewoodRichardson coefficients and branching coefficients.- Affine Hecke algebra and q -Dunkl operators (overview).