The sl_3 Selberg integral

TL;DR

This paper introduces a new sl_3 generalization of the Selberg integral by extending evaluation symmetry and deriving related identities for Macdonald polynomials, enriching the theory of multivariate integrals.

Contribution

It presents a novel sl_3 generalization of the Selberg integral and related identities, extending classical and q-analogues in the context of Macdonald polynomials.

Findings

New sl_3 Selberg integral derived from Macdonald polynomial identities

Extension of q-Selberg integral to transformations between different q-integral dimensions

Connections established between classical limits and Macdonald polynomial evaluations

Abstract

Using an extension of the well-known evaluation symmetry, a new Cauchy-type identity for Macdonald polynomials is proved. After taking the classical limit this yields a new sl_3 generalisation of the famous Selberg integral. Closely related results obtained in this paper are an sl_3-analogue of the Askey-Habsieger-Kadell q-Selberg integral and an extension of the q-Selberg integral to a transformation between q-integrals of different dimensions.