A realization theorem for modules of constant Jordan type and vector bundles

2010/07/22 by David J. Benson, Benson, David J., Julia Pevtsova +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT

paper · pdf · doi:10.48550/arxiv.1007.3827

14 pages

arxiv created 2010/07/22 · openalex publication_date 2010/07/22 · arxiv updated 2010/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E be an elementary abelian p-group of rank r and let k be a field of characteristic p. We introduce functors F_i from finitely generated kE-modules of constant Jordan type to vector bundles over projective space of dimension r-1. The fibers of these functors encode complete information about the Jordan type of the module. We prove that given any vector bundle of rank s on P^r-1, there is a kE-module M of stable constant Jordan type [1]^s such that the functor F₁ applied to M yields the original vector bundle for p=2 and the Frobenius twist of the original vector bundle for p>2. We also prove that the theorem cannot be improved if p is odd, because if M is any module of stable constant Jordan type [1]^s then the Chern numbers c₁, ... ,c_p-2 of F₁(M) are divisible by p.

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