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

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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.

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  1. Department of Mathematics, University of Southern California, 3620 S Vermont Avenue, Los Angeles, CA, 90089, USA

    Sam Armon

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  1. Sam Armon
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Correspondence to Sam Armon.

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Communicated by Alexander Yong.

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Armon, S. A Proof of the \(\frac{n!}{2}\) Conjecture for Hook Shapes. Ann. Comb. 27, 819–832 (2023). https://doi.org/10.1007/s00026-022-00613-3

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