Elliptic triad

TL;DR

This paper explores an elliptic generalization of the Macdonald polynomial framework by embedding it into Noumi-Shiraishi functions and examining linear equations, aiming to extend the theory with elliptic structures.

Contribution

It introduces an elliptic extension of the triad involving Noumi-Shiraishi functions and discusses approaches to formulate elliptic linear equations within this framework.

Findings

Proposes elliptic deformation of the triad

Discusses multiple approaches to elliptic linear equations

Lays groundwork for elliptic Macdonald theory

Abstract

The triad refers to embedding the Macdonald polynomials into the Noumi-Shiraishi functions and their reduction to solutions of simple linear equations at particular values of $t$. It provides an alternative definition of Macdonald theory. We discuss lifting the triad to an elliptic generalization of the Noumi-Shiraishi functions. The central unknown ingredient is linear equations, for which we discuss various possible approaches, including immediate elliptic deformation of periodicity conditions, (elliptic) Ding-Iohara-Miki algebra operators, and elliptic Kostka coefficients.