Related papers: Rationality of blocks of quasi-simple finite group…

We define a new invariant for a $p$-block, the strong Frobenius number, which we use to address the problem of reducing Donovan's conjecture to normal subgroups of index p. As an application we use the strong Frobenius number to complete…

We give a reduction to quasisimple groups for Donovan's conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring $\mathcal{O}$. Consequences are that Donovan's conjecture holds for…

We show that the Morita Frobenius number of the blocks of the alternating groups, the finite groups of Lie type in describing characteristic, and the Ree and Suzuki groups is 1. We also show that the Morita Frobenius number of almost all of…

We give a classification, up to Morita equivalence, of 2-blocks of quasi-simple groups with abelian defect groups. As a consequence, we show that Donovan's conjecture holds for elementary abelian 2-groups, and that the entries of the Cartan…

We construct blocks of finite groups with arbitrarily large Morita Frobenius numbers, an invariant which determines the size of the minimal field of definition of the associated basic algebra. This answers a question of Benson and Kessar.…

Let $G$ be a simple, simply connected linear algebraic group of exceptional type defined over $\mathbb{F}_q$ with Frobenius endomorphism $F: G \to G$. In this work we give upper bounds on the number of simple modules in the quasi-isolated…

We construct blocks of finite groups with arbitrarily large $\mathcal{O}$-Morita Frobenius numbers. There are no known examples of two blocks defined over $\mathcal{O}$ that are not Morita equivalent but the corresponding blocks defined…

We give a reduction of Donovan's conjecture for abelian groups to a similar statement for quasisimple groups. Consequently we show that Donovan's conjecture holds for abelian $2$-groups.

We classify all $2$-blocks with abelian defect groups of rank $4$ up to Morita equivalence. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. An application…

Linckelmann and Murphy have classified the Morita equivalence classes of p-blocks of finite groups whose basic algebra has dimension at most 12. We extend their classification to dimension 13 and 14. As predicted by Donovan's Conjecture, we…

We prove that if $B$ is a $p$-block with non-trivial defect group $D$ of a finite $p$-solvable group $G$, then $\ell(B) < p^r$, where $r$ is the sectional rank of $D$. We remark that there are infinitely many $p$-blocks $B$ with non-Abelian…

Except for blocks with a cyclic or Klein four defect group, it is not known in general whether the Morita equivalence class of a block algebra over a field of prime characteristic determines that of the corresponding block algebra over a…

We classify up to Morita equivalence all blocks whose defect groups are Suzuki $2$-groups. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field, and in fact…

We characterise the Morita equivalence classes of blocks with extraspecial defect groups $p_+^{1+2}$ for $p \geq 5$, and so show that Donovan's conjecture and the Alperin-McKay conjecture hold for such $p$-groups. For $p=3$ we reduce…

In this paper, we introduce a class of blocks which is called hyperfocal abelian Frobenius blocks.This class of blocks is an analogous version of the block with abelian defect group and Frobenius inertial quotient at hyperfocal level and…

Let $k$ be an algebraically closed field of characteristic $p$, and let $\mathcal{O}$ be either $k$ or its ring of Witt vectors $W(k)$. Let $G$ a finite group and $B$ a block of $\mathcal{O}G$ with normal abelian defect group and abelian…

\input amssym.def \input amssym.tex Let $G$ be a connected algebraic reductive group over an algebraic closure of a prime field ${\Bbb F}_p$, defined over ${\Bbb F}_q$ thanks to a Frobenius $F$. Let $\ell$ be a prime different from $p$. Let…

The purpose of this note is to provide a reference for the fact that the strong Frobenius number, in the sense of Eaton and Livesey, of a block of a finite group with a cyclic defect group is equal to one. This answers a question of Farrell…

We give a construction that in many cases gives a simple way to construct infinite families of algebras that are not Morita equivalent, but are all derived equivalent to the same block algebra of a finite group, and apply it to some small…

We introduce the notion of a quasi-Frobenius algebra in a finite tensor category $\mathcal{C}$ and give equivalent conditions for an algebra in $\mathcal{C}$ to be quasi-Frobenius. A quasi-Frobenius algebra in $\mathcal{C}$ is not…