Language:   Search:   Contact
World of
Mathematics Database
»ZBMATH« MSC 2000
MSC 2010 Reviewer
Service Subscription
»ZBMATH«
ZBMATH Database | Simple Search Print
Read more | Try MathML | Hide
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

ZBMATH Database Simple Search Advanced Search Command Search

Simple Search

Query:
Enter a query and click »Search«...
Format:
Display: entries per page entries
Zbl 1024.33001
Macdonald, I.G.
Affine Hecke algebras and orthogonal polynomials. (English)
[B] Cambridge Tracts in Mathematics. 157. Cambridge: Cambridge University Press. x, 175 p. \sterling 35.00; \$ 48.00 (2003). ISBN 0-521-82472-9/pbk

This is a beautiful book, treating in a concise and clear way the recent developments concerning the connection between orthogonal polynomials in several variables and root systems in two or more parameters. The reader should be warned: it is not a simple book! A good knowledge of algebra, geometry of root systems and Weyl groups is indispensable. Combined with the ability for formula manipulation and perseverence, the reader is sure to get a better understanding of orthogonal polynomials situated high in the hierarchy of the generalized Askey scheme [the Askey-Wilson polynomials and the Jacobi polynomials due to Heckman and Opdam]. The first four chapters treat the basic properties of affine root systems and a complete characterization thereof, the extended affine Weyl group, the Artin braid group, the double braid group and the affine and double affine Hecke algebra. After these first chapters, that can be seen as a unified foundation, the main [deep] results on orthogonal polynomials follow in Chapter 5. Finally, in Chapter 6, the author treats the case of a rank $1$ affine root system and makes the results, derived previously, explicit, leading to $q$-ultraspherical (Roger) polynomials and the Askey-Wilson polynomials.
[Marcel G.de Bruin (Delft)]
MSC 2000:
*33-02 Research monographs (special functions)
20-02 Research monographs (group theory)
17-02 Research monographs (nonassoc. rings and algebras)
33D52 Basic orthogonal polynomials and functions assoc. with root systems
20C08 Hecke algebras and their representations
17B22

Keywords: affine Hecke algebras; affine root system; braid group; Weyl group; Dynkin diagram; Hecke relation; Askey-Wilson polynomials; continuous $q$-ultraspherical polynomials; Koornwinder polynomials

Cited in: Zbl 1246.20005 Zbl 1106.33015

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering

Login Username: Password:

Highlights
Scientific prize winners of the ICM 2010
Overhang
Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Master Server

Zentralblatt MATH Berlin [Germany]

© FIZ Karlsruhe GmbH

Zentralblatt MATH master server is maintained by the Editorial Office in Berlin, Section Mathematics and Computer Science of FIZ Karlsruhe and is updated daily.

Other Mirror Sites

Copyright © 2013 Zentralblatt MATH | European Mathematical Society | FIZ Karlsruhe | Heidelberg Academy of Sciences
Published by Springer-Verlag | Webmaster