A proof of the $\frac{n!}{2}$ conjecture for hook shapes

TL;DR

This paper proves the $rac{n!}{2}$ conjecture for hook-shaped partitions by explicitly constructing a basis for the intersection of two Garsia-Haiman modules, advancing understanding in algebraic combinatorics.

Contribution

It provides a proof of the $rac{n!}{2}$ conjecture specifically for hook shapes using a basis from prior work, which was previously unresolved.

Findings

Confirmed the $rac{n!}{2}$ conjecture for hook shapes.

Constructed an explicit basis for the intersection of Garsia-Haiman modules.

Enhanced understanding of the structure of these modules in representation theory.

Abstract

A well-known representation-theoretic model for the transformed Macdonald polynomial $\widetilde{H}_\mu(Z;t,q)$, where $\mu$ is an integer partition, is given by the Garsia-Haiman module $\mathcal{H}_\mu$. We study the $\frac{n!}{k}$ conjecture of Bergeron and Garsia, which concerns the behavior of certain $k$-tuples of Garsia-Haiman modules under intersection. In the special case that $\mu$ has hook shape, we use a basis for $\mathcal{H}_\mu$ due to Adin, Remmel, and Roichman to resolve the $\frac{n!}{2}$ conjecture by constructing an explicit basis for the intersection of two Garsia-Haiman modules.