A Proof of the \(\frac{n!}{2}\) Conjecture for Hook Shapes

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.