Some new applications of the Stanley-Macdonald Pieri Rules

TL;DR

This paper explores new applications of the Stanley-Macdonald Pieri rules for Macdonald symmetric functions, demonstrating their utility in deriving algebraic geometric results through combinatorial methods.

Contribution

It introduces novel applications of Pieri rules to connect algebraic geometry results with combinatorial techniques in Macdonald polynomial theory.

Findings

Simplified combinatorial formulas for Pieri rules involving Macdonald polynomials.

Applications of Pieri rules to derive algebraic geometric results.

Progress in using combinatorial methods to understand Macdonald polynomial properties.

Abstract

In a seminal paper Richard Stanley derived Pieri rules for the Jack symmetric function basis. These rules were extended by Macdonald to his now famous symmetric function basis. The original form of these rules had a forbidding complexity that made them difficult to use in explicit calculations. In the early 90's it was discovered that, due to massive cancellations, the dual rule, which expresses skewing by $e_1$ the modified Macdonald polynomial ${\tilde H}_\mu[X;q,t]$, can be given a very simple combinatorial form in terms of corner weights of the Ferrers' diagram of $\mu$. A similar formula was later obtained by the last named author for the multiplication of ${\tilde H}_\mu[X;q,t]$ by $e_1$, but never published. In the years that followed we have seen some truly remarkable uses of these two Pieri rules in establishing highly non trivial combinatorial results in the Theory of Macdonald polynomials. This theory has recently been spectacularly enriched by various Algebraic Geometrical results in the works of Hikita, Schiffmann, Schiffmann-Vasserot, A. Negut and Gorsky-Negut. This development opens up the challenging task of deriving their results by purely Algebraic Combinatorial methods. In this paper we present the progress obtained by means of Pieri rules.