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

Year 2011, Volume: 9 Issue: 9, 133 - 166, 01.06.2011

Sebastian Burciu Lars Kadison Burkhard Külshammer

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.

Keywords

ring extension, depth, group algebra, Hopf algebra, normal subring, inclusion matrix