The importance of the Selberg integral

By Peter J. Forrester, A. Selberg. 2007

348 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/043ecedec1553a0781977bf870c61f54e0543680

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In Discrete Mathematics and Combinatorics

1. V. Spiridonov
2. S. Ole Warnaar
3. Hjalmar Rosengren

In Mathematical Physics

1. G. Vartanov
2. V. Spiridonov

In Statistics and Probability

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Citations in the corpus

• Toeplitz Matrices and Toeplitz Determinants under the Impetus of the Ising Model: Some History and Some Recent Results (2012, neutral): “j=0 Γ(1 +2α+jγ)Γ(1 +(j+1)γ) Γ(1 +α+β+jγ)Γ(1 +α−β+jγ)Γ(1 +γ) where ℜα>− 1 2 , ℜγ>−min ˆ 1 n , ℜ(2α+1) n−1 ˙ . In a published form, this formula first appeared in the thesis of Morris in 1982 (see [ForWar] for a historic account). The interest in (147) was due to the efforts to prove one of Dyson’”
• The bulk dual of SYK: cubic couplings (2017, neutral): “A more involved relation between the different Sn,p is [38, 39], Sn,p(α, β, γ) = Sn,p−1(α, β, γ) p n− p+ 1 sin (π(n− p+ 1)γ) sin (π(α + β + (n+ p− 2)γ) sin(πpγ) sin (π(α + (p− 1)γ)) .”
• Elliptic Hypergeometry of Supersymmetric Dualities (2009, background): “The Selberg integral plays a fundamental role in mathematics and mathematical physics because of a large number of applications [30].”
• Essays on the theory of elliptic hypergeometric functions (2008, positive): “ of Gustafson [96]. A number of other limiting transitions in parameters leads to the Selberg integral – a fundamentally important integral with a large number of applications in mathematical physics [97]: Z1 0 ··· Z1 0 Y 1≤j≤n xα−1 j (1−xj) β−1 Y 1≤j<k≤n |xj −xk|2γ dx1 ···dxn = Y 1≤j≤n Γ(α+(j−1)”