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# Surveys in Differential Geometry

## Volume 15 (2010)

### Subgroups of depth three

Pages: 17 – 36

DOI: http://dx.doi.org/10.4310/SDG.2010.v15.n1.a2

#### Authors

#### Abstract

A subalgebra pair of semisimple complex algebras $B \subseteq A$ with inclusion matrix $M$ is depth two if $MM^t M \leq nM$ 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 $S^2 \leq nS$ if 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. A tower of Hopf algebras $A \supseteq B \supseteq C$ is shown to be depth-3 if $C \subseteq \mtr{core}(B)$; and this is also a necessary condition if $A$, $B$ and $C$ are group algebras.