An analogue of Schur functions for the plane partitions

TL;DR

This paper explores extending Schur functions to plane partitions using recursive methods, potentially leading to new polynomial families like Macdonald polynomials and revealing complex algebraic structures.

Contribution

It proposes a recursive framework to generalize Schur functions to plane partitions, opening pathways to new polynomial classes and algebraic insights.

Findings

Recursion in partition size can be generalized and deformed.

Potential derivation of Macdonald polynomials from this framework.

Discovery of a rich non-abelian structure with 3-Schurs as eigenfunctions.

Abstract

An attempt is described to extend the notion of Schur functions from Young diagrams to plane partitions. The suggestion is to use the recursion in the partition size, which is easily generalized and deformed. This opens a possibility to obtain Macdonald polynomials by a change of recursion coefficients and taking appropriate limit from three to two dimensions -- though details still remain to be worked out. Another perspective is opened by the observation of a rich non-abelian structure, extending that of commuting cut-and-join operators, for which the discovered 3-Schurs are the common eigenfunctions.