Cut-and-join operators and Macdonald polynomials from the 3-Schur functions

TL;DR

This paper explores how Macdonald polynomials can be derived from 3-Schur functions through projection of plane-partition variables and discusses the role of cut-and-join operators in this process, mainly focusing on level two.

Contribution

It provides a detailed derivation of Macdonald polynomials from 3-Schur functions and introduces the interpolation role of cut-and-join operators between different cases.

Findings

Macdonald polynomials emerge from 3-Schur functions via projection.

Cut-and-join operators interpolate between different polynomial cases.

Analysis primarily focused on level two.

Abstract

We demonstrate in some detail how Macdonald polynomials emerge from the recently introduced 3-Schur functions when the plane-partition vector time-variables are projected onto the ordinary scalar times under non-vanishing angles, which depend on $q$ and $t$. We also explain how the cut-and-join operators smoothly interpolate between different cases. Most of consideration is restricted to level two.