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.

Boltje's plus constructions extend two well-known constructions on Mackey functors, the fixed-point functor and the fixed-quotient functor. In this paper, we show that the plus constructions are induction and coinduction functors of general module theory. As an application, we construct simple Mackey functors from simple restriction functors and simple transfer functors. We also give new proofs for the classification theorem for simple Mackey functors and semisimplicity theorem of Mackey functors. © 2007 Elsevier Inc. All rights reserved

We introduce a new combinatorial object called tower diagrams and prove fundamental properties of these objects. We also introduce an algorithm that allows us to slide words to tower diagrams. We show that the algorithm is well-defined only for reduced words which makes the algorithm a test for reducibility. Using the algorithm, a bijection between tower diagrams and finite permutations is obtained and it is shown that this bijection specializes to a bijection between certain labellings of a given tower diagram and reduced expressions of the corresponding permutation.

The unified treatment of the five module-theoretic notions, transfer, inflation, transport of structure by an isomorphism, deflation and restriction, is given by the theory of biset functors, introduced by Bouc. In this paper, we construct the algebra realizing biset functors as representations. The algebra has a presentation similar to the well-known Mackey algebra. We adopt some natural constructions from the theory of Mackey functors and give two new constructions of simple biset functors. We also obtain a criterion for semisimplicity in terms of the biset functor version of the mark homomorphism. The criterion has an elementary generalization to arbitrary finite-dimensional algebras over a field. © 2008 Elsevier Inc. All rights reserved

The aim of this paper is to describe the group of endo-trivial modules for a p-group P , in terms of the obstruction group for the gluing problem of Borel-Smith functions. Explicitly, we shall prove that there is a split exact sequenceof abelian groups where T (P ) is the endo-trivial group of P , and C b (P ) is the group of Borel-Smith functions on P . As a consequence, we obtain a set of generators of the group T (P ) that coincides with the relative syzygies found by Alperin. In order to prove the result, we solve gluing problems for the functor B * of super class functions, the functor R * Q of rational class functions and the functor C b of Borel-Smith functions. Contents