The importance of the Selberg integral

Details The importance of the Selberg integral The importance of the Selberg integral

2007-10-22

arxiv.org/abs/0710.3981v1

Bull. Amer. Math. Soc. (N.S.) 45 (2008), 489-534.

Mathematics

Classical Analysis and ODEs

43 pages

Scientific paper

It has been remarked that a fair measure of the impact of Atle Selberg's work is the number of mathematical terms which 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 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, 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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