The importance of the Selberg integral

Forrester, P. J. and Warnaar, S. O. (2008) The importance of the Selberg integral. Bulletin (New Series) of the American Mathematical Society, 45 4: 489-534. doi:10.1090/S0273-0979-08-01221-4

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Author Forrester, P. J.
Warnaar, S. O.
Title The importance of the Selberg integral
Journal name Bulletin (New Series) of the American Mathematical Society   Check publisher's open access policy
ISSN 0273-0979
1088-9485
Publication date 2008-07
Sub-type Critical review of research, literature review, critical commentary
DOI 10.1090/S0273-0979-08-01221-4
Open Access Status DOI
Volume 45
Issue 4
Start page 489
End page 534
Total pages 46
Editor S. Friedlander
Place of publication Providence, Rhode Island, U.S.A.
Publisher American Mathematical Society
Collection year 2009
Language eng
Subject 010104 Combinatorics and Discrete Mathematics (excl. Physical Combinatorics)
0101 Pure Mathematics
010101 Algebra and Number Theory
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.
Keyword Selberg integral
Euler beta integral
Random matrix theory
Macdonald conjectures
Q-Index Code C1
Q-Index Status Confirmed Code
Institutional Status Unknown

Document type: Journal Article
Sub-type: Critical review of research, literature review, critical commentary
Collections: 2009 Higher Education Research Data Collection
Excellence in Research Australia (ERA) - Collection
School of Mathematics and Physics
 
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Created: Fri, 12 Dec 2008, 15:18:15 EST by Marie Grove on behalf of Faculty of Science