The importance of the Selberg integral

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.

43 pages

References in corpus (2)

• The Calogero-Sutherland Model and Generalized Classical Polynomials
• Random matrix theory, the exceptional Lie groups, and L-functions

Cited by in corpus (17)

• The Bulk Dual of SYK: Cubic Couplings
• Essays on the theory of elliptic hypergeometric functions
• Knot Homology from Refined Chern-Simons Theory
• Orbifolds and Exact Solutions of Strongly-Coupled Matrix Models
• 2d-4d Connection between q-Virasoro/W Block at Root of Unity Limit and Instanton Partition Function on ALE Space
• A random matrix decimation procedure relating $\beta =2/(r+1)$ to $\beta =2(r+1)$
• On Complex Gamma-Function Integrals
• q-deformations of two-dimensional Yang-Mills theory: Classification, categorification and refinement
• Virasoro irregular conformal block and beta deformed random matrix model
• Toeplitz minors and specializations of skew Schur polynomials
• Classical Virasoro irregular conformal block II
• Higher Spin de Sitter Hilbert Space
• Connection Formula for the Jackson Integral of Type $A_n$ and Elliptic Lagrange Interpolation
• Rolling tachyons for separated brane-antibrane systems
• Twisting perturbed parafermions
• Multivariable Bessel polynomials related to the hyperbolic Sutherland model with external Morse potential
• Vertex Algebras $\mathcal{W}(p{)}^{A_m}$ and $\mathcal{W}(p{)}^{D_m}$ and Constant Term Identities