ON SUBGROUP DEPTH (WITH AN APPENDIX BY S. DANZ AND B. KULSHAMMER)

Abstract

We define a notion of depth for an inclusion of complex semisimple algebras, based on a comparison of powers of the induction-restriction table (and its transpose matrix) and a previous notion of depth in an earlier paper of the second author. We prove that a depth two extension of complex semisimple algebras is normal in the sense of Rieffel, and conversely. Given an extension H ⊆ G of finite groups we prove that the depth of C H in C G is bounded by 2n if the kernel of the permutation representation of G on cosets of H is the intersection of n conjugate subgroups of H. We prove in several ways that the subgroup depth of symmetric groups Sn ⊆ Sn+1 is 2n − 1. An appendix by S. Danz and B. K¨ulshammer determines the subgroup depth of alternating groups An ⊆ An+1 and dihedral group extensions.