Elliptic ${\mathrm{A}}_n$ Selberg integrals

We use the elliptic interpolation kernel due to the second author to prove an ${\mathrm{A}}_n$ extension of the elliptic Selberg integral. More generally, we obtain elliptic analogues of the ${\mathrm{A}}_n$ Kadell, Hua-Kadell and Alba-Fateev-Litvinov-Tarnopolsky (or AFLT) integrals.