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

Abstract

A well-known representation-theoretic model for the transformed Macdonald polynomial ${\tilde{H}}_{\mu }(Z;t,q)$, 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.

arXiv commentFinal version; 12 pages, clarified proof of Theorem 4.3. To appear in Annals of Combinatorics