Overgroups of exterior powers of an elementary group. I. Levels and\n normalizers

2018/01/24 by Roman Lubkov, Lubkov, Roman, Ilia Nekrasov +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1801.07918

openalex publication_date 2018/01/24 · openalex created_date 2022/09/04 · openalex updated_date 2026/08/04

Abstract

In the present paper, we prove the first part in the standard description of\ngroups H lying between m-th exterior power of elementary group E(n,R) and\nthe general linear group GL_ binomnm(R). We study structure of the\nexterior power of elementary group and its relative analog\nE\( binomnm,R,A\). In the considering case n \≥ 3m, the\ndescription is explained by the classical notion of level: for every such H\nwe find unique ideal A of the ring R. Motivated by the problem, we prove\nthe coincidence of the following groups: normalizer of the exterior power of\nelementary group, normalizer of the exterior power of special linear group,\ntransporter of the exterior power of elementary group into the exterior power\nof special linear group, and an exterior power of general linear group. This\nresult mainly follows from the found explicit equations for the exterior power\nof algebraic group scheme GL_n(\_).\n

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