Symmetric polynomials: DIM integrable systems versus twisted Cherednik systems

TL;DR

This paper explores the deep connections between eigenfunctions of DIM algebra Hamiltonians and twisted Cherednik systems, revealing a unified framework for symmetric functions and their eigenstates.

Contribution

It establishes a novel correspondence between eigenfunctions of DIM algebra Hamiltonians and twisted Cherednik eigenfunctions, unifying them through symmetric functions.

Findings

Eigenfunctions of DIM Hamiltonians are twisted Baker-Akhiezer functions.

Eigenfunctions of twisted Cherednik Hamiltonians are twisted non-symmetric Macdonald polynomials.

A unified symmetric function eigenbasis is constructed, linking both systems.

Abstract

We discuss interrelations between eigenfunctions of the Hamiltonians associated with the commutative (integer ray) subalgebras of the Ding-Iohara-Miki algebra and those of the twisted Cherednik system. In the case of $t=q^{-m}$ with natural $m$, eigenfunctions of the first system of Hamiltonians are the twisted Baker-Akhiezer functions (BAFs) introduced by O. Chalykh, while eigenfunctions of the twisted Cherednik Hamiltonians are twisted non-symmetric Macdonald polynomials. Actually, the twisted Cherednik ground state is symmetric and coincides with a peculiar symmetric BAF. We lift this correspondence to excited states, and claim that both Cherednik eigenfunctions and BAF's can be combined to produce symmetric functions, which coincide with each other and are eigenfunctions of the both DIM Hamiltonians and power sums of the twisted Cherednik Hamiltonians at once. This reflects the correspondence between the DIM algebra and the spherical DAHA explicitly.