Communicated by Michel Broué MSC: 16D20 16D90 20C05 20D30 Keywords: Depth of ring extensions Depth of subgroups Group algebra extension Bisets Subgroup lattice We investigate notions of depth for inclusions of rings B ⊆ A, in particular for group algebra extensions R H ⊆ RG for finite groups H G and a non-zero commutative ring R. A group-theoretic (or combinatorial) notion of depth for H in G is defined and used to show that R H ⊆ RG has always finite depth. We compare the depths of H G and R H ⊆ RG, and investigate how the depth varies with R.

So far there exist three independent constructions of two different canonical versions of Brauer's induction theorem for complex characters due to V. Snaith, P. Symonds, and the author. ''Canonical'' in this context means functorial with respect to restrictions along group homomorphisms. In this article we axiomatize the situation in which the above canonical induction formulae are constructed. Mackey functors and related structures arise in this way naturally as a convenient language. This approach allows us to construct canonical induction formulae for arbitrary Mackey functors. In particular we obtain canonical induction formulae for the Brauer character ring, the group of projective characters, the ring of trivial source modules, and the ring of linear source modules. In most cases, it is not difficult to construct such formulae over the rational numbers. A much more subtle question is whether the constructed formula comes from a canonical induction formula defined over the integers. We give a sufficient condition in the general framework of Mackey functors for a canonical induction formula to be integral. As an application we show how canonical induction formulae allow the construction of functorial maps on representation rings in terms of functorial maps on subrings, as, for example, the span of linear characters in the case of the canonical Brauer induction formula. This will be used in a subsequent article in the case of Adams operations and Chern classes.

Let R = 0 be a commutative ring, and let H be a subgroup of finite index in a group G. We prove that the group ring RG is a ring extension of the group ring R H of depth two if and only if H is a normal subgroup of G. We also show that, under suitable additional hypotheses, an analogous result holds for extensions of Hopf algebras over R.

The theory of biset functors, introduced by Serge Bouc, gives a unified treatment of operations in representation theory that are induced by permutation bimodules. In this paper, by considering fibered bisets, we introduce and describe the basic theory of fibered biset functors which is a natural framework for operations induced by monomial bimodules. The main result of this paper is the classification of simple fibered biset functors.

For every composition λ of a positive integer r, we construct a finite chain complex whose terms are direct sums of permutation modules M μ for the symmetric group S r with Young subgroup stabilizers S μ . The construction is combinatorial and can be carried out over every commutative base ring k. We conjecture that for every partition λ the chain complex has homology concentrated in one degree (at the end of the complex) and that it is isomorphic to the dual of the Specht module S λ . We prove the exactness in special cases.