Systems of imprimitivity for wreath products

[摘要] Let Gbe an irreducible imprimitive subgroup of GL(n)(F), where Fis a field. Any system of imprimitivity for Gcan be refined to a nonrefinablesystem of imprimitivity, and we consider the question of when such a refinement is unique. Examples show that Gcan have many nonrefinable systems of imprimitivity, and even the number of components is not uniquely determined. We consider the case where Gis the wreath product of an irreducible primitive H <= GL(d)(F) and transitive K <= Sk, where n = dk. We show that Ghas a unique nonrefinable system of imprimitivity, except in the following special case: d = 1, n = kis even, vertical bar H vertical bar = 2, and Kis a subgroup of C-2 (sic) S-n/2. As a simple application, we prove results about inclusions between wreath product subgroups. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-12-01

[关键词] Group theory;Representation theory;Imprimitive matrix groups;Wreath products