Abstract:
There are normal sub-blocks of nilpotent blocks which are NOT nilpotent or, equivalently, nilpotent extensions of non-nilpotent blocks. In this paper we determine the source algebra structure of the non-nilpotent blocks involved in these situations. Actually, we introduce a new type of blocks - called the inertial blocks - which include the nilpotent blocks and is closed by taking normal sub-blocks.

Abstract:
In "Frobenius Categories versus Brauer Blocks", Progress in Math. 274, we have introduced the Frobenius categories F over a finite p-group P, and we have associated to F - suitably endowed with some central k*-extensions - a "Grothendieck group" as an inverse limit of Grothendieck groups of categories of modules in characteristic p obtained from F, determining its rank. Our purpose here is to introduce an analogous inverse limit of Grothendieck groups of categories of modules in characteristic zero obtained from F, determining its rank and proving that its extension to a field is canonically isomorphic to the direct sum of the corresponding extensions of the "Grothendieck groups" above associated with the centralizers in F of a suitable set of representatives of the F-classes of elements of P.

Abstract:
In "Frobenius Categories versus Brauer Blocks", Progress in Math. 274, we introduce the Frobenius P-categories giving two quite different definitions of them. In this paper, we exhibit a third equivalent definition based on the form of the old Alperin Fusion Theorem; this theorem can be reformulated in our abstract setting, and ultimately depends on the behavior of the so-called F-essential subgroups of P: we call "Alperin condition" a sufficient form of this behavior. Then, we prove that a divisible P-category F is a Frobenius P-category if and only if all the partial normalizers of a suitable set of representatives for the F-isomorphism classes of subgroups of P fulfill both the Sylow and the Alperin conditions.

Abstract:
We show that the refinement of Alperin's Conjecture proposed in "Frobenius Categories versus Brauer Blocks", Progress in Math. 274, can be proved by checking that this refinement holds on any central k*-extension of a finite group H containing a normal simple group S with trivial centralizer in H and p'-cyclic quotient H/S. This paper improves our result in [ibidem, Theorem 16.45] and repairs some bad arguments there.

Abstract:
The weights for a finite group G with respect to a prime number p where introduced by Jon Alperin, in order to formulate his celebrated conjecture. In 1992, Everett Dade formulates a refinement of Alperin's conjecture involving ordinary irreducible characters - with their defect - and, in 2000, Geoffrey Robinson proves that the new conjecture holds for p-solvable groups. But this refinement is formulated in terms of a vanishing alternating sum, without giving any possible refinement for the weights. In this note we show that, in the case of the p-solvable finite groups, the method developed in a previous paper [arXiv.org/abs/1005.3748] can be suitably refined to provide, up to the choice of a polarization, a natural bijection - namely compatible with the action of the group of outer automorphisms of G - between the sets of absolutely irreducible characters of G and of G-conjugacy classes of suitable inductive weights, preserving blocks and defects.

Abstract:
We introduce a new avatar of a Frobenius P-category F in the form of a suitable sub-ring H_F of the double Burnside ring of P - called the Hecke algebra of F - where we are able to formulate the generalization to a Frobenius P-category of the Alperin Fusion Theorem, the "canonical decomposition" of the morphisms in the exterior quotient of a Frobenius P-category restricted to the selfcentralizing objects as developed in the chapter 6 of [4], the "basic P X P-sets" in the chapter 21 of [4], and the generalization by Kari Ragnarsson and Radu Stancu to the virtual P X P-sets in [6]. We also explain the relationship with the usual Hecke algebra a of finite group.

Abstract:
In "Frobenius Categories versus Brauer Blocks" and in "Ordinary Grothendieck groups of a Frobenius P-category" we consider suitable inverse limits of Grothendieck groups of categories of modules in characteristics p and zero, obtained from a so-called "folded Frobenius P-category", which covers the case of the Frobenius P-categories associated with blocks, moreover, in "Affirmative answer to a question of Linckelmann" we show that a "folded Frobenius P-category" is actually equivalent to the choice of a regular central k*-extension of the Frobenius P-category restricted to the set of F-selfcentralizing subgroups of P. Here, taking advantage of the existence of the perfect F-locality L, recently proved, we exhibit those inverse limits as the true Grothendieck groups of the categories of K*\hat G- and k*\hat G-modules for a suitable k*-group \hat G associated to the k*-category obtained from the perfect F-locality L and \hat F, both restricted to the set of F-selfcentralizing subgroups of P.

Abstract:
In a recent paper, Gabriel Navarro and Pham Huu Tiep show that the so-called Alperin Weight Conjecture can be verified via the Classification of the Finite Simple Groups, provided any simple group fulfills a very precise list of conditions. Our purpose here is to show that the equivariant refinement of the Alperin's Conjecture for blocks formulated by Geoffrey Robinson in the eighties can be reduced to checking the same statement on any central k*-extension of any finite almost-simple group, or of any finite simple group up to verifying an "almost necessary" condition. In an Appendix we develop some old arguments that we need in the proof.

Abstract:
The weights for a finite group G with respect to a prime number p where introduced by Jon Alperin, in order to formulate his celebrated conjecture affirming that that the number of G-conjugacy classes of weights of G coincides with the number of isomorphism classes of simple kG-modules, where k is an algebraically closed field of characteristic p. Thirty years ago, Tetsuro Okuyama already proved that in the class of p-solvable groups this conjecture holds. In this paper, for the p-solvable groups, on the one hand we exhibit a natural bijection - namely compatible with the action of the group of outer automorphisms of G - between the sets of isomorphism classes of simple G-modules M and of G-conjugacy classes of weights (R,Y), up to the choice of a polarization. On the other hand, we determine the relationship between a multiplicity module of M and Y. In an Appendix, we show that the bijection defined by Gabriel Navarro for the groups of odd order coincides with our bijection for a particular choice of the polarization.

Abstract:
In "Homotopy decomposition of classifying spaces via elementary Abelian subgroups", Stephan Jackowski and James McClure show, for functors admitting a Mackey complement over categories holding a direct product, a general result on vanishing cohomology. We develop a framework leading to a general result on trivial homotopy which partially generalizes Jackowski and McClure's result in two different directions, one of them proving a result analogous to Conjecture A in "Mackey functors and sharpness for fusion systems", by Antonio Diaz and Sejong Park.