On Integral Forms of Specht Modules Labelled by Hook Partitions

2017/06/09 by Susanne Danz, Danz, Susanne, Tommy Hofmann +1
Mathematics · #20C10 #20C11 #20C20 #20C30 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1706.02860

openalex publication_date 2017/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate integral forms of simple modules of symmetric groups over fields of characteristic 0 labelled by hook partitions. Building on work of Plesken and Craig, for every odd prime p, we give a set of representatives of the isomorphism classes of ℤ_p-forms of the simple ℚ_p \mathfrakS_n-module labelled by the partition (n-k,1^k), where n∈ℕ and 0≤ k≤ n-1. We also settle the analogous question for p=2, assuming that n\not≡ 0\pmod4 and k∈\2,n-3\. As a consequence this leads to a set of representatives of the isomorphism classes of ℤ-forms of the simple ℚ\mathfrakS_n-modules labelled by (n-2,1²) and (3,1^n-3), again assuming n\not≡ 0\pmod4.

Related

• Young's seminormal form and simple modules for S_n in characteristic p
• Vertices of simple modules of symmetric groups labelled by hook partitions
• Elementary divisors of Specht modules
• Specht Modules for Weyl Groups
• Constructing all irreducible Specht modules in a block of the symmetric group