Hunt for 3-Schur polynomials

TL;DR

This paper investigates the recent development of 3-Schur polynomials linked to plane partitions, offering new insights and a simplified framework for their construction, though some details remain unresolved.

Contribution

It introduces a simple definition of time-variables and a cut-and-join operator for 3-Schur functions, advancing understanding of their structure.

Findings

Defined time-variables ${f P}_{i eq j}$ for 3-Schur polynomials

Constructed a cut-and-join operator $ ext{W}_2$ generating 3-Schur functions

Identified unresolved coefficients requiring further research

Abstract

This paper describes our attempt to understand the recent success of Na Wang in constructing the 3-Schur polynomials, associated with the plane partitions. We provide a rather detailed review and try to figure out the new insights, which allowed to overcome the problems of the previous efforts. In result we provide a very simple definition of time-variables ${\bf P}_{i\geqslant j}$ and the cut-and-join operator $\hat W_2$, which generates the set of $3$-Schur functions. Some coefficients in $\hat W_2$ remain undefined and require more effort to be fixed.