Surveys in Differential Geometry

Volume 15 (2010)

Subgroups of depth three

Pages: 17 – 36



Sebastian Burciu (“Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700, Bucharest, Romania; University of Bucharest, Faculty of Mathematics and Computer Science, 14 Academiei St., RO-010014, Bucharest 1, Romania)

Lars Kadison (Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab, 209 S. 33rd St., Philadelphia, PA 19104, U.S.A.)


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.

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