Subgroups of depth three and more

2009/01/20 by Sebastian Burciu, Burciu, Sebastian, Lars Kadison +1 · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Finite Group Theory Research #math.GR #math.RT #msc:16W30 #msc:20D25

paper · pdf · doi:10.48550/arxiv.0901.3039

16 pages, some improvements in notation and the proof of Prop. 1.2

arxiv created 2010/06/09 · arxiv updated 2010/06/10

Abstract

A subalgebra pair of semisimple complex algebras B < A with inclusion matrix M is depth two if MM^t M < nM for some positive integer n and all corresponding entries. If A and B are the group algebras of finite group-subgroup pair H < G, the induction-restriction table equals M and S = MM^t satisfies S² < nS iff the subgroup H is depth three in G; similarly depth n > 3 by successive right multiplications of this inequality with alternately M and M^t. We show that a Frobenius complement in a Frobenius group is a nontrivial class of examples of depth three subgroups. Depth-3 towers of Hopf algebras are also considered: a tower of Hopf algebras A > B > C is shown to be depth-3 if C < core(B).

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