Selberg integrals of Alba–Fateev–Litvinov–Tarnopolsky type

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In 1944 Atle Selberg published a remarkable n-dimensional analogue of Euler's beta integral which now bears his name. The Selberg integral has come to be regarded as one of the most important hypergeometric integrals, a reputation which is upheld by its uses in fields such as random matrix theory, analytic number theory, conformal field theory and enumerative and algebraic combinatorics. In their verification of the AGT conjecture for SU(2), Alba, Fateev, Litvinov and Tarnopolsky (AFLT) discovered a new generalisation of the Selberg integral over a pair of Jack polynomials. The AFLT integral unifies the well-known Kadell and Hua–Kadell integrals. The purpose of this thesis is to investigate generalisations of the AFLT integral in several directions, all based on symmetric function theory. Using new Cauchy-type identities for Macdonald polynomials we present two An Selberg integrals over a pair of Jack polynomials; one directly generalising the AFLT integral, and one generalising the A2 Selberg integral of Warnaar. Following Matsuo and Zhang, we then consider An Selberg integrals with n+1 symmetric functions in the integrand. Here our results are restricted to the Schur case, where we use several new integral formulas for complex Schur functions to evaluate the An integral first considered by Matsuo and Zhang. To conclude, we discuss other recent developments including the elliptic AFLT integral, an AFLT integral for Macdonald polynomials, an Askey–Habseiger–Kadell-type q-AFLT integral, and several open problems.