A proof of the (n!)/(2) conjecture for hook shapes

2022/03/28 by Sam Armon, Armon, Sam
Mathematics · #05A19 #05E05 #05E10 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2203.15146

openalex publication_date 2022/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A well-known representation-theoretic model for the transformed Macdonald polynomial \widetildeH_μ(Z;t,q), where μ is an integer partition, is given by the Garsia-Haiman module H_μ. We study the (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 μ has hook shape, we use a basis for H_μ due to Adin, Remmel, and Roichman to resolve the (n!)/(2) conjecture by constructing an explicit basis for the intersection of two Garsia-Haiman modules.

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