The importance of the Selberg integral

TL;DR

The paper reviews the historical development, significance, and diverse applications of the Selberg integral, highlighting its impact across various fields of mathematics and physics over the past decades.

Contribution

It provides a comprehensive overview of the Selberg integral's rise to prominence and its role in advancing multiple mathematical and physical theories.

Findings

Used to prove a conjecture in random matrix theory

Contributed to the proof of cases of the Macdonald conjectures

Central role in quantum many-body systems and orthogonal polynomials

Abstract

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.