A Notion of Induction-restriction Depth of Multimatrix Algebra Inclusions Applied to Subgroups

We define a notion of depth for an inclusion of multimatrix algebras B ⊆ A based on a comparison of powers of the inductionrestriction table M (and its transpose matrix). The depth of the semisimple subalgebra B in the semisimple algebra A is the least positive integer n ≥ 2 for which M ≤ qM for some q ∈ Z+. We prove that a depth two subalgebra is a normal subalgebra, and conversely. As a corollary, a depth n subalgebra is a normal subalgebra of its (n − 2)’nd iterated endomorphism algebra in a tower above B ⊆ A. Odd and even depth may be expressed in terms of graphical distance between B-simples in the inclusion diagram of B ⊆ A. Applied to subgroups via complex group algebras, the depth of a subgroup H in a finite group G is bounded above by 2n if the core is an intersection of n conjugates of H in G; with upper bound 2n − 1 if the core is trivial. We prove that the subgroup depth of symmetric groups Sn < Sn+1 is 2n − 1. An appendix by S. Danz and B. Külshammer determines the subgroup depth of alternating groups An < An+1 to be 2(n − ⌈ √ n ⌉) + 1.