On subgroup depth

TL;DR

The paper introduces a new notion of depth for algebra inclusions, relates it to existing concepts, and computes subgroup depths for symmetric, alternating, and dihedral groups.

Contribution

It defines a depth concept for multimatrix algebra inclusions, connects it with prior work, and provides explicit subgroup depth calculations for specific groups.

Findings

Depth 2 extensions are normal extensions as per Rieffel.

Subgroup depth of S_n < S_{n+1} is 2n-1.

Determined subgroup depth for A_n < A_{n+1} and dihedral groups.

Abstract

We define a notion of depth for an inclusion of multimatrix algebras B < A based on a comparison of powers of the induction-restriction table M (and its transpose matrix). This notion of depth coincides with the depth from [Kadison, 2008]. In particular depth 2 extensions coincides with normal extensions as introduced by Rieffel in 1979. For a group extension H < G a necessary depth n condition is given in terms of the core of H in G. We prove that the subgroup depth of symmetric groups S_n < S_{n+1} is 2n-1. An appendix by S. Danz and B. Kuelshammer determines the subgroup depth of alternating groups A_n < A_{n+1} as well as dihedral groups.