The importance of the Selberg integral

2008/01/01 by Peter J. Forrester, S. Ole Warnaar · 334 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebra over a field #Combinatorics #Conjecture #Eigenvalues and eigenvectors #Euler's formula #Generalization #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Measure (data warehouse) #Philosophy #Physics #Polynomial #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Random matrix #Riemann hypothesis #Selberg trace formula #TRACE (psycholinguistics)

paper · pdf · doi:10.1090/s0273-0979-08-01221-4

published in Bulletin of the American Mathematical Society 45(4), 489 (American Mathematical Society)

openalex publication_date 2008/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08

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.

Citations

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Cited by

• Local Conformal Structure of Liouville Quantum Gravity
• Toeplitz minors and specializations of skew Schur polynomials
• Selberg integral [wikipedia]