Depth two is a nice little theory for subrings (reminiscent to and generalizing H-separable ring extensions) that combines properties of being normal in the sense of Hopf subalgebras and normal in the sense of Galois field extensions. We will discuss why depth two when applied to Hopf subalgebras is precisely normality; giving a new view of normality in left and right versions (based on a paper by Boltje and Kuelshammer).

When depth two is applied to subalgebra pairs of split semisimple artinian algebras, it is identical with a more experimental notion of normality (Rieffel, 1979). A subalgebra of depth n greater than two is the distance out along a tower of right endomorphism algebras and their left regular representations iterated n-2 times before it becomes normal in this sense. When applied to subgroups and their representations over C, depth n is a condition on the matrix M of the induction-restriction table of the subgroup, its product with its transpose matrix and itself alternatingly.

We will discuss this and why Young diagrams show that S_n in the permutation group of n+1 letters is depth 2n-1 (based on a joint paper with Burciu and Kuelshammer). For certain subgroups that are products of permutations groups, the entries of the matrix M are the Littlewood-Richardson numbers in combinatorics.