AFLT-type Selberg integrals

In their 2011 paper on the AGT conjecture, Alba, Fateev, Litvinov and Tarnopolsky (AFLT) obtained a closed-form evaluation for Selberg integral over product of two Jack polynomials, thereby unifying well-known Kadell Hua--Kadell integrals. this we use variety symmetric functions function techniques to prove generalisations AFLT integral. These include (i) an $\mathrm{A}_n$ analogue integral, containing polynomials in integrand; (ii) generalisation $\gamma=1$ (the Schur or GUE case), $n+1$ functions; (iii) elliptic which role is played by pair interpolation (iv) Macdonald polynomials.