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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



The importance of the Selberg integral

Authors: Peter J. Forrester and S. Ole Warnaar
Journal: Bull. Amer. Math. Soc. 45 (2008), 489-534
MSC (2000): Primary 00-02, 33-02
Published electronically: July 2, 2008
MathSciNet review: 2434345
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Abstract: It has been remarked that a fair measure of the impact of Atle Selberg’s work is the number of mathematical terms that bear his name. One of these is the Selberg integral, an $n$-dimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a question to Selberg from Enrico Bombieri, more than thirty years after its initial publication. In quick succession the Selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the Macdonald conjectures. It further initiated the study of $q$-analogues, which in turn enriched the Macdonald conjectures. We review these developments and proceed to exhibit the sustained prominence of the Selberg integral as evidenced by its central role in random matrix theory, Calogero–Sutherland quantum many-body systems, Knizhnik–Zamolodchikov equations, and multivariable orthogonal polynomial theory.

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Additional Information

Peter J. Forrester
Affiliation: Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
MR Author ID: 68170

S. Ole Warnaar
Affiliation: Department of Mathematics, The University of Queensland, St Lucia, Queensland 4072, Australia
MR Author ID: 269674

Received by editor(s): March 21, 2008
Received by editor(s) in revised form: April 21, 2008
Published electronically: July 2, 2008
Article copyright: © Copyright 2008 American Mathematical Society