A kicking basis for the two-column Garsia-Haiman modules

TL;DR

This paper introduces a new explicit 'kicking basis' for Garsia-Haiman modules with at most two columns, providing a combinatorial tool that could simplify understanding their structure and dimension.

Contribution

It constructs an explicit basis for two-column Garsia-Haiman modules using Orbit Harmonics, advancing combinatorial and algebraic understanding.

Findings

Provides a new explicit basis for the modules

Simplifies the proof of the module's dimension

Connects combinatorial structures with algebraic modules

Abstract

In the early 1990s, Garsia and Haiman conjectured that the dimension of the Garsia-Haiman module is n!, and they showed that the resolution of this conjecture implies the Macdonald Positivity Conjecture. Haiman proved these conjectures in 2001 using algebraic geometry, but the question remains to find an explicit basis for the module which would give a simple proof of the dimension. Using the theory of Orbit Harmonics developed by Garsia and Haiman, we present a "kicking basis" for Garsia-Haiman modules indexed by a partition with at most two columns.