Overgroups of elementary groups in polyvector representations

2022/03/25 by Roman Lubkov, Lubkov, Roman
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2203.13683

openalex publication_date 2022/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We initiate the study of subgroups H of the general linear group GL_\binomnm(R) over a commutative ring R that contain the m-th exterior power of an elementary group \bigwedge^mE_n(R). Each such group H corresponds to a uniquely defined level (A₀,…,A_m-1), where A₀,…,A_m-1 are ideals of R with certain relations. In the crucial case of the exterior squares, we state the subgroup lattice to be standard. In other words, for \bigwedge²E_n(R) all intermediate subgroups H are parametrized by a single ideal of the ring R. Moreover, we characterize \bigwedge^mGL_n(R) as the stabilizer of a system of invariant forms. This result is classically known for algebraically closed fields, here we prove the corresponding group scheme to be smooth over ℤ. So the last result holds over arbitrary commutative rings.

Related

• Overgroups of exterior powers of an elementary group. I. Levels and\n normalizers
• Representations of the general linear groups which are irreducible over subgroups
• The stable representations of \GL_N over finite local\n principal ideal rings
• Overgroups of subsystem subgroups in exceptional groups: inside a sandwich
• A multiplicity one theorem for groups of type A_n over discrete valuation rings