Masatoshi Noumi

Macdonald Polynomials (eBook, PDF)

Commuting Family of q-Difference Operators and Their Joint Eigenfunctions

• Format: PDF

Produktbeschreibung

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 Hall–Littlewood 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 Macdonald–Ruijsenaars 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.

Produktdetails

• Verlag: Springer Nature Singapore
• Erscheinungstermin: 8. September 2023
• Englisch
• ISBN-13: 9789819945870
• Artikelnr.: 68884190

Autorenporträt

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.

Inhaltsangabe

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

---

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